combinatorics ,geometry and probability

COMBINATORICS, GEOMETRY AND PROBABILITY

COMBINATORICS, GEOMETRYAND PROBABILITY

A tribute to Paul Erdos

Edited by

BELA BOLLOBASANDREW THOMASON

_ CAMBRIDGEUNIVERSITY PRESS

PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGEThe Pitt Building, Trumpington Street, Cambridge, United Kingdom

CAMBRIDGE UNIVERSITY PRESSThe Edinburgh Building, Cambridge CB2 2RU, UK

40 West 20th Street, New York NY 10011-4211, USA477 Williamstown Road, Port Melbourne, VIC 3207, Australia

Ruiz de Alarcon 13, 28014 Madrid, SpainDock House, The Waterfront, Cape Town 8001, South Africa

http://www.cambridge.org

© Cambridge University Press 1997

This book is in copyright. Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,

no reproduction of any part may take place withoutthe written permission of Cambridge University Press.

First published 1997First paperback edition 2004

Typeset in 10/13pt Monotype Times

A catalogue record for this book is available from the British Library

ISBN 0 521 58472 8 hardbackISBN 0 521 60766 3 paperback

Contents

Preface Page ixFarewell to Paul Erdos xiToast to Paul Erdos xiiiList of Contributors xvii

Paul Erdos: Some Unsolved Problems 1

Aharoni, R. and R. DiestelMenger's Theorem for a Countable Source Set 11

Ahlswede, R. and N. CaiOn Extremal Set Partitions in Cartesian Product Spaces 23

Aigner, M. and R. KlimmekMatchings in Lattice Graphs and Hamming Graphs 33

Aigner, M. and E. TrieschReconstructing a Graph from its Neighborhood Lists 51

Alon, N. and R. YusterThreshold Functions for //-factors 63

Barbour, A.D. and S. TavareA Rate for the Erdos-Turan Law 71

Beck, J.Deterministic Graph Games and a Probabilistic Intuition 81

Bezrukov, S.L.On Oriented Embedding of the Binary Tree into the Hypercube 95

Biggs, N.L.Potential Theory on Distance-Regular Graphs 107

Bollobas, B. and S. JansonOn the Length of the Longest Increasing Subsequence in a Random Permutation 121

Bollobas, B. and Y. KohayakawaOn Richardson's Model on the Hypercube 129

Cameron, P.J. and W.M. KantorRandom Permutations: Some Group-Theoretic Aspects 139

Chen, G. and R.H. SchelpRamsey Problems with Bounded Degree Spread 145

vi Contents

Cooper, C, A. Frieze and M. MolloyHamilton Cycles in Random Regular Digraphs 153

de Fraysseix, H., P. Ossona de Mendez and P. RosenstiehlOn Triangle Contact Graphs 165

Deuber, W.A. and W. ThumserA Combinatorial Approach to Complexity Theory via Ordinal Hierarchies 179

Deza, M. and V. GrishukhinLattice Points of Cut Cones 193

Diestel, R. and I. LeaderThe Growth of Infinite Graphs: Boundedness and Finite Spreading 217

Dugdale, J.K. and A.J.W. HiltonAmalgamated Factorizations of Complete Graphs 223

Erdos, Paul, R.J. Faudree, C.C. Rousseau and R.H. SchelpRamsey Size Linear Graphs 241

Erdos, Paul, A. Hajnal, M. Simonovits, V.T. S6s and E. SzemerediTuran-Ramsey Theorems and ^-Independence Numbers 253

Erdos, Paul, E. Makai and J. PachNearly Equal Distances in the Plane 283

Erdos, Paul, E.T. Ordman and Y. ZalcsteinClique Partitions of Chordal Graphs 291

Erdos, Peter L., A. Seress and L.A. SzekelyOn Intersecting Chains in Boolean Algebras 299

Fiiredi, Z., M.X. Goemans and D.J. KleitmanOn the Maximum Number of Triangles in Wheel-Free Graphs 305

Gionfriddo, M., S. Milici and Zs. TuzaBlocking Sets in SQS(2v) 319

Haggkvist, R. and A. Johansson(1,2)-Factorizations of General Eulerian Nearly Regular Graphs 329

Haggkvist, R. and A. ThomasonOriented Hamilton Cycles in Oriented Graphs 339

Halin, R.Minimization Problems for Infinite n-Connected Graphs 355

Hammer, P.L. and A.K. KelmansOn Universal Threshold Graphs 375

Hindman, N. and I. LeaderImage Partition Regularity of Matrices 393

Contents vii

Hundack, C, H.J. Promel and A. StegerExtremal Graph Problems for Graphs with a Color-Critical Vertex 421

Komjath, P.A Note on co\ -» co\ Functions 435

Komlos, J. and E. SzemerediTopological Cliques in Graphs 439

Linial, N.Local-Global Phenomena in Graphs 449

Luczak, T. and L. PyberOn Random Generation of the Symmetric Group 463

Mader, W.On Vertex-Edge-Critically n-Connected Graphs 471

Mathias, A.R.D.On a Conjecture of Erdos and Cudakov 487

McDiarmid, C.A Random Recolouring Method for Graphs and Hypergraphs 489

Mohar, B.Obstructions for the Disk and the Cylinder Embedding Extension Problems 493

Nesetfil, J. and P. ValtrA Ramsey-Type Theorem in the Plane 525

Temperley, H.N.V.The Enumeration of Self-Avoiding Walks and Domains on a Lattice 535

Tetali, P.An Extension of Foster's Network Theorem 541

Welsh, D.J.A.Randomised Approximation in the Tutte Plane 549

Wilf, H.S.On Crossing Numbers, and some Unsolved Problems 557

Preface

On Friday, 26 March 1993, Paul Erdos celebrated his 80th birthday. To honour him onthis occasion, a conference was held in Trinity College, Cambridge, under the auspicesof the Department of Pure Mathematics and Mathematical Statistics of the University ofCambridge. Many of the world's best combinatorialists came to pay tribute to Erdos, theuniversally acknowledged leader of their field.

The conference was generously supported both by the London Mathematical Societyand by the Heilbronn Fund of Trinity College. As at former Cambridge Conferences inhonour of Paul Erdos, the day-to-day running of this conference was in the able handsof Gabriella Bollobas, with the untiring assistance of Tristan Denley, Ted Dobson, TomGamblin, Chris Jagger, Imre Leader, Alex Scott and Alan Stacey. The conference wouldnot have taken place without their dedicated work.

On the eve of Erdos' birthday, a sumptuous feast was held in his honour in the Hall ofTrinity College. The words wherein he was toasted are reproduced in the following pages.This volume of research papers was presented to Paul Erdos by its authors as their owntoast, gladly offered with their gratitude* respect and warmest wishes.

Sadly, before this book reached its printed form, Paul Erdos died. Whereas it wasconceived in joy it appears now tinged with sorrow. We feel his loss tremendously.But it is not appropriate that grM should overshadow this volume. Erdos lived to domathematics and he died doing mathematics. So this work remains a tribute to the Erdoswe fondly remember — the living Erdos — the mathematician.

B.B.A.G.T.

IX

Farewell to Paul Erdos(26/3/1913 - 20/9/1996)

(Paul Erdos died in Warsaw on 20th September 1996. A memorial service was held for himon 18th October 1996 in the Kerepesi Cemetery in Budapest, the traditional resting place ofeminent Hungarians. A great number of his friends gathered to mark his passing. Amongthem were colleagues and former students representing mathematics from many countriesand four continents. The orations were given by Akos Csaszar, Paul Revesz, Gyula Katona,Ron Graham, Andras Hajnal, George Szekeres, and by Bela Bollobas, whose tribute isreproduced below.)

Paul Erdos was one of the most brilliant and probably the most remarkable of mathe-maticians of this century. Not only was his output prodigious, with fundamental papersin many branches of mathematics, including number theory, geometry, probability theory,approximation theory, set theory and combinatorics, and not only did he have many morecoauthors than anybody else in the history of mathematics, but he was also a personalfriend of more mathematicians than anybody else. The vast body of problems he has leftbehind will influence mathematics for many years to come.

Many of us are lucky to have known him and to have benefited from his incisivemind, fertile imagination and desire to help. But hardly any one of us knew him in hisprime, from the mid-thirties to the early sixties. He was hardly twenty when he took themathematical world by storm, so that the great Issai Schur of Berlin dubbed him derZauberer von Budapest.

Throughout his life, he lived modestly, despising material possessions and coveting nohonours, and was always somewhat outside the mathematical establishment. Nevertheless,he was showered with honours. Among others, he was an Honorary Member of theLondon Mathematical Society and an Honorary Fellow of the Royal Society. Theseillustrious institutions have sent wreaths to express their grief at his loss. But I am heremainly to represent Paul's many friends, colleagues and, above all, his students.

Thinking of him, David's psalm springs to mind: "surely goodness and mercy shallfollow me all the days of my life." For decades, he was the window to the West for theHungarian mathematicians, and has helped more mathematicians all over the world thananybody else. He was especially kind to young people. I was just over fourteen when hecalled me to him and so changed the course of my life. There is no doubt that I becamea combinatorialist only because of him, and I owe him a tremendous debt of gratitudefor all his kindness and inspiration. Many people owe their careers to him.

As David in his psalm, he could also have said: "though I walk in the valley of theshadow of death, I will fear no evil." Sadly, he was always in the shadow of death. Whenhe was born, his two sisters died; when he was a year-and-a-half his father was takenprisoner of war and spent six years in Siberia; when his father died of a heart attack, hecould not come to Hungary to comfort his mother; most of his relatives perished in the

xi

xii Farewell to Paul Erdos

Holocaust; in the fifties even America abandoned him and he was saved only by Israel;finally, the loss of his mother was a terrible blow to him, from which he never reallyrecovered. But whatever happened, he always had a passionate desire to be free: he couldnot tolerate constraint of any kind, he was never willing to compromise.

Perhaps there were only two happy periods in his adult life: from 1934 to 1939 whenhe was in Manchester and Princeton, and from 1964 to 1971, when he travelled aroundthe world with his beloved mother. I was lucky enough to have known him in this secondhappy period.

The death of Paul Erdos marks the end of an era. No conference will be the samewithout the p.g.o.m., the poor great old man, as he called himself, no mathematicaldiscussion will be as much fun as it was with him. Our beloved Pali Bdcsi has left us allorphans.

This exceptional man did think about what will happen after him. Endre Ady, thefamous Hungarian poet, wrote: "Let him be cursed who takes my place!" Paul's wish wasrather different, reflecting his character: "Let him be blessed who takes my place!"

Now, when we have to say our final goodbye to Paul Erdos, we all know that there isno chance of that. His death is a tremendous loss to us all, and this sense of loss will staywith us for ever. But we should console ourselves that he has had a marvellous life, inwhich he has produced an exceptional amount of outstanding mathematics, and we areprivileged to have known him.

Kerepesi Cemetery, Budapest, 18/10/1996

Bela Bollobas

Toast to Paul Erdos

(The following is the toast of the Banquet for the 80th Birthday of Paul Erdos, held inTrinity College, Cambridge, on 25 March 1993, the eve of the birthday. The banquet wasattended by many of Erdos' other friends, including Lady Jeffreys, Mrs Davenport andPeter Rado, in addition to the conference participants. Trinity College was represented bySir Andrew Huxley, OM, former president of the Royal Society and former Master of theCollege, who presided at the feast. Cambridge mathematics was represented by the presentand former Sadlerian Professors, John Coates, FRS, and J.W.S. Cassels, FRS.)

Professor Erdos, Sir Andrew, Ladies and Gentlemen,

Mathematics is rich in unusual characters, as everyone here at this dinner will know.Nevertheless, most of us would agree that there is none whose achievement and lifestyleare more extraordinary than those of the man we are celebrating tonight, on the eveof his birthday, following a Hungarian custom. For over 60 years, his fertile mind hasmaintained a staggering output in many branches of mathematics: he has made notablecontributions and broken fresh ground in set theory, number theory, probability theory,classical analysis, geometry, approximation theory and combinatorics. Most of us areparticularly aware of his contributions to the last of these subjects: he has done morethan anyone else to establish combinatorics; many branches of the subject find their originin his ideas; the stimulus of his striking theorems and inspiring problems is one that wehave all felt, and for which we owe him an incalculable debt of gratitude. It is also truethat, as well as being so remarkably gifted intellectually, he has the most admirable andattractive personal qualities. He is generous to a fault, gentle, unassuming, always eagerto fight for the downtrodden. Many a young student has been delighted to discover thatthis famous man is so easily approachable and so interested in their work. He has alwaysmade it his business to nurture young talent, possibly his greatest find being Posa.

What anybody, who has ever heard of this unique man, knows is that he is unceasinglyon the move. It is hardly an exaggeration that he has not slept in the same bed for morethan a week in over 50 years. As a constant globe-trotter, he is the living link betweenmathematicians across the world, carrying with him news of theorems, conjectures andproblems.

Paul Erdos was a precocious child: at the age of three he was good at arithmetic tothe point of discovering for himself negative numbers. Much of Paul's education wasdone in private; altogether he spent less than four years in schools. At the age of 17, heproceeded to university, where he soon became the focus of a wonderfully talented groupof mathematicians.

At the age of 21, he completed his degree, and as was the custom, he looked to spenda year abroad. In the world of 1934, the country that most attracted him was Britain. Asan undergraduate, he had corresponded with Louis Mordell, the great American number

xiii

xiv Toast to Paul Erdos

theorist, who by that time had left St John's College, Cambridge to work in Manchester.Mordell offered Erdos a Fellowship in his department, and the offer was gladly accepted.On 1 October 1934, Erdos arrived in London, from where he took the train to Cambridge.At the station he was met by two outstanding young mathematicians who for many yearsto come were to be his closest friends, Harold Davenport and Richard Rado. Sadly,Harold Davenport and Richard Rado are no longer with us, but it is indeed a pleasure tosee Anne Davenport and Peter Rado at this banquet tonight. In fact, it is due to Erdos'sfriendship with Davenport that my own connection with Trinity came about.

At that time Erdos stayed in Cambridge only for a couple of days, but long enoughto meet Hardy and Littlewood, the leading English mathematicians. He then travelledon to Manchester, to Mordell, who became his mentor and friend. In the 1930s Mordellgathered a remarkable group of mathematicians to Manchester: in addition to Erdos,and later Davenport, the group included Mahler, Heilbronn, du Val and Chao Ko. It isextremely fitting that this conference has been supported by Heilbronn's generous bequestto the mathematicians of this college. On looking down on us, Heilbronn must be smilingthat we are celebrating his great friend tonight.

Another prominent member of the Manchester group was the eminent fluid dynamicistMiss Swirles, who befriended Paul soon after his arrival. It is a great pleasure that MissSwirles, by now Lady Jeffreys, can share in this happy celebration tonight.

Paul stayed in Manchester for four years, first as the Bishop Harvey Goodwin Fellow,and then as a Royal Society Fellow. During that time he made frequent visits to Cambridgeand other centres of mathematics. In 1938 Paul left England for the States to take up aFellowship at Princeton. It was to be ten more years before Paul returned to Hungary,and he would never again stay there for more than a few months at a time.

After a year or two at the Institute, the travelling began in earnest, and the now familiarpattern was soon set. In a short space of time, he visited Philadelphia, Purdue, Stanford,Syracuse and Johns Hopkins, and many other universities for even shorter periods.

Since then Paul has been travelling from university to university, from country tocountry, bringing news, inventing problems, writing joint papers, stimulating the minds ofmathematicians everywhere, and generally being the Erdos we know and love so well. Bynow he has over 300 coauthors, and it has often been said that if a train journey is longenough, he will write a joint paper with the conductor. His 1300 research papers placehim in a league of his own among research mathematicians.

It has been said that the world wants geniuses but it wants them to behave just likeother people. Paul found this out when one apocryphal, but not too far-fetched, nightin Chicago he was out walking by himself. Suddenly a police car appeared and theofficers began to question Paul. "So what are you doing out here, all by yourself?" "I amthinking" came the reply. "What do you mean you are thinking? What are you thinkingabout?" "I am proving a theorem." "You'd better come with us back to the station, Sir."

Back at the station, the officer in charge said "Now, what's all this about yourtheorem? Tell me about it." "It doesn't matter anymore" grumbled Paul testily, "I'vefound a counterexample."

In fact, this incident is atypical for, as we know, Paul is remarkably successful in provingtheorems. A striking example is quoted by Mark Kac.

Toast to Paul Erdos xv

"As a mathematician Erdos is what in other fields is called a 'natural'. If a problem can bestated in terms he can understand, though it may belong to a field with which he is not familiar,he is as likely as, or even more likely than, the experts to find a solution. An example of thisis his solution of a problem in dimension theory, a part of topology of which in 1939 he knewabsolutely nothing. The late Witold Hurewicz and a younger colleague, Henry Wallman, werewriting a book on dimension theory which later became an acknowledged classic. They wereinterested in the unsolved problem of the dimension of the set of rational points in Hilbertspace. What all this means is unimportant except that the problem seemed very difficult andthat the 'natural' conjectures were that the answer is either zero or infinity. Erdos overheardseveral mathematicians discussing the problem in the common room of the old Fine Hall atPrinceton. "What is the problem?" asked Erdos. Somewhat impatiently he was told what theproblem was. "What is dimension?" he asked, betraying complete ignorance of the subjectmatter. To pacify him, he was given the definition of dimension. In a little more than an hourhe came with the answer, which, to everyone's immense surprise, turned out to be T!"

In addition to being successful in his own personal research, one of Paul's greatestgifts to mathematics has been his ability to stimulate the creativity of others throughhis fascinating and penetrating conjectures. His offer of monetary rewards for solutionsis legendary. The winner of the largest reward to date is Szemeredi, for finding longarithmetic progressions in sets of positive density. It is a pleasure to see him heretonight. The biggest sum on offer is $10000, for proving that the gap between twoconsecutive primes is rather large infinitely often. Although Schonhage, Rankin, Maierand Tenenbaum have proved exciting results in this direction, they haven't yet managedto claim the prize. Paul is also offering $3000 for finding long arithmetic progressions insequences of natural numbers whose reciprocals diverge, and so, in particular, among theprimes. A group of Swedish computers has just discovered an arithmetic progression of22 primes but I doubt that any payment will be forthcoming from Paul.

Paul worked with most of the leading Hungarian mathematicians, especially the numbertheorist Paul Turan and the probabilist Alfred Renyi, who were his great friends. Turan'swife, Vera Sos, has also been a close friend and collaborator for many years, and it isfitting that she too should be celebrating tonight.

My own friendship with Paul is also of many years standing. We met when I was 14,and I was tremendously impressed by his willingness to talk to me about his fascinatingproblems. To me he seemed to be from a different planet, a flamboyant man with an air ofthe exotic, with his expensive foreign suits and ready cash, brought from the unattainablefree Western world. Now I know better; I think it was Paul who inspired the saying:"The man who leaves footprints on the sands of time never wears expensive shoes."

In those days, I also got to know Paul's mother, Annus neni, a charming lady whoadored Paul, and was, in turn, adored by her son. She kept his reprints in immaculateorder, and sent copies to those who requested them. A year or two later they got to knowmy family, and were frequent visitors to our house whenever Paul was in Hungary.

In 1964, at the age of 84, Annus neni began to travel with Paul. Their first trip was toIsrael; soon Western Europe followed, including England a year later. In 1968, when shewas 88, Annus neni accompanied Paul to Hawaii and Australia. When asked whether sheliked to travel, she used to reply: "You know I don't travel because I like it, but to bewith my son." It was moving too see their affectionate care for each other, catching up

xvi Toast to Paul Erdos

with those lost years, when they couldn't see each other. Annus neni greatly enjoyed herrole as Queen Mother of mathematics, meeting and entertaining all the people coming tosee Paul; her cocoa cake with coffee cream was especially delicious.

Erdos's own tastes in food are well known to be frugal, and he doesn't care for wine,which he calls poison. It has been suggested that the College should on this occasionproduce a meal of bread and water. Unfortunately when I checked with the Kitchens,they could not find the recipe, so we had to use the second choice menu.

Paul Erdos has always kept up his close links with Trinity and Cambridge. Some yearsago he was a Visiting Fellow Commoner of Trinity College, and in 1991 Cambridgeawarded him an Honorary Doctorate - the first citizen of Hungary to receive this honour.At the ceremony it was charming to see the great actor Sir Alec Guinness taking it uponhimself to shepherd Paul through the long ritual.

Since his youth, Paul Erdos has had catholic interests: in particular, he has maintainedan enthusiasm for history and medicine. It is always fascinating to engage him in discussionpf his favourite historical events. Nevertheless, Paul is the quintessential mathematician:he breathes, eats, drinks, and sleeps mathematics, if he sleeps at all. It could have beenErd5s, whom Littlewood had in mind when he wrote:

"There is much to be said for being a mathematician. To begin with, he has to be completelyhonest in his work, not from any superior morality, but because he cannot get away witha fake. It has been cruelly said of arts dons, especially in Oxford, that they believe there isa polemical answer to everything; nothing is really true, and in controversy the object is toprove your opponent a fool. We escape all this. Further, the arts man is always on duty as agreat mind; if he drops a brick, as we say in England, it reverberates down the years. Afteran honest day's work a mathematician goes off duty. Mathematics is very hard work, anddons tend to be above average in health and vigour. Below a certain threshold a man cracksup; but above it, hard mental work makes for health and vigour (also - on much historicalevidence throughout the ages - for longevity)."

If hard mental work be the secret of longevity then Paul Erdos will live forever andcontinue to enrich us all with the brightness of his intellect and the warmth of his heart.In the meantime, we honour him on his 80th birthday.

Ladies and Gentlemen, please rise and toast Paul Erdos.

B.B.

List of Contributors

Ron AharoniDepartment of Mathematics, Technion, Haifa 32000, ISRAEL

Rudolf AhlswedeUniversitat Bielefeld, Fakultat fur Mathematik, Postfach 100131, 33501 Bielefeld,GERMANY

Martin AignerFreie Universitat Berlin, Fachbereich Mathematik, WE2, Arnimallee 3, 1000 Berlin 33,GERMANY

Noga AlonDepartment of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences,Tel Aviv University, Tel Aviv, ISRAEL

A. D. BarbourInstitut fur Angewandte Mathematik, Universitat Zurich, Winterthurerstrasse 190,CH-8057, Zurich, SWITZERLAND

Jozsef BeckDepartment of Mathematics, Rutgers University, Busch Campus, Hill Center, NewBrunswick, NJ 08903, USA

Sergej L. BezrukovFachbereich Mathematik, Freie Universitat Berlin, Arnimallee 2-6, D-14195 Berlin,GERMANY

Norman L. BiggsLondon School of Economics, Houghton St, London WC2A 2AE, UK

Bela BollobasDepartment of Pure Mathematics and Mathematical Statistics, University of Cambridge,16 Mill Lane, Cambridge, CB2 1SB, UK and Louisiana State University, Baton Rouge,LA 70803 USA

Ning CaiUniversitat Bielefeld, Fakultat fur Mathematik, Postfach 100131, 33501 Bielefeld,GERMANY

Peter J. CameronSchool of Mathematical Sciences, Queen Mary and Westfield College, Mile End Road,London, El 4NS, UK

G. ChenNorth Dakota State University, Fargo, ND 58105, USA

Colin CooperSchool of Mathematical Sciences, University of North London, London, UK

xvii

xviii List of Contributors

Walter A. DeuberUniversitat Bielefeld, Fakultat fur Mathematik, Postfach 100131, 33501 Bielefeld 1,GERMANY

Michel DezaCNRS-LIENS, Ecole Normale Superieure, Paris, FRANCE

Reinhard DiestelFaculty of Mathematics (SFB 343), Bielefeld University, 4-4800, Bielefeld, GERMANY

J. K. DugdaleDepartment of Mathematics, West Virginia University, PO Box 6310, Morgantown, WV26506-6310, USA

Paul Erdos^late, Mathematical Institute of the Hungarian Academy of Sciences, Budapest V,HUNGARY

Peter L. ErdosCentrum voor Wiskunde en Informatica, PO Box 4079, 1009 AB Amsterdam,The NETHERLANDS

R. J. FaudreeDepartment of Mathematical Science, Memphis State University, Memphis, TN 38152, USA

Hubert de FraysseixCNRS, EHESS, 54 Boulevard Raspail, 75006, Paris, FRANCE

Alan FriezeDepartment of Mathematics, Carnegie-Mellon University, Pittsburgh, PA 15213, USA

Zoltan FurediDepartment of Mathematics, Massachusetts Institute of Technology, Cambridge, MA02139, USA

Mario GionfriddoDipartimento di Matematica, Cittd Universitaria, Viale A, Doria 6, 95125 Catania,ITALY

Michel X. GoemansDepartment of Mathematics, Massachusetts Institute of Technology, Cambridge, MA02139, USA

Viatcheslav GrishukhinCentral Economic and Mathematical Institute of Russian Academy of Sciences (CEMIRAN), Moscow, RUSSIA

Roland HaggkvistDepartment of Mathematics, University of Umed, S-901 87 Umed, SWEDEN

A. HajnalMathematical Institute of the Hungarian Academy of Sciences, Budapest V, HUNGARY

R. HalinMathematisches Seminar der Universitat Hamburg, Bundesstrafie 55, D-20146, Hamburg,GERMANY

P. L. HammerRUTCOR, Rutgers University, New Brunswick, NJ 08903, USA

List of Contributors xix

A. J. W. HiltonDepartment of Mathematics, University of Reading, Whiteknights, PO Box 220, ReadingRG6 2AX, UK

Neil HindmanDepartment of Mathematics, Howard University, Washington, DC 20059, USA

Christoph HundackInstitut fur Diskrete Mathematik, Universitdt Bonn, Nassestr. 2, 53113 Bonn, GERMANY

Svante JansonDepartment of Mathematics, Uppsala University, PO Box 480, S-751 06, Uppsala,SWEDEN

Anders JohannsonDepartment of Mathematics, University of Umed, S-901 87 Umed, SWEDEN

William M. Kan torDepartment of Mathematics, University of Oregon, Eugene, OR 97403, USA

A. K. KelmansRUTCOR, Rutgers University, New Brunswick, NJ 08903, USA

Daniel J. KleitmanDepartment of Mathematics, Massachusetts Institute of Technology, Cambridge, MA02139, USA

R. Klimmekc/o M. Aigner, Freie Universitdt Berlin, Fachbereich Mathematik, WE2, Arnimallee 3,1000 Berlin 33, GERMANY

Y. KohayakawaInstituto de Matemdtica e Estatistica, Universidade de Sao Paulo, Caixa Postal 20570,01452-990 Sao Paulo, SP, Brazil

Peter KomjathDept. Comp. Sci. Eotvos University, Budapest, Muzeum krt 6-8, 1088, HUNGARY

Janos KomlosDepartment of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA

Imre LeaderDepartment of Pure Mathematics and Mathematical Statistics, University of Cambridge,16 Mill Lane, Cambridge, CB2 1SB, UK

Nathan LinialInstitute of Computer Science, Hebrew University, Jerusalem, ISRAEL

Tomasz LuczakAdam Mickiewicz University, Poznan, POLAND

W. MaderInstitut fur Mathematik, Universitdt Hanover, 30167 Hanover, Weifengarten 1, GERMANY

Endre MakaiMathematical Institute of the Hungarian Academy of Sciences, Budapest V, HUNGARY

A. R. D. MathiasPeterhouse College, Cambridge, UK

xx List of Contributors

Colin McDiarmidDepartment of Statistics, University of Oxford, Oxford, UK

Patrice Ossona de MendezCNRS, EHESS, 54 Boulevard Raspail, 75006, Paris, FRANCE

Salvatore MiliciDipartimento di Matematica, Cittd Universitaria, Viale A, Doria 6, 95125 Catania, ITALY

Bojan MoharDepartment of Mathematics, University of Ljubljana, Jadranska 19, 61111 Ljubljana,SLOVENIA

Michael MolloyDepartment of Mathematics, Carnegie-Mellon University, Pittsburgh, PA 15213, USA

Jaroslav NesetfilDepartment of Applied Mathematics, Charles University, Malostranske ndm. 25, 118 00Praha 1, CZECH REPUBLIC

Edward T. OrdmanMemphis State University, Memphis, TN 38152, USA

Janos PachDepartment of Computer Science, City University, New York, USA and the MathematicalInstitute of the Hungarian Academy of Sciences, Budapest V, HUNGARY

Hans Jurgen PromelInstitut fur Diskrete Mathematik, Universitdt Bonn, Nassestr. 2, 53113 Bonn, GERMANY

Laszlo PyberMathematical Institute of the Hungarian Academy of Sciences, Budapest V, HUNGARY

Pierre RosenstiehlCNRS, EHESS, 54 Boulevard Raspail, 75006, Paris, FRANCE

C. C. RousseauDepartment of Mathematical Science, Memphis State University, Memphis, TN 38152, USA

R. H. SchelpDepartment of Mathematical Science, Memphis State University, Memphis, TN 38152, USA

Akos SeressThe Ohio State University, Colombus, OH 43210, USA

M. SimonovitsMathematical Institute of the Hungarian Academy of Sciences, Budapest V, HUNGARY

V. T. SosMathematical Institute of the Hungarian Academy of Sciences, Budapest V, HUNGARY

Angelika StegerInstitut fur Diskrete Mathematik, Universitdt Bonn, Nassestr. 2, 53113 Bonn, GERMANY

Laszlo A. SzekelyUniversity of New Mexico, Albuquerque, NM 87131, USA

Endre SzemerediMathematical Institute of the Hungarian Academy of Sciences, Budapest V, HUNGARY

List of Contributors xxi

Simon TavareDepartment of Mathematics, University of Southern California, Los Angeles, CA90089-113, USA

H. N. V. TemperleyThorney House, Thorney, Langport, Somerset, UK

Prasad TetaliAT & T Bell Labs, Murray Hill, NJ 07974, USA

Andrew ThomasonDPMMS, 16, Mill Lane, Cambridge, CB2 1SB, UK

Wolfgang ThumserUniversitdt Bielefeld, Fakultdt fur Mathematik, Postfach 100131, 33501 Bielefeld 1,GERMANY

Eberhard TrieschForschungsinsitut fur Diskrete Mathematik, Nassestrafie 2, 5300 Bonn 1, GERMANY

Zsolt TuzaComputer and Automation Institute, Hungarian Academy of Sciences, H-llll Budapest,Kende u. 13-17, HUNGARY

Pavel ValtrDepartment of Applied Mathematics, Charles University, Malostranske ndm. 25, 118 00Praha 1, CZECH REPUBLIC and Graduiertenkolleg Algorithmische DiskreteMathematik', Fachbereich Mathematik, Freie Universitdt Berlin, Takustrasse 9, 14195Berlin, GERMANY

D. J. A. WelshMathematical Institute and Merton College, University of Oxford, Oxford, UK

Herbert S. WilfUniversity of Pennsylvania, Philadelphia, PA 19104-6395, USA

Raphael YusterDepartment of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences,Tel Aviv University, Tel Aviv, ISRAEL

Yechezkel ZalcsteinDivision of Computer and Computation Research, National Science Foundation,Washington, DC 20550, USA

Some Unsolved Problems

P A U L E R D O S 1

During my long life I have written many papers on my favourite unsolved problems (see,for example, Baker et a\. [2]). In the collection below, all the problems are either newones, or they are problems about which there have been recent developments.

Number theory

1. As usual, let us write 2 = p\ < p2 < • • • for the sequence of consecutive primes. I provedin 1934 that there is a constant c > 0 such that for infinitely many n we have

c log n log log nPn+1-Pn>

Rankin [35] proved that for some c > 0 and infinitely many n the following inequalityholds:

c log n log log n log log log log nPn+l — Pn > 7j \ \ 7? • (1)

(log log log n)2

I offered (perhaps somewhat rashly) $10000 for a proof that (1) holds for every c. Theoriginal value of c was improved by Schonhage [38] and later by Rankin [36]. Rankin'sresult was recently improved by Maier and Pomerance [30].

2. Let a\ < a2 < ' * * be an infinite sequence of integers. Denote by f(n) the number ofsolutions of n = at + a,-. Assume that f(n) > 0 for all n > no, i.e. (an)^=l is an asymptoticbasis of order 2. Turan and I conjectured that then

lim f(n) = 00 (2)

and probably lim/(n)/log n > 0. I offer $500 for a proof of (2). Perhaps (2) andlim f(n)/ log n > 0 already follow if we only assume an < en2 for all n.

Let a\ < «2 < "'" and b\ < b^ < • • • be two sequences of integers such that an/bn —• 1and let g(n) be the number of solutions of at + bj = n. Sarkozy and I conjecture that if

2 P. Erdos

g(n) > 0 for all n > no then limg(n) = oo. The condition an/bn —• 1 can not entirely beomitted but 1 — e < an/bn < 1 + 6* (e small) may suffice.

3. I proved that there is an asymptotic basis of order 2 for which

c\ log n < f(n) < c2 log n

(see Halberstam and Roth [26]). I conjecture that

j ^ - - C , (0<C<oo),logn

is not possible and I offer $500 for a proof or disproof of this conjecture. Sarkozy and Iproved that

logncannot hold.

4. Is it true that

is irrational? I conjectured that

2n - 3is irrational. This assertion and its generalizations have been proved by Peter Borwein [6].Denote by co(n) the number of distinct prime factors of n. Is it true that

E cojn)

is irrational?

5. Is it true that if n ^ 0 (mod 4) then there is a squarefree natural number 0 such thatn = 2k + 61 I could only prove that almost all integers n ^ 0 (mod 4) can be written inthe form 2k + 9.

Combinatorics

6. Let m = m(n) be the smallest integer for which there are n-element sets A\, ..., Am suchthat AtnAj ^ 0 for all 1 ^ i < j ^ m, and such that every set S with at most n—\ elementsis disjoint from some At. (Note that the lines of finite geometry have this property.) Iconjectured with Lovasz that m(n)/n —• oo, but it is not even known whether m(n) > 3n ifn sufficiently large. In the other direction, we could prove only that m(ri) < m+e, but JeffKahn [27] very recently proved m(n) < en.

Perhaps more is true: for every C > 0 there is an e > 0 such that if n is sufficientlylarge and m ^ Cn then for every n-element set A\, . . . , Am with At nAj^=0 there is a setS with \S \<n(l—e) which meets all At.

7. Is it true that in a finite geometry there is always a blocking set which meets every linein at most c points where c is an absolute constant?

Some Unsolved Problems 3

More generally: Let \Sf\ = n, A\, . . . , Am be a family of subsets of y , |4I > cjn, c < 1,|4: n 4 / | ^ 1. Is it then true that there is a set B for which BnAt^0 but |fl H4I < c/ forall /? In other words, is there a set B which meets all the 4 ' s but none in many points?

8. Here is a problem of Jean Larson and myself [19]. Is it true that there is an absoluteconstant c so that for every n and \Sf\ = n there is a family of subsets A\9 • • •, Am of y ,| 4 | > ft1^2 ~~ c? 14 ^ ^ / l ^ 1 a n d every x, j ; G y is contained in some 4 ?

Shrikhande and Singhi [39] have proved that every pairwise balanced design on n pointsin which each block is of size ^ n^ — c can be embedded in a projective plane of ordern + i for some i ^ c + 2 if n is sufficiently large. This implies that if the projective planeconjecture (that the order of every projective plane is a prime power) is true then theErdos-Larson conjecture is false. But the problem remains for which functions h(n) willthe condition |4I > ^5 — h(n) make the conjecture true?

Graph theory

9. I offer $500 for a proof or disproof of the following conjecture of Faber, Lovasz andmyself. Let G\, . . . , Gn be complete graphs (each on n vertices), no two of which have anedge in common. Is it then true that x(U?=i ^ ) ^ n?

Jeff Kahn [27] recently proved that the chromatic number is n + o(ri).

10. Is it true that every triangle-free graph on 5n vertices can be made bipartite by theomission of at most 5n2 edges? Is it true that every triangle-free graph on 5n vertices cancontain at most n5 pentagons? Ervin Gyori [25] proved this with l.O3n5.

Gyori now proved n5 for n > no. One could ask more generally: Assume that thenumber of vertices is (2r + l)n and that the smallest odd cycle has size 2r + 1. Is it thentrue that the number of cycles of size 2r 4-1 is at most n2r+l ?

11. Let H be a graph and let Gn be a graph on n vertices which does not contain H as aninduced subgraph. Hajnal and I [13] asked whether there is an absolute constant c = c(H)such that Gn contains either a complete graph or an independent set on nc vertices? If His C4 then | ^ c < ^.

12. Let Qn be the graph of the n-dimensional cube {0,1}". I offered $100 for a proof ordisproof of the conjecture that for every e > 0 there is an no such that, for n > no, everysubgraph of Qn with at least (\ + e)e(Qn) edges contains C4. It is easy to find subgraphswith more than \e(Qn) edges and no C4; Guan (see Chung [9]) has constructed an examplewith (1 + o(l))(n + 3)2n~2 edges. Chung has given an upper bound of (a + o(l))n2"-1,where a « 0.623.

I also conjectured that every subgraph of Qn with ee(Qn) edges contains a Ce, for nsufficiently large. Chung [9] and Brouwer, Dejter and Thomassen [7] disproved this byconstructing an edge-partition of Qn into four subgraphs containing no C^.

13. Suppose that G is a graph of order n with the property that every set of p verticesspans at least q edges. We let H(n;p,q) be the largest integer such that G necessarilycontains a clique of that order. In the case where q = 1 this corresponds to the standard

4 P. Erdos

finite Ramsey problem: the condition is precisely that G contains no independent set ofsize p.

Faudree, Rousseau, Schelp and I investigated the behaviour of H(n;p, q) as a functionof n. We set

(log H(n;p,qyc(P><l) = lim :

t=^o \ log nStandard bounds on Ramsey numbers (see, for example, Bollobas [5]) tell us that

We conjecture that with p fixed, c(p,q) is a strictly increasing function of q for 1 ^ q <(p^1) + 1 . It is easy to see that if g = (p^1) + 1 then c(p, g) = 1, which is as large as possible.For in this case, the complement of G cannot contain any connected subgraphs of sizep, so all components of the complement have size less than p. Hence the complementcontains at least n/(p — 1) independent vertices so G contains a clique of size at leastn/(p-\). On the other hand, we have shown that H(n;p,(p^>1)) ^ cvS9 so c(p,(p^1)) < 1/2.

14. For e > 0, Rodl [37] constructed graphs with chromatic number Ko such that everysubgraph of order n can be made bipartite by omitting en edges, for every n; anotherconstruction was given by Lovasz. Now let f(n) —> oo as slowly as we please. Is therea graph of chromatic number No such that every subgraph of n vertices can be madebipartite by omitting f(n) edges?

Perhaps for every e > 0, there is a graph with chromatic number Ki for which everysubgraph of order n can be made bipartite by omitting en edges, but this seems unlikelyand I would guess that there is a subgraph of size n which cannot be made bipartite byomitting nh(n) edges, where h(n) —> oo. But perhaps h(n) does not have to tend to infinityfast. See also the paper with Hajnal and Szemeredi [17].

Hajnal, Shelah and I [16] proved that if G has chromatic number Ki then for some rc0

it contains a cycle of length n for every n > no. Now if F(n) tends to infinity sufficientlyfast, then is it true that every graph of chromatic number Ki has a subgraph on at mostF(n) vertices with chromatic number n, for all n sufficiently large?

Geometry

15. Let xi, . . . , xn be n distinct points in the plane, and let s\ ^ 52 ^ • • • ^ Sk be themultiplicities of the distances they determine, so

I conjectured [12] thatk

J^sj <cn3(\ognf (3)/=i

for some a > 0. The lattice points show that we must have a ^ 1.In forthcoming papers Fishburn and I conjecture that if xi, . . . , xn form a convex set

then (3) can be improved tok

J2j<cn\ (4)

and that ^ sj is maximal for the regular n-gon, for n ^ 8.A weaker inequality than (3) would follow easily from the following conjecture. Let

A(x\, . . . , xn) be the number of pairs x,-,x;- whose distance is 1, and let f(n) be themaximum A(x\, . . . , xn) over all sets of n distinct points in the plane. I conjecture that

f(n) < n{+c/lozl°zn. (5)

The best bound found to date is due to Spencer, Szemeredi and Trotter [40], who provedf(n) < en5. It would follow from (5) that $ > ? < Cn3+c/lo^°^n.

Is it true that the number of incongruent sets of n points with f(n) unit distancesexceeds one for n > 3 and tends to infinity with w?

Leo Moser and I conjectured that if xi, . . . , xn is a convex n-gon then

A(xu •-., xn) < en. (6)

Furedi [22] proved that A(x\, • • •, xn) < en log n; this gives an upper bound of en3 log n in(4). The inequality (4) would follow from (6).

16. Let xi, . . . , xn be n distinct points in the plane. Denote by Fk(n) the maximum numberof distinct lines passing through at least k of our points and by fk(n) the maximumnumber of lines passing through exactly k of our points. Clearly fk(n) ^ Fk(n). Determineor estimate fk(n) and Fk(n) as well as possible. Trivially fi{ri) = F2(n) = (JJ). The problemwith k = 3 is the Orchard problem, and really goes back to Sylvester. Burr, Grimbaumand Sloane [8] proved that

2 2

/3(w) = ^ - O ( n ) and F3(n) = \ - O(n).o o

Determine lim,i_>00 Fk(n)/n2 and limn_KX)//c(rc)/n2, if they exist. The upper bound Fk(n) ^(2)7(2) follows from an obvious counting argument; a lower bound can be obtained byconsidering a rectangle of k by n/k lattice points. Are the limits attained by the latticepoints?

17. Let f(n) denote the minimum number of distinct distances among a set # = (x,-)" ofpoints in the plane. In 1946 [12] I proved that

4 ~~ 2

and conjectured that the upper bound gave the true order of f(n). So far, the best lowerbound is n^(\ogn)~c, due to Chung, Szemeredi and Trotter [10].

The question also arises whether, in general, a particular point of the configuration isassociated with a large number of distances. I conjecture that in any configuration thereis some point with at least cn/^Jlogn distinct distances to other points. In fact this maybe true for all but a few of the points.

Altman [1] showed that if # is convex then there are at least n/2 distinct distances

6 P. Erdos

between the points; I conjecture that there is some point associated with at least n/2distinct distances. Szemeredi conjectured there are at least n/2 distinct distances amongthe points of # provided only that # has no three points collinear, but could only provethis with a bound of w/3.

18. Consider two configurations ^ = (x,)", *€' = (j^)", and define F(2n) to be the minimumover all # and <€' of the number of distinct distances || xt — yt ||. Is F identically 1 in fourdimensions (in this case f(2n) > n€)l How does f/F behave at dimension 2 or 3? Do wehave f/F —• oo or is the ratio bounded? Possibly it is unbounded in ]R3 but bounded in 1R2.

19. Let ^ be a set of points in the plane such that distinct distances between the pointsalways differ by at least 1. I conjecture that the diameter of ^ is at least n — \ provided nis large enough. Note that if # = ((J,0))"=1 we obtain equality. However for n < 10 someconfigurations have diameter less than n — 1.

The best result in this direction so far is due to Kanold [29], who proved that

20. Let ^ be a set of n points in Euclidean space among which all distances differ byat least 1. A conjecture independent of dimension is that diam^ ^ (1 + o(\))n2. Clearlydiam^ is always at least g).

The conjecture is settled only for ^ cz ]R (not even for ]R2). To prove it for ]R,let ^ = {x\)\ with 0 = x\ < X2 < ... < xn. Further, let ykj = x,-+fc — x, and letYk = YMII yk,i = xn + xn-\ + . . . + xn-t+i < kxn. Because the y^t are all distinct (even over/c), we have

^Y s y ± y ^ , ' 2 n - 25Xn > I\ -h 12

and for k ^ n/2,

4- 1 \

n> Yi+... + YK

Now let k = \yjri]. Roughly, we get

so xn ^ n2(l + o(l)) as desired.

Analysis

21. We let / = [—1,1] and supose / : / —• 1R is a continuous function which we wish toapproximate by a polynomial. Suppose we are given, for each 1 ^ n < oo,

so we have a triangular matrix X = (xjn|). Let /,'"' be the unique polynomial of degree

Some Unsolved Problems

n — 1 satisfying

/(»>(*(»)) = 1 a n d l l n \ x { f ) = O i f 7 ^ 1s o

Then we denote by «£?„(/, X), or simply j£?n(/), the polynomial given by

1=1

so this is the unique polynomial of degree n — 1 agreeing with / on x^\.. x%\It is known that for certain choices of X, if/ is of bounded variation then ifn(/)(x) —•

/(x) uniformly. However, for more general continuous / the behaviour is not so goodand, as we now describe, a number of authors have examined how bad this behaviourcan be.

With a fixed choice of X, we can regard £?n as a linear map from C(I) to itself. Let uswrite down its norm. Let

Then we easily see thatrr

so if we letXn = max Xn(x),

then \\&n\\=kn. _Faber [21] proved that for any choice of X, l im^oo^ = oo. It therefore follows from

the Principle of Uniform Boundedness that there exists an / with lim^oc ||ifw(/)||= oo.This result was strengthened by Bernstein [4] who showed that for any X, there exist

/ G C [ - l , 1] and x0 G [-1,1] such that

lim ||J2\,(/)(x0)||=oo, i-e. \imXn(x0) = 00.n—>oo rc—•oo

In several papers (Bernstein [3], Grlinwald ([23], [24]), Marcinkiewicz [31] and Privalov([32], [33])) it was shown that for particular choices of X, this kind of bad behaviourcan occur almost everywhere and, in certain cases, everywhere. In 1980 Vertesi and I [20]showed that given any X, there exists an / with

hm ||J£\,(/)(x)||= 00 for almost all x.n—+oo

Certainly this result cannot be extended from almost all x to all x. For example, if xoappears in all but finitely many rows of X - i.e. is equal to some x) for all n ^ no -then we have j£?n(/)(xo) = /(xo) for n ^ no. Does there, however, exist an X, such thatfor every / , there is some point xo where divergence would be possible, i.e. where

oo yetn—>oo

22. Let f(z) = zn + ... be a monic polynomial of degree n.

8 P. Erdos

Is it true that the length of {z e C : | /(z) |= 1} is maximal in the case when /(z) = zn—1 ?This problem was posed, along with many others, in my paper with Herzog and Piranian

[18].

23. Let \zn\ = 1(1 ^ n < oo). Put

k=\

and

Mn = max|/n(z)|.

Is it true that limMw = oo? This conjecture was settled by Wagner: he proved that there isa c > 0 such that Mn > (log nf holds for infinitely many values of n. I further conjecturedthat Mn > nc for some c > 0 and infinitely many n and, in fact, for every n we have

n

Y,Mk>nl+c. (7)/ c = l

Inequality (7), if true, may very well be difficult, so I offer $100 for a solution.

24. Let xi,X2,... be a sequence of real numbers tending to 0. We call (yn)%L\ similar to(x«)S=i ^ y« = axn + b f° r some a, b G IR and all n. Is it true that there is a set £ c R ofpositive measure which contains no subsequence (yn)^Li similar to (xn)™=ll

Komjath proved that if xn —>• 0 slowly (xn > c/n) then there is a set of positive measurewhich contains no subsequence similar to (xn)™=1.

Set theory

25.1 have not included our many problems on set theory with Hajnal since undecidabilityraises its ugly head everywhere and many of our problems have been proved or disprovedor shown to be undecidable (this happened most often). However, I think that thefollowing simple problem is still open. Let a be a cardinal or ordinal number or an ordertype. Assume a —• (a, 3)2. Is it then true that, for every finite n, a —> (a, n)2 also holds?Here a —> (a, n)2 is the well-known arrow symbol of Rado and myself: if G is a graphwhose vertices form a set of type a then either G contains a complete graph Kn or anindependent set of type a. See Erdos, Hajnal and Milner [15] and Erdos, Hajnal, Mateand Rado [14].

Group theory

26. Let G be a group. Assume that it has at most n elements which do not commutepairwise. Denote by h(n) the smallest integer for which any such G can be covered by h(n)Abelian subgroups. Determine or estimate h(n) as well as possible. Pyber [34] proved that

for some positive constants c\ and c^. The lower bound was already known to Isaacs.


References

[I] Altman, (1963), On a problem of P. Erdos, Amer. Math. Monthly 70 148-157.[2] Baker, A., Bollobas, B. and Hajnal, A. eds. (1990), A Tribute to Paul Erdos, Cambridge

University Press, xi;-f-478pp.[3] Bernstein, S. (1918), Quelques remarques sur Interpolation, Math. Ann. 79 1-12.[4] Bernstein, S. (1931), Sur la limitation des valeurs d'un polynome, Bull. Acad. Sci. de I'URSS 8

1025-1050.[5] Bollobas, B. (1985), Random Graphs, Academic Press, xjt;+447pp.[6] Borwein, P. B. (1991), On the irrationality of £ ( l / f a n + r)), J. Number Theory 37 253-259.[7] Brouwer, A. E., Dejter, I. J. and Thomassen C. (1993), Highly symmetric subgraphs of hyper-

cubes (preprint).[8] Burr, S.A., Griinbaum, B. and Sloane, N.J.A. (1974), The orchard problem, Geom. Dedicata 2

397-424.[9] Chung, F. R. K. (1992), Subgraphs of a hypercube containing no small even cycles, J. Graph

Theory 16 273-286.[10] Chung, F.R.K., Szemeredi, E. and Trotter, W. (1992), The number of different distances

determined by a set of points in the Euclidean plane, Discrete and Computational Geometry 11-11.

[II] Erdos, P. (1935), On the difference of consecutive primes, Quart. J. Math. Oxford 6 124-128.[12] Erdos, P. (1946), On sets of distances of n points, Amer. Math. Monthly 53 248-250.[13] Erdos, P. and Hajnal, A. (1989), Ramsey-type theorems, Discrete Applied Math. 25 37-52.[14] Erdos, P., Hajnal, A., Mate, A. and Rado, R. (1984), Combinatorial Set Theory: Partition Rela-

tions for Cardinals, North-Holland Publishing Company, Studies in Logic and the Foundationsof Mathematics, Vol. 106.

[15] Erdos, P., Hajnal, A. and Milner, E. C. (1966), On the complete subgraphs of graphs definedby systems of sets, Acta Math. Acad. Sci. Hungaricae 17 159-229.

[16] Erdos, P., Hajnal, A. and Shelah, S. (1974), On some general properties of chromatic numbers,in Topics in Topology (Proc. Colloq. Keszthely, 1977) Colloq. Math. Soc. J. Bolyai 8 243-255.

[17] Erdos, P., Hajnal, A. and Szemeredi, E. (1982), On almost bipartite large chromatic graphs,Annals of Discrete Math. 12, Theory and Practice of Combinatorics, Articles in Honor of A.Kotzig (A. Rosa, G. Sabidussi and J. Turgeon, eds.), North-Holland, 117-123.

[18] Erdos, P., Herzog, F. and Piranian, G. (1958), Metric properties of polynomials, Journald Analyse Mathematique 6 125-148.

[19] Erdos, P. and Larson, J. (1982), On pairwise balanced block designs with the sizes of blocks asuniform as possible, Annals of Discrete Mathematics 15 129-134.

[20] Erdos, P. and Vertesi, P. (1980), On the almost everywhere divergence of Lagrange InterpolatoryPolynomials for large arbitrary systems of nodes, Acta Math. Acad. Sci. Hungaricae 36 71-89.

[21] Faber, G. (1914), Uber die interpolatorische Darstellung stetiger Funktionen, Jahresber. derDeutschen Math. Ver 23 190-210.

[22] Fliredi, Z. (1990), The maximum number of unit distances in a convex n-gon, J. Comb. Theory(Ser. A) 55 316-320.

[23] Griinwald, G. (1935), Uber die Divergenzersheinungen der Lagrangeschen Interpolationpoly-nome, Acta. Sci. Math. Szeged 1 207-221.

[24] Griinwald, G. (1936), Uber die Divergenzersheinungen der Lagrangeschen Interpolationpoly-nome stetiger Funktionen, Annals of Math. 37 908-918.

[25] Gyori, E. (1989), On the number of C5's in a triangle-free graph, Combinatorica 9 101-102.[26] Halberstam, H. and Roth, K.F. (1983), Sequences, Springer-Verlag, xiii+290pp.[27] Kahn, J. (1992), Coloring nearly-disjoint hypergraphs with n + o(n) colors, J. Combinatorial

Theory (Ser. A) 59 31-39.[28] Kahn, J. (1993), On a problem of Erdos and Lovasz. II n(r) = o(r), J. Amer. Math. Soc. 14.

10 P. Erdos

[29] Kanold, (1981), Uber Punktmengen im /c-dimensionalen euklidischen Raum, Abh. Braunschweig.wiss. Ges. 32 55-65.

[30] Maier, H. and Pomerance, C. (1990), Unusually large gaps between consecutive primes, Trans.Amer. Math. Soc. 322 201-238.

[31] Marcinkiewicz, J. (1937), Sur la divergence des polynomes d'interpolation, Ada Sci. Math.Szeged 8 131-135.

[32] Privalov, A. A. (1976), Divergence of Lagrange interpolation based on the Jacobi abscissas onsets of positive measure, Sibirsk. Mat. Z. 18 837-859 (in Russian).

[33] Privalov, A. A. (1978), Approximation of functions by interpolation polynomials, in Fourieranalysis and approximation theory I—II, North-Holland, Amsterdam 659-671.

[34] Pyber, L. (1987), The number of pairwise non-commuting elements and the index of the centrein a finite group, J. London Math. Soc. 35 287-295.

[35] Rankin, R. A. (1938), The difference between consecutive prime numbers, J. London Math. Soc.13, 242-247.

[36] Rankin, R. A. (1962), The difference between consecutive prime numbers. V, Proc. EdinburghMath. Soc. 13 331-332.

[37] Rodl, V. (1982), Nearly bipartite graphs with large chromatic number, Combinatorica 2 377-387.[38] Schonhage, A. (1963), Eine Bemerkung zur Konstruktion grosser Primzahllucken, Arch. Math.

14 29-30.[39] Shrikhande, S. S. and Singhi, N. M. (1985), On a problem of Erdos and Larson, Combinatorica

5 351-358.[40] Spencer, J., Szemeredi, E. and Trotter, W. (1984), Unit distances in the Euclidean plane, Graph

Theory and Combinatorics, Academic Press, London 293-303.

Menger's Theorem for a CountableSource Set

RON AHARONI+ and REINHARD DIESTEL*+Department of Mathematics, Technion, Haifa 32000, Israel

* Mathematical Institute, Oxford University, Oxford OX1 3LB, England

Paul Erdos has conjectured that Menger's theorem extends to infinite graphs in the followingway: whenever A, B are two sets of vertices in an infinite graph, there exist a set of disjointA-B paths and an A-B separator in this graph such that the separator consists of a choiceof precisely one vertex from each of the paths. We prove this conjecture for graphs thatcontain a set of disjoint paths to B from all but countably many vertices of A. In particular,the conjecture is true when A is countable.

1. Introduction

If there is any conjecture in infinite graph theory whose fame has clearly transcended theboundaries of the field, it is the following infinite version of Menger's theorem, conjecturedby Erdos:

Conjecture 1.1. (Erdos) Whenever A,B are two sets of vertices in a graph G, there exist aset of disjoint A-B paths and an A-B separator in G such that the separator consists of achoice of precisely one vertex from each of the paths.

Here, G may be either directed or undirected and either finite or infinite, and 'disjoint'means 'vertex disjoint'. If G is finite, the statement is clearly a reformulation of Menger'stheorem. A set of A-B paths together with an A-B separator as above will be called anorthogonal paths/separator pair.

We remark that the naive infinite analogue to Menger's theorem, which merely comparescardinalities, is considerably weaker and easy to prove. Indeed, consider any inclusion-maximal set & of disjoint A-B paths. If ^ can be chosen infinite, (J ^ , which is trivially anA-B separator, still has size only \g?\. If not, choose & of maximal (finite) cardinality, andthere is a simple reduction to the finite Menger theorem [5]. This was in fact first observedby Erdos, and seems to have inspired his above conjecture as the 'true' generalization ofMenger's theorem.

12 R. Aharoni and R. Diestel

Although Erdos's conjecture has been proved for countable graphs [2], a full proof stillappears to be out of reach. However, no other conjecture in infinite graph theory hasinspired as interesting a variety of partial or related results as this has; see [4] for a surveyand list of references.

The main aim of this paper is to prove a lemma, which, in addition to implying (with[2]) the results stated in the abstract, might play a role in an overall proof of the conjectureby induction on the size of G. Briefly, the lemma implies that if the conjecture is truefor all graphs of size K, where K is any infinite cardinal, then it is true also for arbitrarygraphs, provided the source set A is no larger than K. (In particular, we see that theconjecture holds for any graph if A is countable.) Now if \A\ = \G\= X and the conjectureholds for all graphs of size < 2, the lemma enables us to apply the induction hypothesis toG with A replaced by its smaller subsets A'; we may then try to combine the orthogonalpaths/separator pairs obtained between these A' and B to one between A and B. Wemust point out, however, that such a proof of Erdos's conjecture will be by no meansstraightforward, and it is not the only possible approach.

2. Definitions and statement of the main result

All the graphs we consider will be directed; undirected versions of our results can berecovered in the usual way by replacing each undirected edge with two directed edgespointing in opposite directions. An edge from a vertex x to a vertex y will be denotedby xy. When G is a graph, G denotes the graph obtained from G by reversing all itsedges.

Paths, likewise, will be directed, and we usually refer to them by their vertex sequence.If P = x... y is a path and v9 w are vertices on P in this order, vPw denotes the subpathof P from v to w. Similarly, we write Pv and vP for initial and final segments of P, Pvfor Pv — v, vP for vP — v, and so on. If Q = y... z is another path, and P n Q = {y},then PyQ denotes the path obtained by concatenating P and Q.

Let X, Y be sets of vertices in a graph. An X-Y path is a path from X to Y whoseinner vertices are neither in X nor in Y. If x is a vertex, a set of {x}-Y paths that aredisjoint except in x is an x-Y fan; the fan is onto if every vertex in Y is hit. Similarly, aset of X-y paths that are disjoint except in y is an X-y fan.

A warp is a set of disjoint paths. When W is a warp, we write V[iV] for the set ofvertices of the paths in 1V, and E \iV\ for the set of their edges. Similarly, we write in \W~\for the set of initial vertices of the paths in #", and ter [W] for the set of their terminalvertices. For a vertex x G V\iV\ we denote the path in W containing x by <2>r(x), orbriefly Q(x). For x £ V[i^]9 we put Qir(x) := {x}. A warp consisting of A-B paths is anA-B warp. By "IV we denote the warp in *G consisting of the reversed paths from W*

' Clearly, ?F = iV. We shall use this fact as an excuse to denote warps in *G, if they are introduced afreshrather than being obtained from a warp in G, by iV etc. straight away; their reversals in G will then bedenoted by iV. The idea here is to avoid the counter-intuitive practice of having a warp iV in *G and aresulting warp iV in G. This convention, if not its explanation, should help the reader avoid any warps inhis or her intuition when such things are discussed briefly in Section 5.

Menger's Theorem for a Countable Source Set 13

Let G = (F ,£) be a graph and A,B c 7. Any such triple T = (G,A,B) will be called aweb. The web (G , B,A) is denoted by 1 . An A-B warp ^ with in [W] = A is a linkagein F, and F is linkable if it contains a linkage.

A set S ^ F separates A from 5 in G if every path in G from A to B meets S. Notethat A and 5 may intersect, in which case clearly Af)B ^ S.

A warp W in G is called a wai;e in F if F[#"] n A = in [W] and ter[W] separates Afrom J? in G. The wave {(a) | a e A} is called the trivial wave. If ^ is a wave in F, thenF/#^ denotes the web

(G-(A\in[W])-(V[W]\ter[W]), ter[W], * ) •

In every web F = (G,,4,£) there is a wave T * such that T/W has no non-trivial wave.(This is not difficult to see. If Wo is a wave in F and W\ is a wave in F/Wo, then # 1defines a wave in F in a natural way: just extend its paths back to A along the pathsof Wo. This wave in F is 'bigger' than Wo, and chains of waves in F with respect to thisorder tend to an obvious limit wave W, which consists of the paths that are eventually inevery wave of the chain. If the chain was maximal, then Y/W has no non-trivial wave.See [2] for details.)

A wave W in F is a hindrance if A\in [W] ^=0; if F contains a hindrance, it is calledhindered. Note that every hindrance is a non-trivial wave. The following was observed in[2]:

Erdos's conjecture is equivalent to the assertion that every unhindered web is linkable.

We are now in a position to state the main result proved in this paper. (For the reasonsexplained earlier, and because it is of a technical nature, we call it a lemma, not a theorem.)

Lemma 2.1. Let F = (G,A,B) be a web and / an A-B warp in G (possibly empty). If> \B\ter[f]\, then F is hindered.

Lemma 2.1 will be proved in Sections 3 and 4. Our aim will be to turn the given warp f,step by step, into a hindrance. This will require some alternating path techniques; thedefinitions and lemmas needed are given in Section 3. Section 4 is devoted to the mainbody of the proof of Lemma 2.1. In Section 5 we look at the implications of the lemmafor Erdos's conjecture.

3. Aternating paths

Let F = (G,A,B) be a web, and let f be an A-B warp in G. A finite sequenceP = xoeoxi^i ...en-\xn of not necessarily distinct vertices xt and distinct (directed) edgeset of G will be called an alternating path (with respect to f) if the following threeconditions are satisfied:

(i) for every i < n, either e, = x,-x,-+i G E(G)\E[f] or et = x;+ix; G E[f];(ii) if x,- = Xj for i ± j9 then xt e V[f];(iii) for every i, 0 < i < n, if x, <E V[f], then {e,-_i,et} nE[/]^ 0.

Figure 1 Two alternating paths with respect to £

All the alternating paths we consider in this section will be alternating paths in G withrespect to / . Note that, by (iii) above, an alternating path starting at a vertex of / hasits first edge in / . As the edges of an alternating path are pairwise distinct, it can visitany given vertex at most twice, and this happens in essentially only two ways: if x, = Xjfor some i < j < n, then x,- £ V[f] by (ii), so by (iii)

either ei-\,ej e E[f] and e,-,e/_i ^ E[f] (Figure 1 left)or eueHX e £ [ / ] and <?/_!,£/ £ £ [ / ] (Figure 1 right).

Note that initial segments of alternating paths are again alternating paths, but finalsegments need not be. Finally, an ordinary path which avoids J* or meets it only in itslast vertex is trivially an alternating path.

There are analogous alternating versions of the notions of X-Y path, X-y fan and soon.

Lemma 3.1. If a e A\in[f] and b e B\ter[/], and if P = a...b is an alternating pathwith respect to /, then G contains an A-B warp /' such that in [f] = in [/] U {a} andter[f] = ter[f] U {b} and {Q e / | P nQ = 0} c / ' .

Proof. Consider the graph on V\J\ U V(P) whose edge set is the symmetric differenceof E[f] and E(P). The (undirected) components of this graph are all finite. Consideringtheir vertex degrees, we see that they are either A-B paths or cycles avoiding Au B(possibly trivial). The assertion follows. •

Lemma 3.2. Let P\ = xo^o • • • Cn-i^n and Pi = yo/o • • -fm-iym be alternating paths. If xn =yo, then there exists an alternating path P^ from xo to ym such that V(P^) <= V(P\)Uand E(P3)^E(P{)UE(P2).

Proof. Let i < n be minimal such that there exists a j < m with the following twoproperties:

(ii) if Xi G V[/l then either e^x G E[f] or / , G £ [ / ] .

(Note that such an i exists, because xn = yo and P2 is an alternating path. Moreover, j iseasily seen to be unique.) Then xo^o• • • ei-ixtfj• --fm-iym is an alternating path as desired.

•

4. Proof of the main lemma

We now prove Lemma 2.1. As in the lemma, let F = (G,A,B) be a web, and let / be anA-B warp in F. Let us write

A\ := in [ /] and A2 :=

and

#! := ter[/] and B2 :=

and put /c := |#2|. We assume that \A2\ > K, and construct a hindrance #^ in F. Again, allthe alternating paths considered in this section will be alternating paths in G with respectto f, unless otherwise stated.

Let us quickly dispose of the case when K is finite. Assume that K is minimal such thatthe lemma fails. By Lemma 3.1 and the minimality of K, there is no alternating path fromA2 to B2. For each path Q e </, let x(Q) denote the last vertex of Q that lies on somealternating path starting in A2\ if no such vertex exists, let x(Q) be the initial vertex of Q.We claim that

^ := {Q* I Q e / and x = x(Q)}

is a wave in G; since \A2\ > K > 0 and hence in ]iV\ = in \J\ g A, this wave #^ will be ahindrance and the lemma will be proved.

To show that iV is a wave, we have to prove that ter[if] separates A from B. So letP be any A-B path. Since P is not an alternating path from A2 to B2, it meets V[f]and hence V[i^]; let y be its last vertex in K[#"], and write Q := Q/(y) and x := x(Q).Suppose P avoids ter[i^]. Then x ^ y, so there exists an alternating path R from A2 to ythat ends with an edge of iV. (Indeed, by definition of #^, there is an alternating A2-xpath B!\ if x! is the first vertex of B! on yQx, then KV followed by yQx! in reverse orderis an alternating path from A2 to y.) By definition of #", # avoids V[/]\V[^] and isthus an alternating path with respect to if. Let z be the first vertex of R on yP. Theneither z = j ; or z ^ V[iT], so KzP is again an alternating path with respect to #". Bydefinition of TT, i^zP avoids V[/]\V[iT]. Therefore zP avoids F[ / ] \{y} 3 Bi. (Recallthat y £ B, because y ^ x and hence y G Qx.) Thus KzP is an alternating path from A2

to B2, a contradiction.We shall now assume that K is infinite. To motivate our proof, let us consider the (much

easier) case of / = 0. (This is an important special case, and we recommend that thereader remain aware of it throughout the proof of Lemma 2.1.) Assume Erdos's conjecture

16 jR. Aharoni and R. Diestel

is true, and let S be an A-B separator as in the conjecture. Then \S\ < \B\ = K. Let usthink of the vertices in A as being 'to the left' of S, and of those in B as 'to its right'.Which other vertices of G will be to the left of S ? Surely those that cannot be separatedfrom A by < K vertices, i.e. that are joined to A by a fan of size > K. We shall call thesevertices 'popular'. If a popular vertex is in S, it is the starting vertex of a path to B thatcontains no other popular vertices; let us call such a path 'lonely', and its starting vertex'special'. The special vertices, i.e. the vertices that are popular and from which we canget to B without hitting any other popular vertex, are in a sense 'rightmost' among thepopular vertices, even when they are not in S. As we shall see, they turn out to be 'closeenough' to S that they themselves form the set of endvertices of a hindrance in F, whichis constructable without reference to S.

For the general case, we follow a similar approach, except that now all the relevantpaths and fans will be alternating. Let us call a vertex x G V(G)\A\ popular if eitherx e A2 or there exists an alternating A2-x fan of order > K. An alternating path P endingin B2 and with no inner vertex in A2 is called a lonely path if all its vertices are unpopular,except possibly its starting vertex and any vertices x G V[f] such that, if e is the edgefollowing x on P, then e £ E\J\. (In the latter case, the edge preceding x on P must bethe edge of / starting in x.) Note that a final segment x ^ x ^ ... of a lonely path is againlonely if and only if xt satisfies condition (iii) in the definition of an alternating path, i.e.if and only if et G E[f] when xt G V[f].

Our first lemma is merely a technical argument that will be used twice later and hasbeen extracted for economy. The first time we will use it is in the proof of Lemma 4.2below, and for motivation the reader may prefer to read Lemma 4.2 and its proof firstand then return to Lemma 4.1.

Lemma 4.1. Let cc be a cardinal, ££ = {Lp | /? < a} a family of lonely paths, and Ji ={Mp | /? < a} a family of pairwise disjoint alternating paths starting in A2. Assume that, foreach ft < a, the last vertex of Mp is the starting vertex of Lp. Then OL<K.

Proof. Suppose a > K. For each /? < a, let Pp be an alternating path from the startingvertex of Mp to the final vertex of Lp as provided by Lemma 3.2. Construct an undirectedforest H = \Jp<!XHp from these paths, as follows. Let HQ be the undirected graphunderlying Po- Now let p < a be given, and assume that Hy has been defined for everyy < /?. Let zp denote the first vertex of Pp that is in B2 U V(Hj), where H^ := IJ7<^ Hr,and let Hp be the union of Hj with the undirected graph underlying Ppzp. (If zp occurstwice on Pp, we take Ppzp to stop at the first occurrence of zp.) Since \B2\ < K and everypath Pp ends in B2, H has at most K components. One of these components must havesize > /c, so it contains a vertex z of degree > K. Then z lies on > K of the paths Pp, soz = zp for every /? in some set A ^ a of size > K.

Let

F := {Ppz | p G A}.

Note that the paths in F are pairwise disjoint except for z, so F is an alternating A2-zfan. Hence, z is popular. As the paths Mp are pairwise disjoint, we have z ^ Mp for all

but at most one /? G A; let us delete this one fi from A if it exists. Now for all /? G A, wehave that z £ Lp and z is not the starting vertex of Lp (since this is on Mp).

Now consider any /? G A. Since L^ is a lonely path and z is popular but not thestarting vertex of Lp, we have z G F[/]> and if e denotes the edge following z on Lp,then e ^ E[f]. (Note that e exists, because z G K[/] but Lp ends in 52.) Since Lp isalternating, this means that the edge / preceding z on Lp must be the edge of / startingat z (and such an edge exists). Since z £ Mp, the edge of P^ preceding z is precisely thisedge / .

As ft was chosen arbitrarily, this is true for every /? e A and thus contradicts the factthat for these /? the paths Ppz are disjoint. •

If the starting vertex of a lonely path is popular, this vertex is called special; the setof all special vertices outside V[/] is denoted by S. Special vertices will be our primecandidates for the terminal vertices of the hindrance we are seeking to construct. Since thecorresponding paths of the hindrance will have to be constructed from the fans connectingAi to these terminal vertices (making them popular), it is important that there are fewerspecial vertices to be connected in this way than there are connecting paths availablefrom those fans.

Lemma 4.2. There are at most K special vertices.

Proof. Suppose that {sp | /? < K+} is a set of distinct special vertices, where K+ is thesuccessor cardinal of K. For each /?, let Lp be a lonely path starting at sp. Using thepopularity of the sp, we may inductively choose a family {Mp | /? < K+} of pairwisedisjoint alternating paths Mp from Ai to sp. This contradicts Lemma 4.1. •

Let E denote the set of all those edges in G that lie on some lonely path, and let K bethe graph

K :=|J/-E.Let & be the set of all those (undirected) components of K that contain a special vertexor a vertex from A. (Thus, & is a set of pairwise disjoint subpaths of paths in /.) Let

T := {x G V(G) | x is the last vertex on some P G 0>}.

We shall define our desired hindrance W in such a way that

=SuT.

Lemma 4.3. If P = x...y is a non-trivial component of K and x £ A, then x is special(and hence P G & and y e T).

Proof. Let r be the predecessor and s the successor of x on Q(x). Then rx G E, so thereexists a lonely path starting at x with the QdgQ rx. But preceding this path with s doesnot yield another lonely path (since xs G E(P), and hence xs £ E). Therefore x must bepopular (see the definition of lonely paths), and hence special. •

To construct W, let us start from 9. Let 1V§ be the set of all paths P £ 0> that startin A. (These paths may be entire paths from /, and they may be trivial.) Our aim is tocomplete iV§ to our desired wave iV^ by paths of the form a... xPy, where a £ A2 andP — x... y is a path as in Lemma 4.3, together with paths a... s, where again a £ A2 andeither s £ S or s is a special vertex in V[f] making up a singleton component of K. Itwill not be possible to construct iV in exactly this way, because the required paths mayinterfere with the paths in #o- However, such interference will be limited by Lemmas4.1 and 4.2, and can therefore be overcome by the alternating path tools developed inSection 3.

Let

S' = {s; | £ < v < K)

be a well-ordering of those special vertices that are either in S or are the initial vertexof some (possibly trivial) path P £ 0* (cf. Lemma 4.2). For each £ < v in turn, we shallchoose an alternating path P^ from A2 to s^, with the following properties:

(i) PcnP{ =0 for all £ < £ ;(ii) Pc n Qf(s) c {sc} for all s£Sf;

(hi) if Q G / and £ < £ are such that P^ n Q ^ 0, then P ^ n g c {sc};

Let £ < v be given, and assume that paths P% for all £ < £ have been chosen in accordancewith (i)-(iv). By (ii), none of these paths contains (s =) s^. Since s is popular, there is analternating Ai-s^ fan F of size > K. Clearly, at most K of the paths in F meet any of thepaths P% (£, < £) or Q(s) for s £ Sf, except, for the latter, in s?. Similarly, at most K of thepaths in F meet (in a vertex =/= s^) any path Q £ / that is hit by some P% with £ < £. ByLemma 4.1, at most /c paths of F have an edge in E. (It is straightforward to check thatthe first edge in E on any path in F starts a lonely path.) We may thus choose P~ fromthe paths in F according to (i)-(iv).

Lemma 4.4. For every £ < v, we have E(P^) n E\J\ ^ E[Wo\. Thus, P* is in fact analternating path with respect to #o-

Proof. If e £ E(Pr)nE[/], then, by (iv) above, there is a component P of K containing e.By (ii), the initial vertex of P is not in S", and is therefore not special. By Lemma 4.3,therefore, the starting vertex of P must be in A, and so P £ iV§. •

Applying Lemma 3.1 v times with the paths P^, we now turn # 0 into a warp from Aonto ter[i^o] U Sf with at most K initial points in A2. (Here we use that fact that, by (iii)above, no two of the alternating paths P; use the same path in # 0 to alternate on.) By(ii) above, the paths in @> that start at the vertices in S'\S extend this warp to a warp H\By Lemma 4.3, ter\W\ = S U T as desired.

To prove that iV is a wave in F, it remains to show that the set S U T separatesA from B; note that then iT is also a hindrance, since \in[iT] C\ A2\ = \S'\ < K byconstruction.

In order to prove that S U T separates A from B, consider ally A-B path P = a...b

in G. Suppose P avoids S U T.

Lemma 4.5. Either b G B2, or b is the final vertex of an edge in E HE[f]. In either case,b is not special but starts a lonely path.

Proof. Suppose first that b G B^. Then b is not special, because b £ S. Moreover, {b} is atrivial lonely path.

Suppose now that b G #1, and let P' = x... b be the component of K containing b. Asb £ T, we have P' £ &, so x <£ A and neither x nor b is special. By Lemma 4.3, therefore,P' is trivial, i.e. b = x £ A. The edge e o f / that ends in b is therefore in £, and hencelies on a lonely path. The final segment of this lonely path that starts at b (with e as itsfirst edge) is again a lonely path, because e G E[f]. •

Lemma 4.6. The vertex a does not lie on a lonely path.

Proof. If a G A2, then a is popular by definition, so being on - and hence starting - alonely path would imply a G S. If a G A\ and a lies on a lonely path, then this path usesthe edge of f starting at a. Then {a} is a component of X, and hence a trivial path in# 0 and in #^, giving a G T. •

Let x be the last vertex of P that is not on any lonely path, and let y be the vertexfollowing x on P. Let L be a lonely path containing y. Then

(4.7) x £ L, and xyL is not a lonely path.

Since y ^ T, y can only be in V[f] if the edge of / ending in y is in E. (Recall thatL must use an edge of f incident with y, and apply Lemma 4.3.) We may therefore makethe following assumption:

(4.8) If y G V[f], then L starts at y (with the edge of / that ends in y).

Lemma 4.9. The vertex y is popular.

Proof. If y is not popular, then xyL can fail to be a lonely path only if it fails to be analternating path. By (4.7) and (4.8), this can happen only if x G V[f] and xyL fails tostart with an edge of / . But x £ B, so x has a successor q on Q(x). By (4.7) and (4.8), wehave q ^ y. Now qxyL is a lonely path (possibly containing q twice) that contradicts thechoice of x. •

Let z be the last popular vertex on P. Then z ^= b, because b is unpopular by Lemma 4.5.As z G yP by Lemma 4.9, the choice of x and definition of y imply that z lies on somelonely path. But then z G V[f], say z e Q e /: otherwise the final segment of this lonelypath that starts at z would again be lonely, and the popularity of z would mean thatz G S. Let q be the vertex following z on Q and let t be the vertex following z on P.

Lemma 4.10. zq £ E.

Proof. Let p be the vertex preceding z on Q. (This exists, since z ^= a.) If zq e £, thenpz £ E: otherwise z would be not only popular but special, giving {z} e 0> and z e T.But if pz £ E, then zq e E implies by Lemma 4.3 that z e T, a contradiction. •

Since f G yP, there is a lonely path M containing t. As with y in (4.8), we may assumethe following:

(4.11) If t £ F [ / ] , then M starts at t (with the edge of / that ends in t).

By Lemma 4.10, zq is not an edge of M. By (4.11), this means that t ^ q; in particular,zg and z£ are distinct edges. Moreover, zt is not an edge of M, since then M wouldhave to use its starting edge again. Therefore, qztM is an alternating path. Since t is notpopular (by the choice of z), this means that qztM is even a lonely path. (Note that ztis a 'real' edge, not the reverse of a ^/-edge, so the popularity of z does not prevent thispath from being lonely.) This, however, contradicts Lemma 4.10, completing the proof ofLemma 2.1.

5. Consequences

In this section we apply Lemma 2.1 to deduce some concrete partial results towardsErdos's conjecture. First, we need another lemma.

Lemma 5.1. Let K be an infinite cardinal If Erdos's conjecture holds for all graphs oforder < K, it holds for all webs F = (G,A,B) such that \A\9 \B\ < K.

Proof. Let F = (G,A,B) be a web with \A\, \B\ < K, and assume the conjecture holds forevery graph of order < K. Let G be obtained from G by adding all edges xy such thatG contains a set of > K independent x-y paths (i.e. paths that are disjoint except in xand y). To prove the conjecture for F, it suffices to find an orthogonal paths/separatorpair (^,5) for V := (G\A,B). Indeed, then S is clearly also an A-B separator in G. Asfor the paths in ^ , their foreign edges can be replaced inductively by paths in G whoseinteriors avoid each other and all the paths in & (since \&\ < K), giving an A-B warpin G. We thus obtain an orthogonal pair for F.

Let G" be the union of all minimal A-B paths in G. (A path P = a... b is minimal ifG contains no a-b path Q with V(Q) g V(P).) It is now sufficient to find an orthogonalpaths/separator pair for T":= (G",A,B), which will clearly also be an orthogonal pairfor r . It thus suffices to show that |G"| < K.

Suppose \G"\ > K, and consider a set I c V(G") \ (A U B) of size > K, say X ={xp | f$ < a}. (Recall that \A\, \B\ < K by assumption.) For each /? < a, use the definitionof Gf to find a minimal A-B path P^ in G containing xp. For all /? < a, inductivelydefine P^ as the maximal final segment of Ppxp that meets |Jy<£ Py a t m ° s t in its startingvertex sp. Since \A\ < K, there is a vertex 5 e G" such that 5 = 5 for every /? in some setA g a o f size > K. Then Fi := \JpeAsPpxp is a fan from 5 onto Y := {xp\P e A}.

Similarly, \JpeA xpPp contains a fan F2 from some set Z c Y of size > K to a vertex £.Clearly, F2 may be chosen so that no two of its paths meet a common path of Fi. It isthen easy to combine Fi and ¥2 into a set of > K independent s-t paths in G'. Thus stis an edge of G', by definition of G'. But 5 and t are non-consecutive vertices on somecommon path Pp (take any /? such that xp e Z), which contradicts the minimality of i^.

•

Combining Lemma 2.1 and Lemma 5.1, we can now easily prove the following.

Theorem 5.2. Let K be an infinite cardinal If Erdos's conjecture holds for all graphs oforder < K, it holds for all webs Y = (G,A,B) such that \A\ < K.

Proof. Let F = (G,A,B) be a web with \A\ < K. Let ^ be a wave in *T such thatF' := *T ffi has no non-trivial wave. Let B' := terfiF], and let H c G be such thatFr = (n,B',A). (In other words, take the underlying graph of F' , reverse its edges, andcall the resulting graph H.) Since Y' is unhindered, we have \B'\ < \A\ < K by Lemma 2.1.Now if the conjecture holds for all graphs of size < K, then by Lemma 5.1 it holds for F \and there is a warp $ in Fr together with a Bf-A separator S in H consisting of achoice of one vertex from each path in / . But S is also a B-A separator in G (becauseter[iT] is one), and hence an A-B separator in G. Thus S, together with / followed bya suitable subset of iV, is an orthogonal paths/separator pair for F. •

Corollary 5.3. Erdos's conjecture is valid for all webs F = (G,A, B) in which A is countable.

Proof. By Theorem 5.2 and the fact that the conjecture holds for countable graphs [2]. •

Unsurprisingly, Corollary 5.3 on its own does not need the full strength of Lemma 2.1. Infact, with hindsight, it is not too difficult to deduce the corollary directly from the mainresult of [3].

We conclude this section with an application of Lemma 2.1 to webs that come with apartial linkage.

Theorem 5.4. Let F = (G,A, B) be a web, and assume that G contains an A-B warp #such that A\in[/] is countable. Then Erdos's conjecture holds for F.

Proof. As in the proof of Theorem 5.2, we let iV be a wave in F such that Y' := F / # "has no non-trivial wave. Let Br := ter$F], and let H c G be such that F7 = CH,B\A).

Then Yf is unhindered, and the final segments in *H of the paths in *J form a B'-A warp#' in *H. By Lemma 2.1, we have

\B'\in\J']\ < \A\ter[7']\ = \A\in[/]\ < Ko.

But such unhindered 'countable-like' webs as F' are linkable [3]. Let ^ be a B'-A linkagein *H. The concatenations of the paths in i f with their unique extensions in iV then form


an A-B warp in F, and Br is an A-B separator in G consisting of a choice of one vertexfrom each path in this warp. •

References

[1] Aharoni, R. (1984) Konig's duality theorem for infinite bipartite graphs. J. London Math. Soc.29 1-12.

[2] Aharoni, R. (1987) Menger's theorem for countable graphs. J. Combin. Theory B 43 303-313.[3] Aharoni, R. (1990) Linkability in countable-like webs. In: Hahn, G. et al. (eds.) Cycles and

Rays, NATO ASI Ser. C, Kluwer Academic Publishers, Dordrecht.[4] Aharoni, R. (1992) Infinite matching theory. In: Diestel, R. (ed.) Directions in Infinite Graph

Theory and Combinatorics, Topics in Discrete Mathematics 3, North-Holland.[5] Konig, D. (1936) Theorie der endlichen und unendlichen Graphen, Akademische Verlagsgesell-

schaft, Leipzig (reprinted: Chelsea, New York 1950).

On Extremal Set Partitions in Cartesian ProductSpaces

RUDOLF AHLSWEDE and NING CAI

Universitat Bielefeld, Fakultat fur Mathematik, Postfach 100131, 33501 Bielefeld, Germany

The partition number of a product hypergraph is introduced as the minimal size of apartition of its vertex set into sets that are edges. This number is shown to be multiplicativeif all factors are graphs with all loops included.

1. Introduction

Consider {i^,S), where V is a finite set and $ is a system of subsets of f\ For thecartesian products i^n = f^i an<3 <$n = 111 ^» *et n(n) denote the minimal size of apartition of i^n into sets that are elements of Sn if a partition exists at all, otherwise n(ri)is not defined. This is obviously exactly the case if it is so for n = 1.

Whereas the packing number p(n), that is the maximal size of a system of disjointsets from Sn, and the covering number c(n), that is the minimal number of sets from in

to cover i^n, have been studied in the literature, this seems to be not the case for thepartition number n(n).

Obviously, c(ri) < n(n) < p(n), if c(n) and n(n) are well denned. The quantitylim^oo ^\ogp(n) is Shannon's zero error capacity [11]. Although it is known only forvery few cases (see [7]), a nice formula exists for limw_>oo(l/w)logc(n) (see [1, 10]).

The difficulties in analyzing n(n) are similar to those for p(n). For the case of graphswith edge set $ including all loops, we prove that n{n) = n(l)n (Theorem 3). This resultis derived from the corresponding result for complete graphs (Theorem 2) with the helpof Gallai's Lemma in matching theory [6]. More general results concern products ofhypergraphs with non-identical factors. Another interesting quantity is fx(n), the maximalsize of a partition of Yn into sets that are elements of Sn (again only hypergraphs{f^,S) with a partition are considered). We also call \i the maximal partition number.It behaves more like the packing number (see example 5). Clearly, n(n) < fi(n) < p(n).It seems to us that an understanding of these partition problems would be a significantcontribution to an understanding of the basic, and seemingly simple, notion of Cartesian

24 R. Ahlswede and N. Cai

products. Another partition problem was formulated in [12]. Among the contributions tothis problem, we refer the reader to [5], [9], and [12].

2. Products of complete graphs: first results

For a complete graph # = {V,g), let g* = gU {{v} : v G V) and define the hypergraph<$* = {rn,£% where rn = X[\r and gn = n"<$*•

We study the partition number n(n), first for #n, and in later sections extend our resultsto hypergraphs, which are products of arbitrary graphs including all loops.

First we introduce the map a : gn —• {0, l}n, where

s" = (7(£II) = aog|£i | , . . . , log|£n |) . (2.1)

As weight of En (w(En) for short), we choose the Hamming weight wH(sn) = Y!t=ist-Notice that the cardinality \En\ equals 2w(£").

Instead of partitions, we consider more generally a packing @> of <€n. We set

&i ={Ene&: w(En) = i},Pt = 1^1, (2.2)

and call {P,-}-Lo t n e weight distribution of ^ .

We associate with 0* the set of shadows 1 a £?n defined by

SL = {En e Sn : En c Fn for some F" G ^ } , (2.3)

and its level setsJi = { £ " e i : w(£n) - i},0 < i < n. (2.4)

It is convenient to write Qt = |Jt |. {Q/}f=0 is the weight distribution of £ = shad(^).First we establish some simple connections between these weight distributions.

Lemma 1. For a packing 0* of^n

E ' k Q - = Qt. (2.5)

Proof. Consider any edge En with weight w(En) = i > k. There are exactly 2i~k(lk) edges

contained in En with weight k. Therefore we have always

D

Lemma 2. For a packing 3P of^"n n

l l = E P ' = Z(-1)*2t. (2-7)i=0 /t=0

On Extremal Set Partitions in Cartesian Product Spaces 25

Proof. An edge En e 0>i contributes to J^^i-tfQk the amount

nLemma 3. For a packing 0> of ^n

npo = X(-1)*2*Q* (2.8)

fc=0

and if in addition £? is a partition and S = \V\ is odd,

- 1 > 0. (2.9)k=0

Proof. An edge E" € ^ contributes to £"k=0(-l)k2kQk the amount

fc=O

which equals 1, if i = 0, and 0, otherwise.Therefore (2.8) holds.Furthermore, if S is odd, then so is Sn and there must be an edge in the partition of

odd size, that is, Po > 1 or, equivalently, by (2.8), (2.9) must hold. •

Remark 1. The last two Lemmas can be derived more systematically from Lemma 1by Mobius Inversion. Here this machinery can be avoided, but we need it for the moreabstract setting of [4].

3. Products of complete graphs: the main results

We shall now exploit Lemma 3 by applying it to classes of subhypergraphs, which wenow define. For any / cz {1,2,...,«} and any specification (vj)jeic, where Vj G i^j, we set

\i=i i=i /

where

«) a n d ^ = {,} for ,• e r. ( 3 2 )

Clearly, for a partition 0> of <$n and SL = shad^, the set ^ ( / , (vj)jeIc) = £ n ^n is adownset, and the map

( )iel \i=l / iel

is a bijection.

Write 1 = \p(£n ^n) and let % count the members of 1 of weight I Since J is adownset in Yliei (^i anc^ ^ s m a x m i a l elements form a partition of Yiiei ^ " w e know that% = Sm. This fact and Lemma 3 yield

m

Sm + £(-l)*2*J2k-l >0. (3.4)/ c = l

This is the key to the proof of the following important result.

Theorem 1. For a partition 0> of ^n = {Tn,Sn) with i^n = ]\"=1 *ru \T{\ = S for i =l,2,... ,n, the weight distribution (Qk)l=o of Q = shad& satisfies, for 1 < m < n,

Proof. The map \p preserves inclusions and weights. The total number of pairs (/, (vj)jeic)with \I\ = m equals (^)Sn~m. Moreover, each £ " e i with w(En) = k is contained inexactly (^Z^) sets of the form J ( / , (vj)jei^ and thus for the sets of weight k

We have one equation of the form (3.4) for each pair (l,(vj)jeic)- Summation of theirleft-hand sides gives, therefore,

) s S + Y(l)2(l)Qk(W j£f \m-kj \m

and hence (3.5). D

Now comes the harvest.

Theorem 2. For a partition SP of$n

"Hi •

Proof. Since \En\ < 2", obviously \&>\ > Sn/2n, and for S = 2a even, the result obviouslyholds. Now let S = 2a+ 1.

Summing the left-hand side expressions in (3.5) for m = 1,2,...,rc results in

or in

(2» - i)sn + V(-i)fc2*efc y ( " " ) - [(s +1)" - s»] > o.k=\ m=k

This is equivalent ton

As Qo = 5", we conclude, with Lemma 2,

]P > (5 4- 1)" • 2~n = - , if S is odd.

4. Non-identical factors: a generalization

We now consider hypergraphs ^m with vertex sets irn = n"=i ^t an<^ edge sets Sn =]l"=i f' where the yt\ are finite sets of not necessarily equal cardinalities St. The factors$t are such that (yu St) is a complete graph with all loops included. We shall write, withpositive integers af,

| ^ , | = 2 a r + e,, £ r £ { 0 , l } . (4.1)

Inspection shows that the sizes of factors do not affect the proofs of Lemmas 1 and2. Also (2.8) in Lemma 2 holds and since Po > 1, if £f = 1 for £ = l,2,... ,n, we cangeneralize (2.9) to

l )*2*6k-nfi*^0. (4.2)k=0 k=\

Theorem 1 in Section 3 generalizes to

Theorem T. For a partition 0> of^fn

k=\

Proof. (Sketch) In the proof of Theorem 1, replace Sm by ]JieI Si and inequality (3.4) by

n

n Si+X(-i)*2*s* - n e ^ °- <4-4»is/ /c=l ie/

•Theorem 2'. for a partition 0> of <$'"

\»\ > f l [f 1 • (4-5)1=1

Proof. Summing the expressions on the left-hand side in (4.3) for m = 1,2,..., n results in

zz(::*)(-i)^-z z IMm = l fc=l V 7 m=l / : | / |=m iel jelc

<j)=f=l iel jelc

n n ~~

k=\ I iel

or

I iel jelc

We evaluate the expression on the right-hand side by introducing J = {£ : 1 < f < n,= 1} and/* = J \ / . Then

7 iel jeic

J+e./) a n d (4-5) f ° l l o w s -

Corollary 1. The partition number n(^>fn) equals I~[j=ii "2" I •

Proof. The partition number of (i^j, $j) is -f • Take a product of optimal partitions

for the factors. This construction gives the lower bound in Theorem 2'. •

5. Products of general graphs

We assume now that the factors ^t = (i^t^t) (t — l,2,...,n) are arbitrary finite graphswith all loops included.

Obviously, we have for the partition number

n(9,) = \r,\-v(9t), (5.1)

where v(^t) is the matching number of &t.

Theorem 3. For the hyper graph product J^n = <§\ x ... x&n

{9,). (5.2)(=1

Here only the inequalityn

Y[ (5.3)t=i

is non-trivial. We make use of a well-known result from matching theory.

Gallai's Lemma. ([6] or [8] page 89) If a graph $ = (^", <f) is connected and, for allv € y , v(^ — v) = v(@), then & is factor-critical, that is, for all v G 'V, *§ — v has a perfectmatching.

Proof of 5.3. For every t e {1,2,...,n) we modify <&t as follows: remove any vertex v € i^t

with v(&t - v) < v(&t) and reiterate this until a graph <§\ with v(Tt - v) = v(Tt) for allv e f * is obtained.

Notice that (5.1) ensures that

TT(^) = 71&). (5.4)

Denote the set of connected components of 0* by {^*(y)} G J . Clearly,

7r(^) = ^7c(»; ( / ) ) . (5.5)jeJt

Moreover, by Gallai's Lemma each component $*t{j) has a vertex set i^*^ of odd size

and

v«c/)) = (KC /Vl)2"1=a/, say.Thus,

Now, for ^T*n = l l i y\ we have

T i p H > 7r(^f*n), (5.7)

because the modifications described above transform a partition of Jfn into a partitionof J f *n with no more parts.

Finally, by Theorem 2r, we have for the product %>n of complete graphs with vertex sets

r*tU) that

n(9\Ul) x ... x 9mnW) > n(Vn) = (a{1 + l)...(a^" + 1). (5.8)

Therefore,

7r(^O l ) x ... x

Ul,-Jn)

t=\ jeJ,

t=\ t=\

This and (5.7) imply (5.3). D

6. Examples for deviation from multiplicative behaviour

First we give two examples of product hypergraphs 3f x jf" for which the partitionnumber n is not multiplicative in the factors. They are due to K.-U. Koschnick.Example 1.

f\ ={0 ,1 ,2 , . . . ,6} ,^ , = { £ = *-, : | £ | e { l , 4 } } .

Clearly, n(Jf\) = 4 and the partition

{{/} x {0,1,2,3} :i = 0,1,2} U {{/} x {3,4,5,6} :i = 4,5,6}

U {{0,1,2,3} x{;} :7 = 4,5,6}

U {{3,4,5,6} x {7} :7 = {0,1,2}}

U {{3}x{3}}

has 13 members. Therefore

X Jf \) < 13 < TUpf i)7Epf 1) = 16. (6.1)

While this example seems to be the smallest possible for identical factors, one can dobetter with non-identical factors:

Jfi x jf',, where V\ = [0,1,2,3,4} and S\ = {E c V\ : |£| e {1,3}}.

Here, by a similar construction, n(Jf\ x Jf\) < 11, whereas n(Jf?\) • n(3tf\) = 4 • 3 = 12.Example 2. Since n is multiplicative for graphs, one may wonder whether it is multiplicativeif one factor is a graph.

Consider G = (f\S) with V - {0,1,...,4} and S = {{/, / + 1 mod 5} : / = 0,1,...,4} U{/ : 0 < / < 4}, that is, the pentagon with all loops.

Define J#" = {t\S") with Y r = {1,2,..., 14} and £" = {E a V' : |£ | G {1,9}}.Notice that n(G) = 3, n(J^f) = 7, and that the following construction ensures n{G x

34T) <20<21 =7i(G)

{{/} x {7 + fcmod 14 : 0 < / c < 8} : (1,7) G {(0,0), (1,3), (2,6), (3,9), (4,12)}

U {{1,2} x {7} 17 = 0, 1,2} U {{2,3} x {7} : 7 = 2,3,5}

U {{3,4} x {7} 17 = 6,7,8}

U {{4,0} x {7} : 7 = 9,10,11}

U {{0,1} x {7} :j= 12,13,14}

is a set of 5 + 5 • 3 = 20 edges partitioning V x i'f.

To help orient the reader, we add three examples, which demonstrate that thecovering number c, the packing number p and the maximal partition number /i arenot multiplicative in the factors either.Example 3. r3 = {0,1,2}, <f3 - {E c r : |£ | = 2}

We have

3 = cp f 3 x Jf 3) =£ cpT3) • cp f 3) = 4, (6.2)

because ^{{0,1} x {0,1}, {0,2} x {0,2}, {1,2} x {1,2}} covers f3 x r3 and there is nocovering with 2 edges.

This is the smallest example in terms of the number of vertices.Remark 2. Quite generally, even in the case of non-identical factors #?t — (i\,£%), t e N,with maxf |<fr| < 00, the asymptotic behaviour of c(n) is known [1]:

1 / / \ \l i m - I l o g c (n ) — V l o g I m a x m i n V 1E( I ; )<JE I 1 = 0 ,

where Prob(^f) is the set of all probability distributions on £*, q^ is the probability of Eunder q and 1£ is the indicator function of the set E.Example 4. TT4 = {0,1,2,3,4},^4 = {{x,x+ 1 mod 5} : x e i'\).Here we have

5 = p(j?4 x ^f4) ^ p(rf4)p(J?4) = 4. (6.3)

It was shown in [11] that this is the smallest example in the previous sense. Notice that itis bigger than the previous one.Example 5. To avoid heavy notation, we will write J^s = (^'''5,^5) without an index asjf7 = {V,S). It is made up of the 5 vertex sets

ar{ = {xu : j = 1,2,..., m}, 3 < m(i = 0,1,2,..., 4),

the 6 edge sets

and {#^0, • • •, A}- Thus

4 4

r = {Jwl,$ = {# 0,..., # 4} u (|J/=0 /=0

A look at the pentagon with vertex set {xoi,-Vii,X2i,X3i,x4i} shows that a partitionof Jf must contain at least one of the edges 14 "z- as a member. On the other hand, thevertices 'V \ itr

{ have a maximal partition of size 2m. Therefore we have shown thatf) = 2m + 1. We shall next consider / /pf x jf) . For this we introduce the superedges

in JT, and the superedges ^* x <&*(ij = 0,1,...,4) in tf x Jf\ Whereas ^* can bepartitioned into m edges, they can be partitioned into m2 edges.

First we divide TT X 1T into 25 parts {nT,- x # ^ : j , j ' = 0, l , . . . ,4}. Then we pack 5

superedges (as in Shannon's construction) into f" x f . They cover 20 parts, and the

remaining 5 parts are packed with 5 edges of type iVi x IVi>. Finally, we partition the 5

superedges into the edges of Jf x Jf. Thus we obtain a desired partition with 5 + 5m2

edges. Notice that \i(tf x jtf) > 5 + 5m2 > (2m + I)2 = M ^ ) 2 for m > 3. The smallest

example in this class has 15 vertices.

Remark 3. The construction was based on the pentagon. Its vertices were replaced by

sets of vertices iVi with a numbering. The vertices with the same number in the IV \%

form a pentagon. Thus we obtained m = \iVi\ many pentagons. Then we added the

iV\ as further edges. Finally we used the superedges to mimic the original small edges.

We can make this construction starting with any hypergraph Jf = {i^.S). If it has the

property p(3tf?)2 < p(J4f x j f ) , then for m large enough our construction gives an associated

hypergraph for which \i is not multiplicative.

7. Acknowledgement

The authors are very much indebted to Klaus-Uwe Koschnick for constructing beautiful

examples.

References

[I] Ahlswede, R. (preprint) On set coverings in Cartesian product spaces, Manuscript 1971.Reprinted in SFB 343 Diskrete Strukturen in der Mathematik, Preprint 92-034.

[2] Ahlswede, R. (1979) Coloring hypergraphs: A new approach to multi-user source coding, Pt I.Journ. of Combinatorics, Information and System Sciences 4 (1) 76-115.

[3] Ahlswede, R. (1980) Coloring hypergraphs: A new approach to multi-user source coding, PtII. Journ. of Combinatorics, Information and System Sciences 5 (3) 220-268.

[4] Ahlswede, R. and Cai, N. (preprint) On POS partition and hypergraph products. SFB 343Diskrete Strukturen in der Mathematik, Preprint 93-008.

[5] Ahlswede, R., Cai, N. and Zhang, Z. (1989) A general 4-words inequality with consequencesfor 2-way communication complexity. Advances in Applied Mathematics 10 75-94.

[6] Gallai, T. (1963) Neuer Beweis eines Tutte'schen Satzes. Magyar Tud. Akad. Mat. Kutato Int.Kozl. 8 135-139.

[7] Lovasz, L. (1979) On the Shannon capacity of a graph. IEEE Trans. Inform. Theory IT-25 1-7.[8] Lovasz, L. and Plummer, M. D. (1986) Matching Theory, North-Holland Mathematics studies

121, North-Holland.[9] Mehlhorn, K. and Schmidt, E. M. (1982) Las Vegas is better than determinism in VLSI and

distributed computing. Proceedings Nth ACM STOC 330-337.[10] Posner, E. C. and Me Eliece, R. J. (1971) Hide and seek, data storage and entropy. Annals of

Math. Statistics 42 1706-1716.[II] Shannon, C. E. (1956) The zero-error capacity of a noisy channel. IEEE Trans. Inform. Theory

IT-2 8-19.[12] Yao, A. (1979) Some complexity questions related to distributive computing. Proceedings 11th

ACM STOC 209-213.

Matchings in Lattice Graphs and Hamming Graphs

M. AIGNER and R. KLIMMEK

II. Mathematisches Institut, Freie Universitat Berlin, Arnimallee 3, D-14195 Berline-mail: [email protected]

In this paper we solve the following problem on the lattice graph L(m\,...,mn) and theHamming graph H(mi,...,mn), generalizing a result of Felzenbaum-Holzman-Kleitmanon the n-dimensional cube (all m, = 2): Characterize the vectors (si,...,sn) such that thereexists a maximum matching in L respectively H with exactly s, edges in the i-th direction.

1. Introduction

One of the classical enumeration results concerns the number of maximum matchings(1-factors) in the 2n x 2n-lattice graph (see e.g. Lovasz [2] or Montroll [3]). The answerfor higher dimensional lattices is unknown; in fact, no good estimates are known. Inthis paper we consider and solve a closely related problem concerning the structure ofmaximum matchings.

Consider the simplest case, that of the hypercube Qn. The vertices are all n-tuples(MI,...,MW) with Ui G {0,1}. An edge e joins two vertices if they differ in exactly onecoordinate. If this coordinate is in the i-th dimension, then we call e an /-edge. Let Mbe a maximum matching. We associate to M its type (si,...,sn), where s,- denotes thenumber of /-edges in M. Which sequences are types? This question was answered byFelzenbaum-Holzman-Kleitman [1]. They proved that a sequence (s\,...,sn) is the type ofa maximum matching in Qn, n 2, if and only if

(ii) all Si are even.

The purpose of this paper is to generalize this result to arbitrary lattice graphs andHamming graphs.

The vertex set V(L) of the lattice graph L(m\,..., mn) is the set of all n-tuples (u\,...,un)with 0 ^ ut ^ mt — 1. (Thus \V(L)\ = I lLi m»-) ^ w o vertices (u\,...,un) and (v\,...,vn)are joined by an edge if and only if they differ in exactly one coordinate, say /, and\ut — vt\ = 1. The Hamming graph H(mi,...,mn) has the same vertex-set as L(mi,...,mn),and two vertices are joined if they differ in exactly one coordinate.

34 M. Aigner and R. Klimmek

Thus, keeping all but coordinate i fixed, L(mi,...,m,1) induces along dimension i thepath Pm., and H(m\,...,mn) induces the complete graph Kmr Clearly, L(mi,...,m,,) is asubgraph of / /(mi, . . . , mn). Again, we speak of the type (si,.. . , sn) of a maximum matchingin L(m\,...,mn) or //(mi,...,mn).

2. The results

For the rest of the paper we shall assume that mi,...,mn ^ 2 and n ^ 2. In the nextsection we shall see that the hypercube result can be generalized in a straightforward wayto the case when all m, are even. But when some of the m,'s are odd, then the situationis different. First we note that in the case when all mf are odd, a maximum matching Mleaves one vertex uncovered; hence in general, we have \M\ = \\ YXl=i m,-J.

Lemma 1. Suppose there is a maximum matching M of type ( s i , . . . , s n ) in L(mi,. . . ,m,,) or

/ / (mi , . . .,mn). Let A c: { l , . . . , n } and suppose m, is odd for all i ^ A. Then

Proof. The assertion is trivially true for A = 0 since J ^ = 0 and f]^ = 1. Let0 ^ A c {l , . . . ,n}, and consider L(mi,...,m,,). We decompose L(mi,...,mn) along thedimensions in X into YlieA nit partial lattices Lr = L(mi : i ^ A). Every such L' hasby the hypothesis an odd number of vertices. Hence in every L' there exists at leastone vertex which is not covered by an M-edge within L'. The total number of thesevertices is therefore at least YlieA m<> a n d ^ follows. The same proof holds verbatim for

. . , m , A •

Definition. A sequence (si,...,sn) is called (mi,...,mn)-admissible if it satisfies (1) and

When all m, are even, then (1) is vacuously satisfied. Our main results state that apartfrom a parity condition (like (ii) for hypercubes) admissibility is all we need to characterizethe types.

Theorem 1. A sequence (s\,...,sn) is the type of a maximum matching in L(mi,..., mn) if

and only if

(A) (si,...,sM) is (mi,...,mn)-admissible, and(B) if not all m^ are odd, then

Theorem 2. A sequence ( s i , . . . , s n ) is the type of a maximum matching in / / ( m i , . . . ,m,2) ifand only if

(C) (si,...sn) is (mi,...,mn)-admissible.(D) if mt = 2, then st = Ylj^t mj (mod 2), and(E) ifY\j^imj is even, then s,• =£ 1.

Matchings in Lattice Graphs and Hamming Graphs 35

Remark 1. Let us prove the necessity of the conditions in Theorem 1. We have seen(A) in Lemma 1. Suppose Yijîmj *s even. Decompose L(m\,...,mn) along dimension iinto the (n — l)-dimensional lattices To,..., Tm/_i, where the /-edges connect T7 and Tj+\(7 = 0,... ,mt — 2). Since \V(Tk)\ = Yl#imj *s even> there is an even number of matchingedges between To and T\, hence an even number between T\ and T2, and so on, andwe conclude st = 0 (mod 2). If J] .^. ra; is odd, then we may assume m, even, and weconclude by an analogous argument st = ^ (mod 2), i.e. condition (B).

Remark 2. Let us prove the necessity of the conditions in Theorem 2. Again, (C)holds by Lemma 1. If m, = 2, then decomposing H(m\,...,mn) along dimension 1 yieldsSi = Yijî mj (m°d 2). Condition (E) is obvious.

Remark 3. If all mt are odd, then only conditions (A) and (C) apply, and the characteri-zations are in this case the same.

Definition. A sequence (si,...,sn) satisfying the conditions of Theorem 1 or of Theorem2 is called an L-sequence, respectively, an H-sequence for (m\,...,mn).

The hard part of the proof is then to show that a sequence is the type of a maximummatching in L(m\,...,mn) or of H(m\,...,mn) provided it is an L-sequence or, respec-tively, an M-sequence. Note that every L-sequence is an //-sequence, as it must be sinceL(mi,..., mn) is a subgraph of / /(mi, . . . , mn).

3. All m{ are even

We consider first the easy case when all m, are even. Condition (B) in Theorem 1 and (D)in Theorem 2 reduce to s,- even, and condition (E) in Theorem 2 to st =£ 1, for all 1.

Proposition 1. Suppose all mt are even. The sequence ( s i , . . . , s n ) is the type of a maximum

matching in L(m\,...,mn) if and only if

( \>\ \^n c 1 T~\n m.

(Bf) all Sf are even.

Proof. We have already seen the necessity. We can decompose the vertex-set of

L(m\,..., mn) into m = ^ •... • hypercubes Q\p in an obvious way. Since all s, are even,

we can clearly split ( s i , . . . , s n ) into m sequences (t^',...,t^) with t\^ even for all i and 7,

and YJJ^P = si 0' = 1, • - -, n), Yît\j) = 2n~l (7 = l , . . . ,m) . By the result of Felzenbaum,

Holzman and Kleitman, (r ,...,t\p) can be realized by a matching Mj within Q^ for all

7, and we obtain the desired matching in L(m\,...,mn) by putting the m matchings Mj

together. •

Proposition 2. Suppose all mi are even. A sequence (s,-,...,sw) is the type of a maximum

matching in H(m\,...,mn) if and only if

(C) ZllSi = L2nil>m>

(Dr) if mi = 2, then s,- is even,(E') si ±1 for alii.

Proof. Again it suffices to prove the sufficiency. We use induction on the number k ofindices i e {l, . . . ,n} with mt ^ 4. The case k = 0 is that of the hypercube, so let k ^ 1.We may suppose that mn ^ 4.

Case 1. sn is even. We split (si,...,sw) into two sequences (tu...,tn) and (ri,. . . ,rn) asfollows. Set tj = 3 for S, odd (i.e. s, ^ 3) and £,- = 0 for s, even. Now raise each tt by evennumbers such that tt ^ st (i ^ n — 1), tn ^ 2 and X)"=i *i = ^(EKr/ ^/" 2). This is possiblesince by (D') and sw even

n-\

3(fc - 1) < 4 / c"1

The sequence (u) satisfies (C), (D'), (Er) for (mi,...,mn_i,2) and hence, by the inductionhypothesis, is the type of a maximum matching in //(mi,...,mn_i,2). The sequence (r,)with rt = Si — tt satisfies YM=I

rt = \ \\Xi=l mi) (mn ~ 2) with all r, even, and hence is thetype of a maximum matching in L(m\,...,mn-\,mn — 2) and thus in if(mi,...,mn_i,mn — 2)by Proposition 1. Now put the matchings together.

Case 2. sn is odd. Since Y%=i si = ® (mod 2), there must be an index j < n withSj odd, so s7 ^ 3 by (Er) and m; ^ 4 (by (Dr)). Consider now the sequence (sj) =(si,...,s7- + 1,..., sw — 1) and split it as in case 1 into sequences (£,), (r,-). Since sn — 1 ^ 2,Sy + 1 ^ 4 and both are even, we may assume tj > 0 and r7 > 0, rn > 0. As in case1, (tt) is the type of a maximum matching Mt in Hf = i/(mi,...,mw_i,2) and (r,-) is thetype of a maximum matching Mr in Lr = L(mi,...,mn_i,mw — 2). By Proposition 1, Mr

can be obtained from hypercube matchings. Because of m; ^ 4 there are two neighboringhypercubes Ql,Q2 along dimension j9 and since r7 > 0 and rn > 0 we may assume thatQl contains n-edges in Mr and Q2 7-edges in Mr.

Choose u G ^(Q1) and x <E F(g2) such that the edge ux is in Lr. By the edge-transitivityof hypercubes we may assume that the Mr-edge uv covering u in Q} is an n-edge, andsimilarly that the Mr-edge xy in Q2 is a y'-edge. The matching Mt in Hr contains a j-edgewz since tj > 0, where by the edge-transitivity in Ht and mn ^ 4 we may assume that wand z differ from u and, respectively, x just in the n-th coordinate (see Figure 1).

Now replace the matching edges uv, xy, wz by the edges wv, uy, zx of H(m\,..., mn) asin Figure 1. This raises the number of the n-edges in the matching by 1 and decreases thenumber of the 7-edges by 1, whence we obtain our desired matching of type (si,.. . , sn). •

4. Admissible sequences

The main difficulty in the proof of Theorems 1 and 2 is to check admissibility. Bydefinition, this requires checking (1) for subsets A^{l,...,n}. The following result showshow this check can be restricted to a linear number of subsets.

n-\

Figure 1

L e m m a 2 . L e t m i , . . . , m r be even, m r + i , . . . , m n odd. A sequence ( s i , . . . , s n ) with YH=i s» ~V\ nr=i mi\ and sr+\ ^ • • • ^ sn is (mi,..., mn)-admissible if and only if

'-J foral1 k=r,...,n-l. (2)

Proof. The necessity was established in Lemma 1. Suppose (2) is satisfied, and (si) is notadmissible. Then there exists A with { l , . . . , r } c i c { l } i i i j j j - i } such that

(3)ieA ieA

Among all such sets A choose one of maximal size. By (2), there are indices j , k withk G A, j G {r + 1,..., n} \ A and j < k. From the maximality of \A\ we infer

ieA

and thus by (3) and ra7 ^ 3 odd

This, however, implies

a contradiction to (3).

£' > [[mi,ieA

u

To check the parity conditions (D) and (E) we will make frequent use of the followinglemmas.

L e m m a 3 . L e t m \ , . . . , m r be even, m r + i , . . . , m n odd, and suppose ( s \ , . . . 9 s n ) is ( m i , . . . , m n ) -admissible with s r + \ ^ ... ^ s n . L e t d e { 1 , 2 } , and j < k ^ n. Then (s-) = ( s \ 9 . . . , S j +d , . . . , S k — d,..., sn) is ( m i , . . . , m n ) - a d m i s s i b l e whenever Sk ^ d, r ^ 1 or Sk > d, r = 0.

Proof. Assume the opposite and let A be a set violating (1), {1,...,r} ^ A c {1, . . . , w— 1},i.e.

iGi4 ieA

Since (5/) is admissible, we must have j ^ A, k e A. This implies j > r, and thus my, m ^ 3.Set B = A \ {k}, then sj = s, for all i e B, and thus

From (4) and (5) we infer

Sk=s'k+d =ieA ieB

ieA ieB

2

Since s; ^ Sk, this implies by (4) and (5)

+ SJ ^ / JSi + Sk=/ ^s'i + Sk+dieA ieA ieA

ieA ieB

2mk -

hence„ / 1 Id \ _

/• (6)ieA

If \B\ ^ 1, then (6) yields with nij ^ 3, mk ^ 3

and thus J2ieA si + sj < L^ YlieA mi\ s m c e the difference in (7) is at least 2. So in this casewe have a contradiction to the admissibility of (s,).

Hence suppose B = 0, i.e. A = {k} and r = 0. Note that our assertion is thereforeproved for r ^ 1. Inequality (4) now reads by our assumption sk > d

which implies mk ^ 4 and thus m/c ^ 5. It follows that

2sk ^yyik + 2d — 3.

On the other hand, from the admissibility of (s/) we infer

j 1) ^ sj + Sk ^ 2sk ^mk+2d-3,

hence

(mj - 2)mk < 4rf - 5 s£ 3,

in contradiction to m7 ^ 3, mk 5. D

L e m m a 4 . L e t m i , . . . , m r fee ez;en, m r + i , . . . , m n o d d wff/z r ^ 1, a n d suppose ( s \ , . . . , s n ) isan (mi,...,mn)-admissible sequence with sr+i ^ ... sn. Let d e {1,2}, and assume

2i=\ i=l

(sj) = (si,...,Sj — d,...,sr, sr+i + d,...,sn) is (mi,..., mn)-admissible for every j ^ r

Proof. If sr+i + d sr+2, then s'r+1 ^ ... ^ s'n is preserved, and we are done by (8) andLemma 2. Suppose (sj) is not admissible. Then there exists /c 2 with

Sf-|_i ^ "^r+2 ^ • • • ^ $r+k "^ ^r+l i w,

for whichr r+k . r r+k

i={ i=r+2 i=\ i=r+2

This implies

r+k / r r+k \ A r+k

\i=l i=r+2

= a f m r + i - 1 J ^ a,

since mr+i ^ 3, a ^ 3, ^ 2. But this implies

r+k

^2 si >(k- l )s r + i ^ (fe - l)a ^ a,i=r+2

in contradiction to (9), since YM=I si ^ d. •

5. The decomposition lemma

To prove Theorems 1 and 2 we apply induction on the number k of odd m^s. The basecase, that of k = 0, is supplied by Propositions 1 and 2. The major part of the inductionstep is provided by the following decomposition lemma.

Decomposition lemma. Suppose mi,...,mr are even amd rar+i,...,mn are odd for some r,0 ^ r < n. Let (s\,..., sn) be an H-sequence for (mi,..., mn) with sn ^ max(s; : r + 1 ^ i ^n) — 2. Furthermore,

n-\

sn= LyJ l I m < (m o d 2)- (10)i = l

Then we can split ( s \ , . . . , s n ) into two sequences ( a i , . . . , a n ) , ( b \ , . . . , b n - i ) such that

( I ) at + bi = Si for 1 ^ i ^ n — 1, and an = sn,(II) (at) is an L-sequence for (mi,.. . ,mn-\,mn — 1),(III) (bi,...ftw_i) /s an H-sequence for (mi,...,mn_i), and(IV) b r + 1 ^ . . . O * _ i .

Remark 4. Note that the derived sequences (mi,..., mn — 1) and (mi,...,mn-\) have oneodd length less than (mi,...,mn).

Remark 5. As a corollary, the decomposition lemma also holds if we replace an H-sequence by an L-sequence in the hypothesis and the conclusion (III). Indeed, if all m,- areodd, then //-sequences and L-sequences are the same. If one of the m,'s is even (i.e. r ^ 1)and (st) is an L-sequence, then the a/'s of (II) are even by condition (B) of an L-sequence.Since mn is odd, we conclude by condition (B) for (s;)

n-\

bi = st - at = Sj = L y J J J my- = L y J Y[ mJ ( m o d 2)-

Proof of the decomposition lemma. We are given m\,...,mr even, mr+i,...,mn odd with0 ^ r < n, and an //-sequence (si,...,sn) for (mi,...,mn) with sn ^ max(sr+i,...,sn) — 2.We may assume that sr+i ^ ... ^ sw_i. The proof proceeds by induction on YM=I

5«-

Claim 1. The conclusion holds if

either r = 0 (i.e. all mi are odd)

or r ^ 1 and ]T[=1 s; = ^ n[=i m'-

Note that in the latter case at = 0, bt = 5, for 1 < i ^ r must hold because of

Proof of Claim 1. We construct the //-sequence (b\,..., bn-\) as follows. First we set up apreliminary //-sequence (/?i,...,/?n-i) for (mi,...,mn_i) which does not necessarily satisfyPi 5j for all i, and then modify (/?,-) suitably.

The jSi's are defined in the following way:

la. If r = 0, then ft is minimal with j82 = si (mod 2), ft ^ Lf"J-lb. If r ^ 1, then ft = s* for i = 1,..., r.2. Suppose Pi,..., fa are already defined, then /^+i is minimal with /?^+i = Sk+i (mod 2)

and/c+l . /c+1

It follows thatk / c - l

i=\ i=\

for k = r + 1,..., n — 1, and thus

i=\ i=\

Note that because of fa = Sk for all /c, (12) implies ^ = 1 ft ^ ^ = 1 s,- for fe = r + 1 , . . . , n— 1.Using (12), we have for fe > r

fe / c - l . / c - l

i = l / = 1 i = l

where <5r+1 ^ 0 since ELi ft = ELi ^ = U Il[=i ^J-We want to show next that (j8i,...,/Jw_i) is an //-sequence for (mi , . . . ,m n _i) with

pr+\ ^ . . . ^ ft_i. Condition (D) holds because of ft = st and mn = 1 (mod 2). (E) is true

by definition lb and (13). By (12) and Lemma 2, it remains to show that

n—\ . n—\

YJP"-1 = tj Ii^-I' Le' C"-! = 0' (14)

i=\ i=l

^ ^ . . . O S , , - ! . (15)

To see (14), we note by (10)

41=1 i=\ i=i

n-\1

from which cn-\ = 0 follows by (12).To prove (15) we have for k ^ r + 2 according to (13)

. / c - l . k-2

fa-fa-\ = ——

i Z^"2

2 (n\i=l

since mk-\,mk ^ 3.

From the //-sequence (/?i,...,ft,_i) we now construct the //-sequence (b\,...,bn-\)

satisfying (III), (IV) and bi < 5, for all i. First we note ft ^ s,- for / = l , . . . , r + 1. This is

clear for i ^ r by definition lb , and for i = r + 1 we have ft+i ^ sr+\ if r = 0 by definition

la. For r ^ l we infer from the assumption on ^ = 1 s,- and (13)

1 r+1 r 1 r1 ^—> tYlr.L.\ — 1

2S =2 2i = l i=\ »=1

and hence sr+i ^ ft+i because of sr+i = ft+i (mod 2).The following algorithm generates the sequence (b\9...,bn-\):

1. Set bt = ft for i = 1,..., r + 1, and &,- = 0 for i = r + 2, . . . , n - 1.2. For i = r + 2, . . . ,n — 1 distribute j?/ step by step on b,-, bi-\,...,br+\, by raising the

current bj (i ^ j r ^ r + 1) by the maximal possible increment. The i-th distributionstep is described as follows:

initialization j <— i, /?,, st, bi = 0as long as /?,• > 0 repeat

c = min(j8,-, s7- — bj)Pi <- ft- - c, ^ ^ ^ + c

Since ^ = 1 /?, < ^ = 1 5/ for all fe, the j8,-'s are fully distributed, and the loop for the i-thstep stops for j = r + 1 since br = f$r = sr. Furthermore fe,- s/ for all /.

bi = st (mod 2) for all i. (16)

This is trivially true for i ^ r + 1 . Consider what happens in the i-th step distributing /?,-.At the start we have bt = 0. If pt < s,- tken j8,- <— 0, b, <— ft and thus ft,- = ft = 5, (mod 2).If s; < ft, then ft <— ft — Si = 0 (mod 2) and bt <— s,-. Any bj (j < i) is thus raised byan even number and hence keeps the same parity as Sj. Note that after the i-th step,bt = min(si9Pi).

We have br+l ^ . . . ^ f c n _ i - (17)

Assume inductively br+\ ^ ... < bi-\ before the i-th step, i ^ r + 2. Then at the start ofthe i-th step either bi = ft or bi = s,-. In the first cast we infer by (15)

bi = ft > ft-i > minfo-i.ft-i) = b^{ > ... ^ ftr+1.

In the second case, we have at the start

Now assume inductively that (17) holds before we reach bj (j < i). If bj is raised, thenby the set-up of the algorithm bj+\ = Sj+\ and thus

£ Sj ^ bj (after) ^ bj (pre) ^ ... ^ fer+i.

(bi,...,6n_i) is (mi,...,mn-\)-admissible. (18)

This follows immediately from X^ii bj = S^i i /^/' and Xw=i ^; = Yl)=i Pj after step iand (12), (14), (17), and Lemma 2.

The conditions (D), (E) for (fti,..., ftw-i) are easy. (D) is clear since ft,- = s, (mod 2) andmn = \ (mod 2). As for (E) we have r ^ 1. If i ^ r, then ft, = s,- and there is nothing toprove. For i r + 1 we see by (17) and (13)

m - 1 r

bi ^ ftr+i > pr+i > — ^ JJ m < ^ 2>

Our proof that (fti,...,ftw_i) satisfies (III) and (IV) is thus complete.Now we set a, = s, — ft, for i n — 1, an = sn, and show that (a,) is an L-sequence for

(mi,.. .,mn_i,mn — 1).First we note that all a,-(f ^ n — 1) are even because of (16), and an = sn =

^ 2 ^ Fl^r/ w,- (mod 2) by the assumption (10). So it remains to show that (a,) is(mi,...,mn_i,mn — l)-admissible.

The condition „ „ B-i , w-i

i = l i = l i = l i = l

is clear from the construction of the ft,'s. Since we do not know the size-ranking of thea,'s, we must verify (1). Note that a, = 0 for i r. Since mn — 1 is even, the subsets underconsideration contain 1,...,r,n. Suppose there exists A ^ {r + l,...,n — 1} with

ieA

Then ^ £ {r + 1,..., n - 1} because of (19).Set a = n [ = 1 mf with a = 1 if r = 0. We infer

iG^ ieA i=r+l

and, with«—1 n—1 r / n—1

\i=r+l

n-\((mn-l)JJmf+ JJ m;-l I . (21)

(iG/l i=r-hl /

With B = {r + 1,. . . ,n - 1} \ A £ 0 and E"=r+1 s." = f (EIL.+i »«.• - 1), this yieldsji=r+l \ ieA

n n-\( n n\

JJ m,- - 1 - (mn - 1) JJiWi - JJ m, + 1i=r+l ieA i=r+l

= 2 ( n m i ) (m« J J mj ~ m" ~ ~ n m ' )) (

ieA / \ieB

Since sn ^ max(s; : / ^ r + 1) — 2, this latter inequality implies

2

mr2- 2,/ .A. .A.

in contradiction to (20). •Our claim 1 is thus proved. In particular, the decomposition lemma holds for r = 0.

We therefore assume from now on r ^ 1.

Claim 2. 77ie conclusion holds ifr 1 r

" " +1 . (22)

Proof of Claim 2. Let (si,...,sw) be a sequence which satisfies the hypotheses of thedecomposition lemma, where again sr+i ^ ... ^ sn_i. Note that sn is even by (10), andthus r + 1 < n by (22).

TTzere ex/sts j ; r with nij ^ 4 ami sy ^ 3. (23)

Indeed, if r = 1, then si = ^ + 1 by (22). If mx = 2, then s\ = 2. On the other hand,condition (D) for //-sequences implies s\ = n?=2 mi = ^ (mod 2), a contradiction tos\ = 2. Thus m\ ^ 4, si ^ 3. In case r ^ 2, we infer 5ZI=i 5* = ^ (mod 2) from (22). Hencethere exists j ^ r with s;- odd. Condition (E) now implies Sj ^ 3, and (D) implies m7 ^ 4.

Consider the sequence (sj) = (si,...,s7- — 1,...,sr,5r+i + 1,...,sn). Invoking Lemma 4, wesee that (sj) is (mi,..., mn)-admissible whenever 5r+i ^ 5W. In the case sn < sr+\ ^ sn + 2, theadmissibility follows by a simple calculation using (22). Condition (D) is satisfied becauseof nij ^ 4, and (E) is satisfied since Sj — 1 ^ 2 and sr+i ^ ^ (IlLi m 0 (mr+i — 1) — 1 ^ 1.Hence, (5J) is an //-sequence for (mi,...,mn) with

4 = 5 IKTo apply Claim 1 we have to check that s'n^- s\ — 2 (r + 1 ^ f ^ n — 1). Suppose, on thecontrary, sw < sr+\ — 1. Then 5r+i = ... = sn-\ = sn + 2 = 0 (mod 2), and thus

n 1 r

n\ n 1 r

S];=2 n m ' ! = 2 nin contradiction to (22).

By our Claim 1, there exist appropriate sequences (a\), (b\). Now consider the sequence

Since b\ = sft for 1 ^ r, we have ft^+1 ^ ^ (flLi m 0 (mr+i - 1) ^ 4 by (23), and thus

br+\ ^ 3. The admissibility follows now from Lemma 3, and conditions (D) and (E)hold because b\ = s- (i ^ r) and thus bj = b'}; + 1 = s^; + 1 = Sj ^ 3, br+x ^ 3. Notebr+\ ^ ... ^ bn-\ since b'r+l ^ ... ^ b'n_x holds by Claim 1.

The sequences (a't) and (bi) thus form a desired decomposition, and Claim 2 is proved.•

Claim 3. Assume inductively that the conclusion of the decomposition lemma holds for all

sequences (s-) with YM=I 5! ^ 7> 7 ^ i 111=1 m< + *• ^ ^ n ^ holds for a sequence (s,-) w/t/z

£•=! s." = 7 + 1-Proof of Claim 3. Let (s,) be a sequence satisfying the hypotheses of the decomposition

lemma, and sr+i ^ . . . ^ sn_i, Xw=i s,- = y + 1. If r = 1, then s\ = y + 1 ^ ^ +2^ 3. For

r ^ 2 w e have s, 7 1 (i ^ r) by condition (E). Suppose st e {0,3} for all i ^ r. Since s,- = 3

implies mt ^ 4 (1^ r) by (D), we conclude for t = \{i ^ r : 5/ = 3}|

which is impossible for t ^ 1.Hence, if r ^ 2, there exists some j ^ r with 57 = 2 or s;- ^ 4. With this 7 respectively

j = 1 if r = 1, we construct a new sequence (sj) as follows.If 5r+i ^ sw (case A), then we set

By Lemma 4, (s-) is (mi,...,m^)-admissible. If sr+i > sn (case B), then sr+i,...,sn_i G{5 + l,sw + 2}, and we consider instead

(s'i) = (si,...,Sj - 1,...,sr,sr+i,...,sn + 2).

The admissibility is easily shown, and in both cases condition (D) is satisfied sincethe parities are unchanged, and (E) is satisfied by the choice of Sj. Note, finally, thatsf

n ^ max(sj : r + 1 / ^ n — 1) — 2 holds in either case. By induction, in either case thereexist sequences (a-)> (b\) satisfying the conclusions of the decomposition lemma.

Let t = r + l(case A) or t = n(case B), then a[ i {0,1,3} or b\ $ {0,1,3}. (24)

To see this, note that all a!i are even, since r ^ 1 and mn — 1 even (condition (B) ofL-sequences). However, if a't = 0, then b\ = s[ = st + 2. Since 0 ^ st =fc 1 (condition (E)),b\ i {0,1,3} follows.

Case 1. a't $. {0,1,3}. Consider the sequence (a,-) = (a\,..., a'j + 2, . . . , a'r,..., a't — 2, . . . , a'n).Rearranging the aj's (f > r) in increasing fashion, we use Lemma 3, and conclude that (at)is (mi,...,mn_i,mw — l)-admissible. Since the parities of at and a\ are the same, (a,-) is anL-sequence, and (a/), (&•) give a desired decomposition of (s ).

Case 2. aj = 0, i.e. fcrr ^ {0,1,3}. In this case t = r + 1 (Case A), and we use the new

seqence (6/) = (b\,...,bfj + 2,...,b'r,b'r+l — 2,...,fcJ1_1) and conclude from Lemma 3 that (/?,)is an //-sequence for (mi,...,mn_i). Finally, note frr+i ^ ... ^ bn-\ since

(2, 3, 3, 5)

(2,3,3,4) (2,3,3,4)

(2, 3, 2, 4) (2, 3, -, 4) (2, 3, 2, 4) (2, 3, -, 4)/ \ / \ / \ / \

( 2 , 2 , 2 , 4 ) ( 2 , - 2 , 4 ) ( 2 , 2 , - 4 ) ( 2 , - , - , 4 ) ( 2 , 2 , 2 , 4 ) ( 2 , - 2 , 4 ) ( 2 , 2 , - 4 ) ( 2 , - , - , 4 )

Figure 2

by induction. The sequences (a-), (fc,-) thus form a desired decomposition of (s,), and theproof of the decomposition lemma is complete. •

6. Proof of the theorems

Proof of Theorem 1. Let (si,..., sn) be an L-sequence for (mi,..., mn). As before, mi,.. . , mr

are even, mr+i,...,mn odd, and w.l.o.g. sr+i ^ ... < sn. We proceed by induction on thenumber k = n — r of odd m,'s. Proposition 1 settles the case k = 0. Hence suppose k ^ 1.Assume first r ^ 1. Then sn = [^yj Yl^ll ™i by condition (B), and we can apply thedecomposition lemma (see Remark 5). The two L-graphs La = L(m\,...,mn-i,mn — 1) andLb = L(mi,...,mn_i) have fc — 1 odd lengths m;, and we have r ^ 1 in both cases. Hencethe derived sequences (at) resp. (fc;) are by induction the types of a maximum matching inLa resp. Lb, and can thus be put together to yield a desired matching of type (si,...,sn).

We can conveniently represent the inductive process for r ^ 1 by means of a decom-position tree. Consider as an example (mi,m2,m3,m4) = (2,3,3,5). The tree is depicted inFigure 2, where the (a,)-sequences are in the left subtree and the (fc,)-sequences in the rightsubtree.

Remark 6. The subsequences (mi,...,mn) have the same number of odd lengths on eachlevel. The leaves correspond to the subsequences with all m, even. Note also that thesubsets of missing lengths (-) at the leaves correspond precisely to all 2k subsets of indicesi with mi odd in the starting length sequence (mi,...,mn), k = n — r.

It remains to consider r = 0. In this case any maximum matching leaves one vertexuncovered. To use induction we have to extend the hypothesis.Claim. Let all mi be odd, and suppose (si,...,sn) is an L-sequence for (mi,...,mn). Then

there exists a maximum matching M of type (s,) such that the uncovered vertex u satisfies

) if Si = 5^1 (mod 2)[ otherwise.

Proof of the claim. We use induction on n. For n = 1, L(m\) is a path, s\ = tmf^L, and wechoose as uncovered vertex u = 0. Let n ^ 2, and assume w.l.o.g. s\ < ... < sn.

If sn = mn^L (mod 2), then (10) is satisfied, and we may apply the decomposition lemmato obtain the sequences (at) and (fc,-). Now we use induction on n, and infer the existenceof appropriate matchings of type (at) resp. (fc,-). Let La = L(mi,...,mn-i,mn — 1) with the(fli)-matching have vertex-set Va = {(vi,...,vn) : vn > 0}, and let Lb = L(mi,...,mn-i)with the (fc,)-matching have vertex-set Vb = {(vi,...,vn) : vn = 0}. The uncovered vertex

n-\

Figure 3

(u\,...,un) e V\y satisfies un = 0, and since st = bf (mod 2) for i ^ n — 1, (25) is satisfied.

Thus we may put the matchings together.

Suppose now sn ^ mn^- (mod 2) and consider the sequence (s-) = (si , . . . ,sw_i + l,sw — 1).

By Lemma 3 and sn ^ 1, (sj) is admissible and hence an L-sequence for (mi, . . . ,mw) with

s'w ^ 5 ^ — 2 (1 < f < n — 1). We can therefore apply the decomposition lemma. Let (a,),

(b;) be the sequences constructed from (s-) according to the decomposition lemma, with

Fa and V\, the vertex-sets of La and L& as before. Since La has an even length mn — 1, we

can decompose the matching M a of type (a,) in La eventually into hypercube matchings

as explained above. At least one of these hypercubes, say Q, has full dimension n (see

Remark 6). Suppose the induced type of Q is (c i , . . . , c n ) , where we may assume w.l.o.g.

that the vertex-set of Q is VQ = {(vu...,vn) : vt e {0,1} for i^n—1, and vn e {1,2}}.

Case 1. cn-\ ^ 2, i.e. there are Ma-edges in Q along dimension n—1. Let u = (wi,...,wn_i,0)

be the uncovered vertex in L&. Note that because of bt = s,- (mod 2), (25) is satisfied for

L[,. Let wr = ( M I , . . . , M W _ I , 1) be the neighbor of u in Fa, thus wr € F^. By edge-transitivity

we may choose an (n — l)-edge in Ma as u'w with w = (MI , . . . ,M W _2 , 1 — MW-I, 1). Replace

now the Ma-edge M'W by the edge u'u (see Figure 3). Since u'u is an n-edge, our desired

matching of type (s i , . . . , sw) satisfying (25) results, since the uncovered vertex w satisfies

Case 2. cn-\ = 0, and Cj ^ 2 for some j < n — 1, hence n ^ 3. Since bn_i =

i ^ n — 1), we infer

for w ^ 3. Hence (b'j) = (b\,...,bj + 2,bj+\,...,bn-\ — 2) is admissible and thus an L-

sequence for (mi , . . . ,mw_i) by Lemma 3. By induction there exists a matching My of type

(b\) with uncovered vertex u = ( M I , . . . , M W _ I , 0 ) . N O W change the type of the hypercube

matching in Q to (c\) = (ci,. . . ,c7- — 2,c /+i , . . . ,c n_i + 2,cn) and make an exchange of an

(n — l)-edge and an n-edge as in case 1. The resulting matching of L(mi , . . . , mn) has type

(si , . . . , sw) and satisfies (25).

Case 3. All cj = 0 (7 ^ n - 1 ) , i.e. (c i , . . . , c n ) = (0 , . . . ,0 ,2w"1 ) . In this case there exis ts ; < n

with aj ^ 2, since otherwise an = sn = ^ ^ Il^Ti mu m contradiction to the assumption

sn ^ ^ ^ - (mod 2). We conclude that there are Mfl-edges in La along dimension j . Let

Qf be a hypercube with matching edges along j . Note that Q' contains dimension n

since mn — 1 is even (see Figure 2). Suppose (#i,. . .<?/,. . . ,gw) is the type of the matching

in Q\ qj ^ 2, then replace it by (<?•) = (qu...,qj - 2,...,qn + 2). Now change (a) to(cj) = (ci,.. . , Cjr + 2, . . . , cn - 2) = (0,... , 2, . . . , 2""1 - 2). If j = n - 1, then we are in case 1,otherwise in case 2. Proceeding as before, our desired matching results, and the theoremis proved. •

Proof of Theorem 2. If all m, are odd, then the notions of L-sequence and //-sequenceare equivalent (see Remark 3), and we may apply Theorem 1. Hence we assume r ^ 1.As before, the proof proceeds by induction on the number k = n — r of odd m,'s. Theinduction start is provided by Proposition 2. Let k ^ 1, and (si,...,sn) be an //-sequencefor (mi,..., mn) with sr+\ ^ ... < sn. If sn = mn^-Yl"ll ^ (mod 2), i.e., sn even, then (10)is satisfied, and the theorem follows inductively by an application of the decompositionlemma.

So it remains to prove the induction step for sn odd, where sn ^ 3 by condition (E).

There exists j ^ n — 1 with Sj ^ 3, m7 ^ 3. (26)

This is certainly true for j e {r + l , . . . ,n — 1} by conditions (E) and (D) whenever Sjis odd, since r ^ 1. Hence suppose r — n — \ or r < n — 1 and all s, (r + 1 ^ * ^ n — 1)are even. If r ^ 2, then X^=i 5* ~ \ I lLi m« = ^ (mod 2). Hence there is another oddsj U ^ r) beside sn. Conditions (D) and (E) now imply m7 ^ 4, 5y ^ 3. If r = 1 and^ = 0 (mod 2), i.e. mi ^ 4, then si is odd with s{ ^ f- ^ 2, i.e. si ^ 3. Finally, if r = 1and ^ = 1 (mod 2) then s{ is even, mi ^ 6 (by (D)), and thus s{ ^ f- ^ 3.

Now choose 7 as in (26) where we take j = n — 1 whenever sn_i ^ 3. The sequence

is (mi,...mn)-admissible by Lemma 3, and thus //-sequence since (D) and (E) are triviallysatisfied. Since s'n ^ sj — 2 (r + 1 ^ i ^ n — 1) we may again apply the decompositionlemma obtaining sequences («/) resp. (bi).

Case 1. m7 odd, i.e. j = n — 1. First we note that we may assume bn-\ > 0. Supposebn_i = 0. If r ^ 2, then /v = 2 or fc/ ^ 4 for some / ^ r, since otherwise bi e {0,3} for i ^ r(condition (E)) which implies for t = \{i ^ r : bf = 3} by condition (D), ^ - = 1 bt = 3f ^ ^.Hence f = 1 and we obtain J^ = 1 b, = 3 ^ ^r = 4 by r ^ 2, a contradiction. If r = 1, thenb\ = /?i + ftn-i ^ mi^"~1 ^ 3, ^nd we set / = 1. Consider the new sequence

(fcj) = ( & i , . . . , & , - 2 , . . . A _ i + 2 ) .

We have

Since bn-\ = 0 = min(fct : i > r), we may apply Lemma 4, infering that (ftj) is an//-sequence for (mi,.. . ,mn-\). Furthermore,

(a-) = (ai, . . . , a{ + 2, . . . , an-i - 2, an)

is L-sequence for (mi,...,mn — 1) because of an-\ = sfn_{ = sn-\ + 1 ^ 4 and Lemma 3.

Hence (aft), (b[) form a decomposition of (s-) with bf

n_{ > 0.

Thus we may assume that (a,), (hi) is a decomposition of (s-) with bn-\ > 0. Nextwe show that we may also assume an-\ > 0. Suppose an-\ = 0. Then bn-\ = s'n_x ^ 4.Furthermore, there must be an index «f < n — 1 with ^ > 0, since otherwise we wouldobtain an = sf

n = sn — 1 = ^f^ n ^ / m " an(^ hence

n—1 -j n .j «—1

^ 5 , = - Y[mr- sn = -Y[mi - 1,i=\ i = l i = l

a contradiction to the admissibility of (s/). Now consider the sequences

(flj) = (fli,..., ^ - 2,...,an-i + 2, an).

The sequence (bj) is an //-sequence for (m\,...,mn-\) by Lemma 3 and bn-\ ^ 4, whereas(aj) is an L-sequence for (mi,... ,mn — 1) by Lemma 4 (if / ^ r) resp. Lemma 3 (if £ > r),since a^ > 0 even, i.e. a/ ^ 2. Furthermore, b'n_x > 0,af

n_{ > 0.

Hence we may assume that (a,), (bt) is a decomposition of (sj) with an-\9bn-\ > 0. Nowwe proceed in a similar fashion as in the proof of Theorem 1. Let Ha = //(mi,. . . , mn — 1)be the Hamming graph with lattice-matching Ma of type (a,), and Ht, = //(mi,...,mn_i)the Hamming graph with Hamming matching Mt of type (fc,-). As explained above,Mfl consists of even length lattice matchings which contain all even length dimensionsl , . . . , r ,n and a subset of the other dimensions {r + l , . . . ,n — 1}. Each of these subsetsappears exactly once. Hence in this decomposition there are lattices L(mi,...,mr,mn —1) and L(mi,...,mr,mn_i,mn — 1), and they lie w.l.o.g. next to each other in directionn — 1. In particular, there is an (r + l)-dimensional hypercube Q\ with dimensions1,2,...,r,n, and an (r + 2)-dimensional hypercube Qi with dimensions l, . . . ,r ,n— l,nwhich are neighbors in direction n — 1. For the vertex-sets V(Q\), V(Q2) we may assumew.l.o.g.

V(Q\) = {(t>i,...,i>w) :*;i,.. . ,tv,^ G {0,l},z;r+i = . . . = t;w_i = 0 }

V(Qi) = {(vu...,vn) :vu.--,vr,vn e {0,l},t;n_i e {1,2},^-= 0 otherwise}.

Let (ci,...,cr,cw) be the type of the matching in gi and (qi9...,qr,qn-i,qn) the type ofthe matching in Qi. If cn = 0, then we choose k ^ r with Q > 0, and another hypercubeQi which contains Ma-edges in direction n (recall an = s'n = sn — 1 ^ 2). Now delete two/c-matching edges in Qi and add two n-matching edges, and do the opposite in Q3. Ifqn-\ = 0, then we may apply an analogous exchange since an-\ > 0 by assumption.

Let ux be an (n — l)-edge connecting Q\ and Q2 and choose the matchings of type(ct) resp. (qt) so that u is incident to an n-matching edge uv in Qu and x is incident toan (n — l)-matching edge xxv in Q2 (possible by edge-transitivity). Putting the Hamminggraph Ha and Hb together we may choose w.l.o.g. an (n— l)-matching edge yz in M^ suchthat y is adjacent to u and v in / /(mi, . . . , mn) and z is adjacent to x, both along dimensionn. We thus arrive at the situation shown in Figure 4.

H-\

Figure 4

Replacing the matching edges uv, xw, yz by the non-matching edges uw, vy, xz, wedecrease the (n — l)-edges by 1, increase the n-edges by 1, and thus arrive at a desiredmatching of type (s,).

Case 2. m7 even, i.e. j < r. By an analogous argument as in case 1 we may assume aj > 0,bj > 0. Consider again the Hamming graphs Ha and Hb with matchings Ma and Mh. Thelattice graph L(mi,...,mn — 1) contains in the usual way a sublattice L(mi,...,mr,mn — 1)and therefore, because of m; ^ 4, two (r -f l)-dimensional hypercubes Qu Q2 which areneighbors along dimension j . Let (cu...,cr,cn),(qi,...,qr,qn) be the types of Q\ resp. Qi.As in case 1 we may assume cn > 0, qj > 0 because of s'n > 0, aj > 0. By the assumptionbj > 0 we can exchange matching edges and non-matching edges as in case 1 with j inplace of n — 1. We thus obtain a desired matching in H(mi,...,mn) of type (s,-), and theproof is complete. •

References

[1] Felzenbaum, A., Holzman, R. and Kleitman, D. J. (1993) Packing lines in a hypercube, DiscreteMath. 117 107-112.

[2] Lovasz, L. (1977) Combinatorial Problems and Exercises. North-Holland.[3] Montroll, E. Lattice Statistics. In: Applied Combinatorial Mathematics (Beckenbach, E. F. ed.),

Wiley.

Reconstructing a Graph from its Neighborhood Lists

MARTIN AIGNERf and EBERHARD TRIESCH*+ Freie Universitat Berlin, Fachbereich Mathematik, WE 2, Arnimallee 3, 1000 Berlin 33, Germany

•^Forschungsinsitut fur Diskrete Mathematik, NassestraBe 2, 5300 Bonn 1, Germany

Associate to a finite labeled graph G(V, E) its multiset of neighborhoods ^(G) = {v G V}. We discuss the question of when a list Ar is realizable by a graph, and to what

extent G is determined by ^V{G). The main results are: the decision problem Jf = A (G)is NP-complete; for bipartite graphs the decision problem is polynomially equivalent toGraph Isomorphism; forests G are determined up to isomorphism by Ar{G); and if G isconnected bipartite and JV(H) = ^V(G), then H is completely described.

1. Introduction

Consider a finite labeled undirected simple graph G(V,E). Let us associate to G a finitelist P(G) of parameters. P(G) may consist of the chromatic number, of the degrees, of thelist of cliques, of one-vertex-deleted subgraphs, or whatever we like. To any given list ofparameters there arise two natural problems:

(1) Realizability. Given P, when is P graphic, i.e. when does there exist a graph G withP = P(G)?

(2) Uniqueness. If P (G) = P (H) for two graphs G and H on the same vertex-set V, whatdoes this tell us about G and / / ? In particular, when is G = H or G = HI

The problem when P consists of the degree sequence has been well studied. The famousErdos-Gallai Theorem [1] gives a (polynomial) characterization of graphic sequences,thereby answering question (1). As far as uniqueness is concerned, more or less anythingcan happen. In particular, G and H need not be in the least isomorphic. For a recentresult in this direction, see [2].

We treat a closely related question, raised by V. Sos at the conference [3]. Associateto every u € V its neighborhood N(u) = {v e V : uv G £}, and denote by JV(G) ={N(u) : u e V} the neighborhood list of G. (The mathematical object is, strictly speaking,

52 M. Aigner and E. Triesch

a multiset: the sets N(u) are counted with multiplicities and their order does not matter.)In fact, Sos considered the stars S(u) = N(u) U {w}, but the two problems are easily seento be essentially the same. In fact, a list Jf is the neighborhood list of G if and only if{V \ N : N G J^} is the system of stars of G. At the same conference, L. Babai remarkedthat the readability problem for Jf is at least as hard as the graph isomorphism problem.We will show in Section 2 that it is, in fact, NP-complete. In Section 3 we address theuniqueness question. It may well happen that non-isomorphic graphs G and H on thesame labeled vertex-set have the same lists JV9 but for some classes of graphs we canassert uniqueness up to isomorphism, e.g. for complete /c-partite graphs and for forests.Furthermore, the question whether for a given connected bipartite graph G there exists anon-isomorphic graph H with the same list JV is also shown to be NP-complete.

2. Readability of neighborhood lists

Let JV(G) be the neighborhood list of a graph G with vertex-set V = {l , . . . ,n}. HenceJV(G) consists of n subsets of V (with possible multiplicities). The following characteriza-tion of graphic lists is immediate.

Lemma 1. Let JV be a list of n subsets o /{ l , . . . ,n} . Jf is graphic if and only if there isa numbering N\,...,Nn such that for all i and j

(1) ieNj<=^> jeNit

(2) i^Ni.

If we write Jf as an n x n incidence matrix F with the rows corresponding to the sets,and the columns to the vertices, then Lemma 1 says the following: F is graphic iff thereexists an n x n permutation matrix R such that

(1) RF is symmetric,

(2) all diagonal elements of RV are zero.

We consider the decision problem NL (neighborhood lists) in this matrix form.

Input: An n x n matrix F with {0, l}-entries.

Question: Is F graphic?

Theorem 2. NL is NP-complete.

Proof. It is clear that NL is in the class NP. To show completeness, we provide atransformation from the problem ORDER 2 FIXED-POINT-FREE AUTOMORPHISM(O2FPFA), which was shown to be NP-complete by A. Lubiw in [4]. This latter problemcan be stated as follows:

Input: A graph G.

Question: Is there an involution in the group Aut (G) without fixed points?

Lubiw gives a transformation from 3-SAT. Her construction provides graphs in whichevery vertex has degree at most \V\ — 2, hence we can restrict the input of O2FPFA tosuch graphs without losing NP-completeness.

Reconstructing a Graph from its Neighborhood Lists 53

Now suppose B is the vertex-edge incidence matrix of such a graph, B e Matnxm, anddenote by Permn the set of n x n permutation matrices of order n. Note that

Aut (G) = {P e Perm,, : 3Q e Permm with PBQ = £},

and further

(lr) P involution <=> PT = P9

(2f) P fixed-point free <=> all diagonal elements are zero.

We denote the n x n identity matrix by /„, and the r x s matrix with all entries equalto t by (t)rxs.

Now consider the (w + m + 1) x (n + m + 1) matrix

r -

In

BT

( l ) l x »

( )fflXffl

(O)lxm

(Dnxl

(0)mxl

(O)lxl

Suppose R e Permn+m+i satisfies (1) and (2). Since no row of B contains more thann — 3 ones, the last row of F is the only row with n ones. Hence, if RF = (RT)T holds, Rmust fix the last row. But then R permutes the rows 1,...,n, resp. n + l , . . . , n + m, amongthemselves, and we can write

R =

with P e Permn and Q e Permm.The condition RT = (RT)T is thus equivalent to the equations

P=PT, PB = BQT,

i.e. P is an involution with PBQ = £. Note that Q~l = QT holds for a permutationmatrix. Furthermore, RV has zero-diagonal if and only if P has no fixed points. Hencethe proof is complete. •

There is a natural variant of NL for bipartite graphs. Consider a bipartite graph Gon defining vertex-sets U and V, \U\ = m, \V\ = n. The neighborhood list (multiset)

p

0

0

Q

0

0

1

= {N(u) :ueU} consists of m subsets of V, and the list JVV(G) = {N(v) : v e V}of n subsets of U.

The BIPARTITE NEIGHBORHOOD LISTS (BNL) problem reads as follows:

Input: Lists Jfjj of m subsets of V and Jfv of n subsets of U.

Question: When is the pair (J^u^v) graphic?

BNL might be easier than NL, as the following result suggests.

Theorem 3. BNL is polynomially equivalent to GRAPH ISOMORPHISM.

The GRAPH ISOMORPHISM problem is perhaps the most famous decision problemin the class NP, which is neither known to be polynomially solvable nor to be NP-complete.A recent account of the state of the problem is given in [5].

For the proof of Theorem 3, it is convenient to use the following:

Lemma 4. GRAPH ISOMORPHISM (GI) and HYPERGRAPH ISOMORPHISM (HI)are polynomially equivalent.

Proof. Since each graph is a hypergraph, it suffices to show that HI oc GI. So choosetwo hypergraphs J-f = (V,E) and Jf7' = (V',Ef) and assume, without loss of generality,that V = V',\V\ = n, and E = {eu...,em}, E' = {e\,...,e'm}. Choose sets of new

elements X = {x\,...,xm},Y = {y,yi,...,yn+\}, and construct the graph G = G(Jf) (resp.

G = Gffi)) on V U X U Y as follows: connect xt to all points in et (resp. e[) and to y(1 <i < m); and y to all y-} (1 < j <n+ 1). No other edges are added.

We claim that G and G are isomorphic if and only if the corresponding hypergraphsare isomorphic. The 'if-direction being clear, suppose that G and G are isomorphic withisomorphism (j). By looking at the degrees, we see that 0 fixes y, and that each orbit of (j)is contained in one of the sets V,X, {y} and Y \ {y}. From this it follows readily that Jfand Jf7' are isomorphic. •

Proof of Theorem 3. In view of the Lemma, it suffices to prove the polynomial equivalenceof BNL and HI. By identifying neighborhood systems as well as hypergraphs with theirincidence matrices, we see that an input to BNL consists of an (n x m)-matrix T and an(m x n)-matrix T. The question is do there exist P £ Permn and Q G Permm such that(PT)T = QT1 But (PT)T = QT if and only if TT = QYP if and only if the hypergraphswith incidence matrices T and TT are isomorphic. The result follows. •

Another interesting decision problem related to neighborhood lists is MATRIX SYM-METRY (MS):

Input: An n x n matrix A with {0,1 gentries.

Question: Does there exist P e Permn such that PA = (PA)T holds?

It is easy to see that MS is at least as hard as GRAPH ISOMORPHISM by consideringa matrix of the type

0

Bi

1

0

0

1

0

where B\ and B2 are the incidence matrices of graphs (each with maximum degree< \V\ — 3). MS is equivalent to the variant when arbitrary entries are allowed, but we donot know whether MS is NP-complete.

3. Uniqueness of neighborhood lists

Consider two graphs G and H on the same labeled vertex-set V, \V\ —n. We call G andH hypomorphic, denoted by G « H, if Jf{G) = JV(H). Thus G « H iff there exists abijection cp : K —• V with NG(M) = NH((pu) for all w G V. If wr is an edge of G, then forbrevity we write uv e G.

Proposition 5.

(i) If G & H by means of q>, then

(A) u,cpu # G

(B) uv G G <=> cpu,cp~xv G G.

If(p:V —>V satisfies (A) and (B), then G^H, where E(H) = {cpu.v : uv e G}

(/?M, 1; G H. Since = No(u),

Proof.

(i) By the definition of cp, we have uv e Gwe infer w, cpu ^ G. To prove (B), we have

uv G G <=> c w, 1; G H <=> (pw G iV/fW = NG((p~lv) <=> cpu,cp~xv G G.

(M) for all u e V, and show that(ii) We set up the list {N(cpu) : u e V} with JV(<pw) =it defines a graph //, i.e.

cpu $ N(cpu)

cpu G N(cpv) <=> (p G N(cpu).

We have (pw ^ N(cpu), since N(<pu) = NG(M) and cpw ^ NG(w) by (A). To prove thesecond claim, we infer

cpu G N(cpv) cpu G

<pt; G NG{u)

(B)cpu, v G G <=> u,cpv

cpv G N(cpu).

This completes the proof. •

Proposition 5 tells us that the possible graphs H, hypomorphic to G, are completelydescribed by mappings cp : V —• V satisfying conditions (A) and (B) within G. Let uscall these mappings cp admissible. We will write H^ for the hypomorphic graph inducedby cp, and we note

(*) uv G G <=> cpu, v G H(p <=> u, cpv G H(p.

Lemma 6. Let cp be an admissible mapping of the graph G, and O\,...,Om its orbits onV. Then for all ij

(i) u, v G Ot = > uv ^ G,(ii) Let \Ot\ = k, \Oj\ = / , u G Ot, v e Oj, uv e G. Then the subgraph of G induced

by Oi U Oj is a disjoint sum of complete bipartite graphs Kr,s, where r = k/gcd(k,£),

Proof.(i) Let v = cpju. If j = 0 (mod 2), then by repeated application of (B) it follows that

M, cpju G G <=> cpj/2u, cpi/2u G G, which is impossible. If j = 1 (mod 2), then by thesame argument u,cpJu G G <=^ cp^'^û.cp^^û G G <^> cp^-^û^cpicp^'^û) G G,contradicting (A).

(ii) Easily seen by iterating condition (B). •

Corollary 7. Let cp be an admissible mapping of G. If all the induced (bipartite) subgraphsG(O(, Oj) on the orbits 0,, Oj are either empty or complete bipartite, then G = H^.

Corollary 8. / / G is a complete k-partite graph, then H « G implies H = G.

Before going on, let us look at some examples. The smallest graph G that admits ahypomorphic graph H =fc G is the 5-path. In the following figure, G is shown with fulllines and H with dashed lines. The corresponding mapping is cp = Q \ \ \ 5

{). Of course,

The smallest graph G that admits a non-isomorphic hypomorphic graph H is the 6-cycle(1,2,3,4,5,6), with H being the sum of two disjoint triangles (1,3,5) + (2,4,6). The orbitsof G are {1,4}, {2,5}, {3,6}.

We can immediately generalize this example. Let Ho be a graph and H = Ho + HQ bethe disjoint union of two copies of Ho. Denote by v! the vertex in HQ corresponding tou G Ho. The involution cp : u <—• 1/ is then admissible, and hence H « G^. If #o = ^4,the reader may verify that the corresponding hypomorphic graph G^ is the 3-cube Q3,where again Q3 ^ X4 4- K4. Note that the graphs G^ arising in this way are bipartite.

Proposition 9. Let cp be an admissible mapping of G with orbits (9 = {O\,...,Om}. Denote

by Gc> the graph with vertex-set (9 and 0(0j G G^ iff there exist u € Oif v G Oj with uv G G.

If Go is bipartite, G = H<p.

\p(u) =

Proof. Let R and S be the color classes of G^, and define the bijection \p : V —• V by

u if ueOte R

cpu if u G 07 G <S.

Consider MI; G G. By Lemma 6(i), M and v are in different orbits O,-,07, hence we maysuppose without loss of generality that 0,- G #, 0 ; G S, i.e. tpw = u, xpv = cpv. By (*), thisyields

uv G G <=> w, (jpu G //^ <=> tpM, tpi; G Z/^,

implying that t/; is an isomorphism from G onto H^. •

Corollary 10. 7/G is a forest, G « / / imp/to G = //.

Proof. Just observe that any cycle in G gives rise to a cycle in G, by iterating condition(B) in Proposition 5. •

Now let G be an arbitrary connected bipartite graph. In our examples above, we havealready seen that G may have non-isomorphic hypomorphs of the form G « H + H. Wenow show that these are essentially all the possibilities. First we prove an easy lemma.

Lemma 11. Let cp be an admissible mapping of G. If u and v are connected in G by a trailof even length, the same holds in H^.

Proof. The trail

U = Xo,X\,X2,...,X2t = V

in G gives rise by (*) to the trail

in Hy, D

Theorem 12. Let G be a connected bipartite graph on the defining vertex-sets Vo, VQ.If H = Hep is a non-isomorphic hypomorph of G, then H = Ho + HQ, where Ho andHQ are connected graphs on Vo and VQ, respectively. Furthermore, (p restricted to Vo is abisection between Vo and VQ, and, by identifying u = cpu (u G VQ), HO is hypomorphic to HQ.Conversely, if Ho and HQ are connected hypomorphic graphs, H = Ho + HQ is hypomorphicto a connected bipartite graph.

Proof. Let cp be an admissible mapping of G. Since any two vertices in Vo (resp. VQ)are connected in G by a trail of even length, we infer from the Lemma that H cpu, v G H<p = i/0 + HQ, we infer that cpu G VQ foru G VQ, and vice versa. Hence cp restricted to Ko is a bijection between VQ and VQ. Let usidentify u = cpu (u G VQ)9 i.e. we regard both Ho and HQ as graphs on Vo. We claim thatxp = (p~2 is an admissible mapping of Ho giving rise to HQ (on Vo). Indeed, by (*) we have

uv G Ho <==> u, (p~lv G G <=> cpu, cp~lv G //Q (on VQ),

which by the identification u = cpu means u, <p~2v G HQ (on Fo), i.e. uv G Ho <=> w, i/w G HQ(on Fo). Hence \p is an admissible mapping for Ho generating HQ. The proof of the conversestatement follows the same lines.

Suppose now that H = H^ is connected with H ^ G. There must be an edge uv e H^with u € Vo, v e VQ and MI; ^ G, since otherwise H^ = G. By (*), this means cp~lu,v G G,u,cp~lv G G, hence (p-1M G Fo, cp"1^ G KQ and cp~lu 7 w, cp"1!; ^ v. Consider the orbits of cp.With w = (/)-1M, <pw = u € Vo and x = cp~lv, cpx = v G KQ, we know that there is an orbitOj = {W,M,...} containing two consecutive K0-vertices, and similarly an orbit Oi (possiblyequal tq Q\) containing two consecutive VQ-vertices. If all orbits are monochromatic, i.e.they consist pf Fo-vertices or KQ-vertices only, then G<c is bipartite, and we are done, byProposition 9. Hence, assume there is an orbit Oj containing both Vo- and ^-vertices.Since G connected implies G<n is connected, we infer by Proposition 5(B) that any orbitcontains vertices of both color classes. Consider again the orbit O\ = {w,u = cpw,...},w,u G Vo. We now derive a contradiction by showing that w and u are not connected inG.

Let w = xi,X2,X3,...,X2r+i = u be a path in G. Then xi,X3,...,X2r+i G Ko andX2,X4,...,X2r £ FQ. From wx2 G G and w = cpw, we conclude u,cp~xX2 G G, cp~{X2,(px3 GG,...,cp~lX2t,(pu G G, hence (/)~1X2 G KQ,(/)X3 G V0,...,(pu G KO. Since the orbit Oialso contains ^-vertices, we infer cpu ^ W,M. NOW we start the next round. Becauseof u,cp~lx2 G G, we have cpu,cp~2X2 G G, (p~2X2,(p2xi G G,...,cp~2X2t,(p

2u G G with(p~2x2 G KQ, (/>2X3 G Ko,...,<p2M G Fo. Hence (/)2M ^ w,u,cpu. Continuing in this way, wecome to the conclusion that the orbit Oi is infinite, which is absurd. •


Our final theorem is again an NP-completeness result.

Theorem 13. The decision problem (NIH) whether a connected bipartite graph has a non-isomorphic hypomorph is NP-complete.

For the proof, we need some preparations. Suppose F is an n x m matrix. We call Findecomposable if there is no partition AuB of {l,...,m} into nonempty subsets such thatthe column indices of the nonzero entries of each row of F are either all contained in Aor all in B. We have the following supplement to Theorem 2:

Lemma 14. The decision problem NL is NP-complete even if the input is restricted toindecomposable {0,1}-matrices.

Proof. Consider again the matrix F constructed in the proof of Theorem 2. Add as the(n + m -f 2)nd row and column, the vectors e and eT to F, where e = (1 , . . . , LO), and call

n+m+l

the new matrix F'. It is easy to see that V is indecomposable, and that the transformationfrom O2FPFA works with V instead of F as well. •

Proof of Theorem 13. We give a transformation from NL with input restricted to inde-composable matrices. Note that a {0, l}-matrix F is indecomposable if and only if thebipartite graph with adjacency matrix

is connected.

Claim: An n x n matrix F with {0, l}-entries is graphic via R e Permn if and only if FT isgraphic via RT.

To see this, assume that F = (ylV/) satisfies (1) and (2). RTR = F T , hence TR = RTFT,and thus (RTTT)T = TR = RTTT. If R is the permutation matrix (Sff^)j) with permutationa, the element in position (i,i) in RT and RTTT is ya^ and y^-i^, respectively. HenceF T satisfies (1) and (2), and the claim follows in view of F r r = F.

Now suppose that F is an indecomposable input for NL. We claim that the connectedbipartite graph G with adjacency matrix A(T) has a nonisomorphic hypomorph if andonly if F is graphic. Suppose first that F is graphic. Choose R e Permn satisfying (1) and(2). Then the graph with adjacency matrix

0 RT \ . _ / RTTT 0R 0 )Air)={ 0 RT

is a non-isomorphic hypomorph of G (cf the claim above). If, on the other hand, theconnected, bipartite graph G has a non-isomorphic hypomorph //, then by Theorem 12

there is a permutation matrix

such that

is the adjacency matrix of H. Hence RT satisfies (1) and (2), and is thus graphic. •

4. Concluding remarks

Finally, we should like to indicate some possible directions of further research.First, it would be interesting to know more about the complexity status of the MS

problem (see Section 2). Is it NP-complete? Is it ISOMORPHISM-complete?Our second suggestion is to study the structural differences between hypomorphic

graphs G « H. Since they have the same neighborhood lists, their degree sequences areidentical. What about other parameters? Lemma 6 implies that the number of componentsof H is at most twice the number of components of G. On the other hand, Theorem12 shows that the coloring numbers can be arbitrarily far apart, and our final exampledemonstrates that planarity is also not preserved:

Let Ho be the non-planar graph K33 with two edges subdivided, and H = //0 + HQ asbefore. Then the bipartite graph G arising from the usual involution v <—• v' is planar.

f

References

[1] Erdos, P. and Gallai, T. (1960) Graphs with prescribed degrees of vertices (Hungarian). Math.Lapok 11 264-274.

[2] Erdos, P., Jacobson, M.S. and Lehel, J. (1991) Graphs realizing the same degree sequencesand their respective clique numbers. In: Alavi et al. (eds.) Graph Theory, Combinatorics andApplications, John Wiley, 439-449.


[3] Hajnal, A. and Sos, V. (1978) Combinatorics. Coll. Math. Soc. J. Bolyai 18, North-Holland.[4] Lubiw, A. (1981) Some NP-complete problems similar to graph isomorphism. SIAM J. Com-

puting 10 11-21.[5] Kobler, J., Schoning, U. and Toran, J. (1993) The graph isomorphism problem. Progress in

Theoretical Computer Science, Birkhauser.

Threshold Functions for if-factors

NOGA ALON1 and RAPHAEL YUSTER

Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences,

Tel Aviv University, Tel Aviv, Israel

Let H be a graph on h vertices, and G be a graph on n vertices. An H-factor of G isa spanning subgraph of G consisting of n/h vertex disjoint copies of H. The fractionalarboricity of H is a(H) = max{ ,yV{}, where the maximum is taken over all subgraphs(V',Ef) of H with \V'\ > 1. Let S(H) denote the minimum degree of a vertex of H. It isshown that if S(H) < a(H), then n " 1 / " ^ is a sharp threshold function for the property thatthe random graph G(n,p) contains an //-factor. That is, there are two positive constantsc and C so that for p(n) = cn~l/a{H\ almost surely G{n,p(n)) does not have an //-factor,whereas for p{n) = Cn~l^a{H\ almost surely G{n,p{n)) contains an //-factor (provided hdivides n). A special case of this answers a problem of Erdos.

1. Introduction

All graphs considered here are finite, undirected and simple. If G is a graph of order nand H is a graph of order /z, we say that G has an H-factor if it contains n/h vertexdisjoint copies of H. Thus, for example, a X2-factor is simply a perfect matching.

Let G = G(n,p) denote, as usual, the random graph with n vertices and edge probabilityp. In the extensive study of the properties of random graphs, (see [5] for a comprehensivesurvey), many researchers have observed that there are sharp threshold functions forvarious natural graph properties. For a graph property A and for a function p = p(n), wesay that G(n,p) satisfies A almost surely if the probability that G(n,p(n)) satisfies A tendsto 1 as n tends to infinity. We say that a function f(n) is a sharp threshold function for theproperty A if there are two positive constants c and C so that G(n,cf(n)) almost surelydoes not satisfy A and G(w, Cf(n)) satisfies A almost surely.

Let H be a fixed graph with h vertices. Our concern will be to find the thresholdfunction for the property that G(n,p) contains an //-factor, (assuming, of course, that hdivides n). The case H = K2 has been established by Erdos and Renyi in [7]. They showed

f Research supported in part by a United States Israeli BSF grant

64 N. Alon and R. Yuster

that log(n)/n is a sharp threshold function in this case, and there are many subsequentpapers by various authors that supply more detailed information on this problem. In thegeneral case, however, it is much harder to determine the threshold function. Even for thecase H = K3 the threshold is not known (cf [3, Appendix B]). In [3, page 243], P. Erdosraised the question of determining the threshold function when H = He is the graph onthe 6 vertices <3i,a2, #3,^1,^2,^3 whose 6 edges are a\b\â2bâ^bi and a\a2,aiai,a\a^. Itturns out that in this case n~2/3 is a sharp threshold function for the existence of an//-factor. In fact, the graph //6 is just an element of a large family of graphs H for whichwe can determine a sharp threshold function for the existence of an //-factor. In order todefine this family we need the following definition.

For a simple undirected graph H that contains edges, define the fractional arboricity ofH as

a(H) = max\V'\- 1

where the maximum is taken over all subgraphs (V',Ef) of// with \V'\ > 1. Observe thatby the well-known theorem of Nash-Williams [13], [#(//)] is just the arboricity of//, i.e.,the minimum number of forests whose union covers all edges of //. Denote by S(H) theminimum degree of a vertex of //, and let 3F be the family of all graphs H for whicha(H) > S(H) (or, equivalently, the family of all graphs with arboricity bigger than theminimum degree). Our main result is the following

Theorem 1.1. Let H be a fixed graph in J*\ Then the following two statements hold:

1 There exists a positive constant c such that if p = cn~lâ^H\ then almost surely G(n,p)does not contain an H-factor.

2 There exists a positive constant C such that if p = C n ~ ] ^ H \ then almost surely G(n,p)contains an H-factor, assuming h divides n.

Thus Theorem 1.1 asserts that for every H e J^, n~lâ^ is a sharp threshold functionfor the property that G contains an //-factor. In particular, the theorem shows that n~2/3

is a sharp threshold function in the special case H = H6 mentioned above.Our method yields the following extension of Theorem 1.1 as well.

Theorem 1.2. Define the set $ recursively as follows:

1 ^ c ^.2 If C\,Ci are two of the connected components of some H' e *& and H is obtained from

H' by adding to it less than a(H') edges between C\ and Ci, then H €.<&.

If H £$, then n~xâ^ is a sharp threshold function for the existence of an H-factor.

The proofs are presented in the next section. They rely on the Janson inequalities (cf[6] and [8]), and on a method used by Alon and Furedi in [2].

The last section contains some concluding remarks and open problems.

Threshold Functions for H-factors 65

2. The proofs

Proof of Theorem 1.1. We begin by establishing the first statement in Theorem 1.1, whichis not difficult and, in fact, holds even when H is not a member of 3F.

Lemma 2.1. Let H be any fixed graph that contains edges, and let a = a(H). There existsa positive constant c = c(H) such that almost surely G(n,p) does not contain an H-factorfor p = cn~xla.

Proof. Let H' = (V'9Ef) be any subgraph of H for which \E'\/(\V'\ - 1) = a. Denote

\E'\ = e' and \V'\ = vf. Let {At : i e 1} denote the set of all distinct labeled copies of H'in the complete labeled graph on n vertices. Let Bt be the event that At a G(n, /?), and letXt be the indicator random variable for Bt. Let X = ^-G / Xt be the number of distinctcopies of Hf in G(n,p). It suffices to show that almost surely X < n/h. The expectation ofX clearly satisfies

E[X]<

and yet clearly E[X] = Q,(n) —• oo. Choosing an appropriate constant c, we obtainE[X] < n/2h. We next show that Kar[X| = o(E[X]2). This suffices, since by Chebyschev'sinequality it implies that almost always X < n/h. For two copies At and Aj, we say thati ~ j if they share at least one edge. Let A = £ . . Pr[£; ABj], the sum taken over orderedpairs. Since Var[X] < E[X] + A and E[X] - • oo, it remains to show that A = o(E[X]2).The intersection of any At and A-} is a subgraph H" = (V'\E") of H' (not necessarily aninduced subgraph). We can therefore partition A into partial sums A" corresponding tothe various possible H". It suffices to show that for each typical term A!' , A" = o(E[X]2).Denote |K"| = v" and \E"\ = e". Then

r — v"

since £^1 > ^ i . Hence A" = o(£[X]2). D

Note that by considering the minimal H' for which |£ ' | / ( l^ l — 1) = ^(^)» w e couldobtain A = o(E[X]) but our estimate suffices.

In order to prove the second part of Theorem 1.1, we need to state the Jansoninequalities in our setting. Let Q be a finite universal set, and let R be a random subsetof Q, where Pr[r e R] = pr: these events being mutually independent over r e Q. Let{A( : i G /} be subsets of Q, with / a finite index set. Let Bt be the event A[ a /?, and letXt be the indicator random variable for £,-, and X = ^ieI Xt, the number of At c R. Forf, j G / we write i ~ y if f j and X/ n A-} ^ 0. We define

where the sum is taken over ordered pairs. Note that if the Bi were all independent, wewould have A = 0. The Janson inequalities state that when the Bi are 'mostly' independent,then X is still close to a Poisson distribution with mean \i = E[X]. The first inequality(cf [6]) applies when A is small relative to \i,

Lemma 2.2. Let Bt, A, \i be as above, and assume that Pr[Bi] < e for all i. Then

( 1 A= 0] <exp - j u + - -

V 1 - 6 2When A/2 > Ml — e)> t n e bound in Lemma 2.2 is worthless. Even for A slightly less, it

is improved by the second Janson inequality (cf. [6])

Lemma 2.3. Under the assumptions of Lemma 2.2 and the further assumption that A >

The Janson inequalities play a crucial role in the proof of the next lemma.

Lemma 2.4. If H is any fixed graph with h— 1 vertices and fractional arboricity a = a(H),there exists a constant C = C(H) such that almost surely G(n,p) contains n/h vertex disjointcopies of H, where p = Cn~l/a.

Proof. It suffices to show that almost surely every subset of n/h vertices contains a copyof H. Fix such a subset of vertices, A a {1,2,...n}. We use a similar notation to that inthe proof of Lemma 2.1. That is, X denotes the number of labeled copies of H in A, andA" denotes the sum on all ordered pairs of copies of H whose intersection corresponds toa fixed subgraph H" of H. However, this time we need to bound fi = E[X] from below;

li = E[X] > ("/k \cn-x/a)e > (h(h - l))l-hCe

where e is the number of edges of H. Note that E[X] = Q(n), since e/(h — 2) =e/((h — 1) — 1) < a. We now bound A" from above,

A" * (*"_ i){h -!)! ( t - i V ) { h ~l ~ v"] ^~Ua)le-e"< Qle-e"n2h-2-v"-{\/a){2e-e")

Claim 1. IfC> (h(h - \))2h-22h2+\ then

Proof. Note that e" > 1 in each term A". Hence

in2 (h(h-l))2-2hC2en-2e^a+2h-2

~fcT> — Qle-e" n2h-2-v"-(l/a)(2e-e")

> (h(h - \))2-2hCnv"-e"/a > (h(h - \))2-2hCn > 2h2+ln.

There are less than 2h2 subgraphs of H, so the last inequality (which holds for any A")

implies (1). This completes the proof of the claim. •

Returning to the proof of the Lemma with C selected as in the above claim, we proceedas follows. If A < fi, we use Lemma 2.2. Note that in our case we may pick e as anarbitrary small constant, and the lemma implies (since [i > 3n for our C)

Pr[X = 0] < exp(-/i/3) < exp(-n).

If A > \i, we use Lemma 2.3. Picking e = 0.1 and using the above claim we obtain

= 0] < exp(-0.9w).

In any case, ( ^ )Pr[X = 0] tends (even exponentially) to zero when n tends to infinity,and this completes the proof of the Lemma. •

Armed with Lemma 2.4 we can now complete the proof of Theorem 1.1. Given H e F,let d be a vertex of minimal degree in H and denote the set of its neighbors by N(d).Set H' = H \ {d}. Note that a{H') = a(H), since H e J*\ We apply Lemma 2.4 by firstsetting p' = C'n~1/a(H), where C is chosen as in Lemma 2.4. Almost every G(n,pf) willhave n/h vertex disjoint copies of H'. We now need to match every remaining vertex in Gto a copy of H' in such a way that there is an edge between the assigned vertex and eachvertex in the set corresponding to N(d) in the matched copy. We use (a modified versionof) the method from [2] to do so. We choose the edges of G(n,p) once again (but stillkeeping the edges of the first selection) with probability p" = n~l/aW. Note that this isthe same probability space as G(n,p), where (1 — p) = (1 — p')(\ —p"). We define a randombipartite graph with one side being the n/h pairwise disjoint copies of H\ and the otherside being the remaining vertices of G. There is an edge of the bipartite graph betweena copy and a remaining vertex if the vertex can be matched to the copy using only thenew randomly chosen edges. The edge probability in this bipartite graph is n~d^a, whereS is the degree of d. Moreover, crucially, the edges of this bipartite graph are chosenindependently, since their existence is determined by considering pairwise disjoint subsetsof edges of our random graph. Since 5 < a, it follows from the result in [7] that almostalways there is a perfect matching. Note also that p = p' + p" — p'p", and since p" < p',we may generously set C = 2C'. This completes the proof of Theorem 1.1. •

Proof of Theorem 1.2. The fact that the threshold function for the existence of an H-factor is at least n~1/a(H) follows directly from Lemma 2.1. Let H e &. We must showthat there is a constant C = C(H) such that almost always G(n, Cn~1/a(H)) contains an//-factor. We prove this by induction on the minimal number of applications of rule 2in the definition of ^ needed to demonstrate the membership of H in ^. If no such

application is needed, then H e 3F and the result follows from Theorem 1.1. Otherwise,there is an H' e & and two connected components C\ and Ci of it such that H is obtainedfrom H' by adding a set R of edges between C\ and C2, where r = \R\ < a(H'). It iseasy to check that this implies that a(H) = a(H'). By the induction hypothesis, thereexists a constant C such that almost surely G(n, C'rc~1//a(H)) contains an //'-factor. As inthe proof of Theorem 1.1, we choose the edges of G(n,p) once again with probability

n-\/a(H) ^ e define a r a n dom bipartite graph with one side being the n/h pairwise disjointcopies of C\ and the other side being the n/h pairwise disjoint copies of Ci (that arealso pairwise disjoint with the copies of C\). There is an edge in the bipartite graphbetween a vertex corresponding to a copy of C\ and a vertex corresponding to a copy ofC2 if the edges corresponding to R exist in G(n,p) among the freshly selected edges. Theedge probability in this bipartite graph is n~r/a{H) and the choices of distinct edges aremutually independent. Since r < a(H) it follows, as in Theorem 1.1, that G(n,p) almostsurely contains an if-factor, where (1 - p) = (1 - C'w-1/fl(H))(l - n-{/a{H)). We can nowset C = C + 1 and p = Cn-{/a{H) to complete the proof. •


Somewhat surprising is the fact that there are many regular graphs H that fall intothe category of Theorem 1.2. As an example, consider three arbitrary cubic graphs, andsubdivide an edge in each of them. Add a new vertex and connect it to the verticesof degree 2 that were introduced by the subdivisions. The resulting graph H is cubic,and satisfies the properties of the graphs in Theorem 1.2, which supplies the appropriatethreshold function for the existence of an //-factor.

Our theorems raise a natural algorithmic question. Suppose H is a graph in the family^ defined in Theorem 1.2 and a(H) = a. Then, by the theorem, there is a positive constantC = C(H) such that for p(n) = Cn~xla, the random graph G(n,p) contains, almost surely,an //-factor, provided |K(//)| divides n. Can we find such an //-factor efficiently? Theproof easily supplies a polynomial time algorithm for every fixed H. Moreover, thisalgorithm can be parallelized. To see this, observe that in the first step of the proof itsuffices to find a maximal set of vertex disjoint copies of an appropriate graph H' inour random graph G(n,p), where the maximality is with respect to containment. Sucha set can be found in NC (i.e., in polylogarithmic time, using a polynomial number ofparallel processors) using any of the known TVC-algorithms for the maximal independentset problem (see, e.g., [10], [11], [1]). The rest of the algorithm only has to find perfectmatchings in appropriately defined graphs, and this can be done in (randomized) NC bythe results of [9] or [12]. Thus, the //-factors whose existence is guaranteed almost surelyin Theorems 1.1 and 1.2 can actually be found, almost surely, efficiently (even in parallel).

The methods used in the proofs of the theorems can be used to compute the thresholdsfor the existence of spanning graphs other than //-factors. For example, let H be the 4vertex graph consisting of a vertex of degree 1 joined to a triangle. Let Q be the graphobtained from n/4 pairwise disjoint copies of //, / / 1 , . . . , / / * , where at is the vertex ofdegree 1 in //,, by adding a cycle of length n/4 on the vertices at. By Theorem 1.1,p = Cn~2^ is a threshold for the existence of an //-factor. Suppose we now draw edges


again with probability of n~2/3 (which is much more than needed) in the subgraph ofthe n/4 vertices of degree 1. By the result of Posa in [14], we will almost always havea Hamilton cycle in this subgraph. Therefore n~2/3 is a sharp threshold function for theproperty that Q is a spanning subgraph of G(n,p). Various similar examples can be given.Here, too, the proof is algorithmic, by applying the result of [4].

The following conjecture seems plausible.

Conjecture 3.1. Let H be an arbitrary fixed graph with edges. Then the threshold for theproperty that G(n,p) contains an H-factor, (if h divides n) is n-

l/aW+°(l)m

We note that Lemma 2.1 shows that the above threshold is at least n~l/a{H). Also,the o(l) term cannot be omitted entirely, because, for example, \og(n)/n is the thresholdfor a ^-factor, although a(K2) = 1. Similarly, the threshold for a X3-factor is at leastlog(n)1/3n~2/3, since as proved by Spencer [15], this is the threshold for every vertex to lieon a triangle, which is an obvious necessary condition in our case.

Note added in proof. We have recently learned that A. Rucinski, in 'Matching andcovering the vertices of a random graph by copies of a given graph' Discrete Math.(1992) 105 185-197, proved, independently (and before us), Theorem 1.1, using similartechniques. He did not prove the more general Theorem 1.2.

References

[I] Alon, N., Babai, L. and Itai, A. (1986) A fast and simple randomized parallel algorithm for themaximal independent set problem. J. Algorithms 7 567-583.

[2] Alon, N. and Furedi, Z. (1992) Spanning subgraphs of random graphs. Graphs and Combina-torics 8 91-94.

[3] Alon, N. and Spencer, J. H. (1991) The Probabilistic Method, John Wiley and Sons Inc., NewYork.

[4] Angluin, D. and Valiant, L. Fast probabilistic algorithms for Hamilton circuits and matchings.J. Computer Syst. Sci. 18 155-193.

[5] Bollobas, B. (1985) Random Graphs, Academic Press.[6] Boppana, R. B. and Spencer, J. H. (1989) A useful elementary correlation inequality. J.

Combinatorial Theory, Ser. A 50 305-307.[7] Erdos, P. and Renyi, A. (1966) On the existence of a factor of degree one of a connected

random graph. Acta Math. Acad. Sci. Hungar. 17 359-368.[8] Janson, S. (1990) Poisson approximation for large deviations. Random Structures and Algorithms

1 221-230.[9] Karp, R. M., Upfal, E. and Wigderson, A. (1986) Constructing a perfect matching in random

NC. Combinatorica 6 35-48.[10] Karp, R. M. and Wigderson, A. (1985) A fast parallel algorithm for the maximal independent

set problem. J. ACM 32 762-773.[II] Luby, M. (1986) A simple parallel algorithm for the maximal independent set problem. SI AM

J. Computing 15 1036-1053.[12] Mulmuley, K., Vazirani, U. V. and Vazirani, V. V. (1987) Matching is as easy as matrix inversion.

Proc. \9th ACM STOC 345-354.[13] Nash-Williams, C. St. J. A. (1964) Decomposition of finite graphs into forests. J. London Math.

Soc. 39 12.


[14] Posa, L. (1976) Hamiltonian circuits in random graphs. Discrete Math. 14 359-364.[15] Spencer, J. H. (1990) Threshold functions for extension statements. J. Combinatorial Theory,

Ser. A 53 286-305.

A Rate for the Erdos-Turan Law*

A. D. BARBOURt and SIMON TAVAREJ

tlnstitut fur Angewandte Mathematik, Universitat Zurich, Winterthurerstrasse 190, CH-8057,Zurich, Switzerland

t Department of Mathematics, University of Southern California, Los Angeles, CA 90089-1113

The Erdos-Turan law gives a normal approximation for the order of a randomly chosenpermutation of n objects. In this paper, we provide a sharp error estimate for theapproximation, showing that, if the mean of the approximating normal distribution isslightly adjusted, the error is of order log"1/2 n.

1. Introduction

Let a denote a permutation of n objects, and O(a) its order. Landau [13] proved thatmax^log 0{a) ~ {nlogn}112. In contrast, if a is a single cycle of length n, logO(cr) = logw,such cycles constituting a fraction \/n of all possible cr's. In view of the wide discrepancybetween these extremes, the lovely theorem of Erdos and Turan (1967) comes as somethingof a surprise: that, for any x,

-^ # {tr:logO(tr) < £log2w + *{§log3w}1/2} ~ O(JC),

where O denotes the standard normal distribution function. In probabilistic terms, theirresult is expressed as

-±\og2n) < x] ~ O(x), (1.1)

with a now thought of as a permutation chosen at random, each of the n! possibilities being

equally likely. They remark that

'Our proof is a direct one and rather long; but a first proof can be as long as it wants to be. It wouldbe however of interest to deduce it from the general principles of probability theory.'

* This work was supported in part by NSF grant DMS90-05833 and in part by Schweizerischer NF ProjektNr 20-31262.91.

72 A. D. Barbour and S. Tavare

They also entertain hopes of finding a sharp remainder for their approximation.Shorter probabilistic proofs of (1.1) are given by [5], [6] and [1], the last exploiting the

Feller coupling to a record value process. Stein (unpublished) gives another coupling proof,with an error estimate of order Iog~1/4fl{log log«}1/2, which he describes as 'rather poor'.In fact, [16] sharpens the approach of Erdos and Turan, showing that the first correctionto (1.1) is a mean shift of — logn log log n, and that the error then remaining is of order atmost (9(log~1/2«log loglog«). Nicolas also conjectures that the iterated logarithm in theerror is superfluous. Our birthday present is to show this, by probabilistic means, not onlyfor the uniform distribution on the set of permutations, but also under any Ewens samplingdistribution. Since many combinatorial structures are, in a suitable sense, very closelyapproximated by one of the Ewens sampling distributions (see [4]), the result carries overeasily to many other contexts. A typical example is the l.c.m. of the degrees of the factorsof a random polynomial over the finite field with q elements, thus improving upon atheorem of [15].

Consider the probability measure /id on the permutations of n objects determined by

d

where k(cr) is the number of cycles in <x, 0 > 0 is a parameter that can be chosen at will, andwhere rising factorials are denoted by

x(0) = 1.

If 6 = 1, the uniform distribution is recovered. Under /i0, the probability of the set ofpermutations having a^ cycles of lengthy, \ ^j ^n, is given by

j-r (1-3)Uj-

as may be verified by multiplying the probability (1.2) by the number of permutations thathave the given cycle index.

The joint distribution of cycle counts given by (1.3) is known as the Ewens samplingformula with parameter 0. It was derived by Ewens [8] in the context of population genetics.Ewens [9] provides an account of this theory that is accessible to mathematicians.

Under the Ewens sampling formula, the joint distribution of the cycle counts convergesto that of independent Poisson random variables with mean Q/i as n -> oo. Indeed, using theFeller coupling, the cycle counts for all values of n can be linked simultaneously on acommon probability space with a single set of independent Poisson random variableswith the appropriate means. The following precise statement of this fact comes essentiallyfrom [2].

Proposition 1.1. Let {£pj ^ 1} be a sequence of independent Bernoulli random variablessatisfying

^ ( L 4 )

A Rate for the Erdos-Turdn Law 73

Define (Zjm,j>\) by

zjm= t &(i-£.+i)-(i-&+,-i)&+,. d-5)i=m+l

and set Z} = Zj0 and Z = (Zp y > 1). Define Cw = (C/«), j > I) by

C,(«) = "f g,(l - g , + 1 ) . . . (1 - g , ^ ) & + , + gB_,+1(l - g H _ , + 2 ) . . . (1 - g n )* = 1

= Z, - Z3, „_, + gn_j+l{ 1 - £B_i+2)... (1 - 1 J (1.6)

for 1 ssy ^ «, setting C,(n) = Oforj > n. Then V\(Cx(n),...,Cn(ri)) = (a1,...,an)] is given by

(1.3), and the Z} are independent Poisson random variables with EZ} = 0/j. Furthermore, for

Z}-Zjn-I[Jn + Kn =j+ 1] < C,(«) < Z} + l[Jn =j\, (1.7)

where Jn and Kn are defined by

yn = min{/>l:gn_,+ 1 = l} and *„ = min{/> l:g,,+, = 1}. (1.8)

With this representation, the order of the random permutation is On(Cm), where, for any

aeUrj,

On(a) = l.c.m. {/: 1 < / < n, a, > 0} Pn(a) = f\ i"<.f = l

On the other hand, from (1.6), C^n) is close to Zj for eachy when ^ is large, so log On(C{n))

might plausibly be well approximated by logOn(Z). Now functions involving Z are verymuch easier to handle than are the same functions of C{n\ because the components Zj ofZ are independent and have known distributions. In particular, logOn(Z) is close enoughfor our purposes to log Pn(Z) — 0 log/dog log «, and

logPn(Z)=£z, logii=i

is just a sum of independent random variables. The classical Berry-Esseen theorem [10,p. 544, Theorem 2] can thus be invoked to determine the accuracy of the normalapproximation to its distribution.

The above arguments, justified in detail in Section 2, lead to the following result.

Theorem 1.2. If C{n) is distributed according to the Ewens sampling formula (1.3) withparameter 6,

sup Ll3 J \ 2 J " J= 0({logw}-1/2).

It would not be difficult to give an explicit bound for the constant implied in the error term.Indeed, the leading contributions arise from a Berry-Esseen estimate, for which the

necessary quantities are estimated in Proposition 2.4, from inequality (2.1), for which (2.2)and Lemma 2.5 already provide a bound, and from the next mean correction, whichrequires a more careful asymptotic evaluation following (2.4).

A process variant of Theorem 1.2 can also be formulated. Let Wn be the random elementof D[0,1] defined by

1 - 1 / 2 t

Theorem 1.3. It is possible to construct C{n) and a standard Brownian motion W on the sameprobability space, in such a way that

EJsupV\ogn

2. Proofs

As previously indicated, the proof of Theorem 1.2 consists of showing that log On(C(n)) is

close enough to log On(Z), which in turn is close enough to log Pn(Z) — Ologn log log n. TheBerry-Esseen theorem then gives the normal approximation for log Pn(Z).

For vectors a and b, define \a — b\ = ^ \a{ — bf\. Since On(a) ^ On(b)nlb~a] whenever a andb are vectors with a ^ b, it follows from (1.7) that

logOn(Z)-(Yn+I)logn^ logOn(C{n)) ^logOn(Z) + logn, (2.1)

where Yn = Yul-i^jn *s independent of C(n), and

/-I

(2.2)

Inequality (2.1) combined with (2.2) is enough for the closeness of log On(C(n)) and

log On(Z).Next, we can compute the difference between logOn(Z) and logPn(Z) using a formula

of [5] and [14]:

(2.3)

where J] and J] denote sums over /?r/ra<? indices, and

j « n : 11;

Considering first its expectation, observe that, since ( J - 1)+ = rf— 1 +I{d = 0},

= 0]

AnkA±AlkX (2.4)

where

and \jr{r+\) = El=i7 1- Hence

fin := E{logPw(Z)-log On(Z)} = E ' X

= L'llog/KAfip.-l+exp{-Anj(.})= £ ' 0pp s ^ 1 p ^ log w

= 0 log n log log « + O(log «),

where the estimates use (2.4), integration by parts, and Theorems 7 and 425 of [11].For the variability of logOw(Z) —logPn(Z), we now need two lemmas.

Lemma 2.1. For p =N q prime and s, t ^ 1,

Proof. Set

K = If1, A2 = E r1 and i = E / - < (1+!°fH),

psb' ?fK psot\i

and write Dj = DnpS and Z>2 = Dn^. Then, in the expansion

Cov ((D1 - 1)+, (D2 - 1)+) = Cov (£>!, D2) + Cov (D19 7[D2 = 0])+ Cov (I[D1 = 0], Z)2) + Cov (I[Dl = 0], I[D2 = 0]),

the first contribution is evaluated as

Cov(Z)1,Z)2) = E | E E ( Z j - . f 1

PS\J Q*\i

because of the independence of the Z/s. For the second contribution, we have

Cov(Z)157[Z)2 = 0]) = P [Q{Z , = 0}l {E(Z)11 D2 = 0)-E7)1} = - ^ - ^ ,

and similarly for the third, and for the last we have

Cov (I[D1 = 0],7[Z)2 = 0]) = ^(Ai+A^{

Hence

Cov ((/>!- l)+,(£>2- 0+) = 0£{1 - ^ A i - e - ^ + ^ ( A i + A ^ } ^ 6>£,

proving the lemma.

Lemma 2.2. For \ ^ s t,

Cov((Dnps-l)\(Dnp,-iy) ^ 6p-'(l+\ogn).

Proof. The argument runs as for Lemma 2.1, with Ax denned as before, but now with

The computations now yield

Cov (D19D2) = 0g; Cov (D1,I[D2 = 0]) - -dge-0X*; Cov (I[D1 = 0],D2) = -6A2

and

Cov (I[D1 = 0],I[D2 = 0]) = e~d\\ -e~dX*\

and thus

The two lemmas enable us to control the difference between log On(Z) and log .Pn(Z) asfollows.

Proposition 2.3. For any K > 0,

P[|log/>w(Z)-logOw(Z)-/.J > Klogn] =

Proof. Write

\p ^ log2 n p > log2 n

— 1/ _L 7/ _|_ 1 /

say. Lemmas 2.1 and 2.2 give

: £ g^ X3? < log2 n P p =t= <? ^ log2 n PQ

= O(\ogn(\og\ognf);

it follows from (2.4) that

E ^ y £'^ /r2log/>(l+logn)2 =^ p > log2 n

and Lemmas 2.1 and 2.2 imply that

V a r F < V ' W 2 n V 00+kg"), ^ , „ . , „ . V 00+tog/!)V s,t>2 P

Thus, by Chebyshev's inequality,

P[ | V2 - E V2\ > § tflog «] - O (log-1*),

and

P[ | F3 - E F3| > I tflog «] = O (log"1«),

proving the proposition.We now use the closeness of the quantities \ogOn(C

{n)), log On(Z) and logPn(Z)—/in toprove Theorem 1.2. To do so, we introduce the standardized random variables

logP n (Z)- | log 2 / i logOn(Z) + ^ - | l o g 2 ^— l S 2 n = -

and

whose distributions we shall successively approximate. Since the quantity log Pn(Z) can bewritten in the form ^j^Z^logy as a weighted sum of independent Poisson randomvariables, the normal approximation for Sln follows easily from the Berry-Esseen theorem.

Proposition 2.4. There exists a constant c1 = cx(6) such that

sup|P[Slw ^ x\ — O(x)| ^ cxlog'll2n.X

Proof. It is enough to note that

tuzjiogj) = opthat

£var(Z,logy) = #f;

and that

78 A. D. Barb our and S. Tavare

indeed, for j ^ 6,

J J J

and hence, for 6 2,

t E|Z,- EZ/log3/ < #[1 +2^1] trMog3; = ^ ' ~ Jqog4w + Q(l)). (2.5)

In order to show that S2n and 5I3w have almost the same distribution as Sln, because of

Proposition 2.3 and (2.1), one further lemma is required.

Lemma 2.5. Let U and X be random variables with s\xpx\P[U ^ x] — O(x)| ^ TJ. Then, forany e > 0,

> e]. (2.6)V27T

If W and Y are independent random variables with EF < oo, and if \ W— U\ ^ Y, then

sup\P[W^ x]-®(x)\ ^ 3L + =\. (2.7)

Proof. The first part is standard. For the second, let Sy = P[W^y]-Q>(y) and setA = supjtfj. Write p = 3EF and p = P[Y> p], so that p < 1/3. Then, since, for any x,{U^x}=> {W+ Y< x}, it follows that

P[W^x-y]FY(dy)J[0,oo)

where FY denotes the distribution function of Y. Hence, comparing as much as possible toQ>(x — p), it follows that

V/77 V2.7T

implying that

A similar argument starting from {U ^ x} cz {W— Y x} then gives

The choice of x being arbitrary, it thus follows that

4Er

also, and hence that

as claimed.To complete the proof of Theorem 1.2, apply (2.6) with Sln for U and S2n — Sln for X,

taking rj = c^og^^n from Proposition 2.4 and e = log~1/2rc. By Proposition 2.3,

- S l n | > e] = p\\logPH(Z)-logOH(Z)-pn\ > e / | log3 J =L v 3 J logw / '

and hence, from (2.6),

sup|P[S2w ^ x]-O(x)\ ^ c2\og~1/2n

for some c2 = c2{6). Now we can apply (2.7) with U = S2n and W = S3n, since (2.1) impliesthat \U-W\^Y, with Y = {(6/3)\ogn}-1/2(Yn + \l giving

sup|P[S3n ^ x] —O(x)| = O(\og~1/2n(l + EYn)) = O(log"1/2«),

in view of (2.2). This is equivalent to Theorem 1.2.

To prove Theorem 1.3, we use essentially the same estimates. First, from (2.1),

|log O[nt] (C<»>) - log O[nt](Z)\ ^ (1 + Yn) log n

for all 0 < r < 1, and then, from (2.3),

0<logP[ n t ] (Z)-logO[ n.

= y/ v—< La

Hence

EJ suplo < t < I

Now

s log ndBn(s),3=1 j=l J0

wherein*]


can be realized as

+ 1)) - W l + 1)}

using a Poisson process P with unit rate. Also, since

s[dBn(s)-dB(s)] - I'{Bn(s)-B(sJo

t[Bn(t)-B(t)]- I{Bn(s)-B(s)}dsJ

sup \Bn(t)-B(t)\,0 ^ t ^ 1

the uniform approximation of Bn by a standard Brownian motion B, in the form

E{ sup |5n(0-5(0l} = O({log«}-1/2loglog/i),l J

as carried out using the theorem of Komlos, Major and Tusnady [12] in the case 6 = 1 in[3], now implies the conclusion of Theorem 1.3: take W{f) = V3^sdBn(s).

References

[I] Arratia, R. A. and Tavare, S. (1992) Limit theorems for combinatorial structures via discreteprocess approximations. Rand. Struct. Alg. 3 321-345.

[2] Arratia, R. A., Barbour, A. D. and Tavare, S. (1992) Poisson process approximations for theEwens Sampling Formula. Ann. Appl. Probab. 2 519-535.

[3] Arratia, R. A., Barbour, A. D. and Tavare, S. (1993) On random polynomials over finite fields.Math. Proc. Cam. Phil. Soc. 114 347-368.

[4] Arratia, R. A., Barbour, A. D. and Tavare, S. (1993) Logarithmic combinatorial structures.Ann. Probab. (in preparation).

[5] Best, M. R. (1970) The distribution of some variables on a symmetric group. Nederl. Akad.Wetensch. Indag. Math. Proc. Ser. A 73 385-402.

[6] Bovey, J. D. (1980) An approximate probability distribution for the order of elements of thesymmetric group. Bull. London Math. Soc. 12 41-46.

[7] Erdos, P. and Turan, P. (1967) On some problems of a statistical group theory. III. Acta Math.Acad. Sci. Hungar. 18 309-320.

[8] Ewens, W. J. (1972) The sampling theory of selectively neutral alleles. Theor. Popn. Biol. 387-112.

[9] Ewens, W. J. (1990) Population genetics theory - the past and the future. In: Lessard, S. (ed.)Mathematical and statistical developments of evolutionary theory, Kluwer Dordrecht, Holland,177-227.

[10] Feller, W. (1971) An introduction to probability theory and its applications, Volume II, 2ndEdition, Wiley, New York.

[II] Hardy, G. H. and Wright, E. M. (1979) An introduction to the theory of numbers, 5th Edition,Oxford University Press, Oxford.

[12] Komlos, J., Major, P. and Tusnady, G. (1975) An approximation of partial sums of independentRV'-s, and the sample DF. I. Z. Wahrscheinlichkeitstheorie verw. Geb. 32 111-131.

[13] Landau, E. (1909) Handbuch der Lehre von der Verteilung der Primzahlen. Bd. I.[14] De Laurentis, J. M. and Pittel, B. (1985) Random permutations and Brownian motion. Pacific

J. Math. 119,287-301.[15] Nicolas, J.-L. (1984) A Gaussian law on FQ[X]. Colloquia Math. Soc. Jdnos Bolyai 34 1127-1162.[16] Nicolas, J.-L. (1985) Distribution statistique de l'ordre d'un element du groupe symetrique.

Acta Math. Hungar. 45 69-84.

Deterministic Graph Games and a ProbabilisticIntuition

JOZSEF BECKf

Department of Mathematics, Rutgers University,Busch Campus, Hill Center,

New Brunswick, New Jersey 08903 U.S.A.e-mail: [email protected]

There is a close relationship between biased graph games and random graph processes. Inthis paper, we develop the analogy and give further interesting instances.

1. Introduction

We shall examine the following class of combinatorial games. Two players, Breaker andMaker, with Breaker going first, play on the complete graph Kn of n vertices in such a waythat Breaker claims b (> 1) previously unselected edges a move, and Maker claims onepreviously unselected edge a move. Maker wins if he claims all the edges of some graphfrom a family of prescribed subgraphs of Kn. Otherwise Breaker wins, that is, Breakersimply wants to prevent Maker from doing his job.

As a warm-up consider the following three particular cases. Let Clique(n;b, l ;r) denotethe game where Maker wants a complete subgraph of r vertices (from his own edges ofcourse). Denote by Connect(n;b,l) and Hamilt(n;b,l) the games where Maker's goal isto select a spanning tree (i.e. a connected subgraph of Kn) and a Hamiltonian cycle of Kn,respectively. Clearly if fo is large enough with respect to n, Breaker has a winning strategy;if b is small, Maker has a winning strategy. The following crude heuristic argument, dueto Paul Erdos, predicts the asymptotic behaviour of the 'breaking point' with surprisingaccuracy. The duration of a play allows for approximately n2/2(b + 1) Maker's edges.In particular, if b = n/2c log n, Maker will have the time to create a graph with en log nedges. A random graph with n vertices and en log n edges is almost certainly connected(and Hamiltonian as well) for c > 1/2, and almost certainly disconnected (if fact, it hasmany isolated points) for c < 1/2. Now one suspects that the breaking points for the

Supported by NSF Grant No. DMS-9106631.

82 J. Beck

games Connect(n;b,l) and Hamilt(n;b,l) are around b = n/\ogn. Similarly, the largestclique in a random graph of n vertices and of parameter p = 1/2 has approximately2 log nj log 2 vertices with probability tending to one as n tends to infinity. This suggeststhat Maker wins the fair game Clique(n; 1, l ;r) if r is around logn. The following resultssupport this probabilistic intuition.

Theorem A. (Erdos-Selfridge [7] and Beck [1]) Breaker has a winning strategy in thefair game Clique{n\ 1,1; 2 log n/log 2). On the other hand, given e > 0, if n is sufficientlylarge, Maker has a winning strategy in Clique(n; 1,1;(1 — e)logn/log2).

Theorem B. (Erdos-Chvatal [6] and Beck [2],[3])

(i) If b > (1 + e)n/\ogn, Breaker has a winning strategy in Connect(n;b,l) if n is largeenough,

(ii) Ifb> (log2 — e)n/\ogn, Maker has a winning strategy in Connect(n;b, l)ifn is largeenough.

(iii)Ifb > (log2/27 — e)n/ log n, Maker has a winning strategy in Hamilt(n\b,\) if n islarge enough.

The object of this paper is to point out further instances of this exciting analogybetween the evolution of random graphs and biased graph games. The following fourtheorems were motivated by some well-known results in the theory of random graphs (seethe monograph [5] by Bollobas). For example, Theorem 1 below is the game-theoreticanalogue of a result of Bollobas [4] on long paths in sparse random graphs'. Needless tosay, our proof is essentially different from Bollobas' argument in [4].

The basic idea is that Maker's graph possesses some fundamental properties of randomgraphs (mostly 'expanding' type properties) provided Maker uses his best possible strategy.

Let us begin with a trivial observation: if b = 2n, Breaker can easily prevent Maker evenfrom getting a path of two edges (Breaker blocks the two endpoints of Maker's edge). Ifb = en, e > 0 constant, then, in view of Theorem B(i), Breaker can force Maker's graph tobe disconnected. In fact, Breaker can force at least (e/2)e~1/en isolated points in Maker'sgraph, as shown by the following argument, which is a straightforward adaptation of theErdos-Chvatal proof of Theorem B(i). Breaker proceeds in two stages. In the first stage,he claims all the edges of some clique K*m with m = en/2 vertices, such that none ofMaker's edges has an endpoint in this K^. In the second stage, he claims all the remainingedges incident with at least (e/2)e~l^'n vertices of K^, thereby forcing at least [e/2)e~x^nisolated points in Maker's graph.

The first stage lasts no more than m = en/2 moves, and goes by a simple induction onm. During his first i — 1 (1 < i < m) moves, Breaker has created a clique K*_{ with i — 1vertices, such that none of Maker's edges has an endpoint in K*_{. At this moment thereare /— 1 < en/2 Maker's edges, hence there are at least two vertices u9v in the complementof V(K*_{) that are incident with none of Maker's edges. On his ith move, Breaker claimsedge {u,v}, and all the edges joining u and v to the vertices of K*_{, thereby enlargingK*_{ by two vertices. Then Maker can kill one vertex from this clique K*+1 by claimingan edge incident with that vertex. Nevertheless, a clique of i vertices still 'survives'.

Deterministic Graph Games and a Probabilistic Intuition 83

In the second stage, Breaker has m = en/2 pairwise disjoint edge-sets: for every u €V(K*m), the edges joining u to all vertices in the complement of V(K*m). It is easy to seethat Breaker can completely occupy at least e~1/£m of these m disjoint edge-sets by thesimple rule that he has the same (or almost the same) number of edges from all the'surviving' edge-sets at any time.

Our first result says that Maker is able to build up a cycle of length at least (1— e~l/2OOe)n.That is, if Breaker claims en edges a move, then Maker has an 'almost Hamiltonian cycle'in the sense that the complement is 'exponentially small' (the constant factor of 200 inthe exponent is of course very far from the best possible). In this paper we do not makeany effort to find the optimal (or even nearly optimal) constants.

Theorem 1. IfO<s< 1/200 and Breaker selects en edges for each move of Maker, thenMaker can build a cycle of length at least (1 — e~l^2OOe)n on a board Kn.

Note that if Maker just wants a path Pm of m = (1 — const^Ji)n edges, he can do itin the shortest way. in m moves. Indeed, Maker can employ the following simple greedystrategy: keep extending the path by adding that available point (as a new endpoint)that has minimum degree in Breaker's graph. Trivial calculation shows that this greedyprocedure does not terminate in m = (1 — const^/e)rc moves.

However, if Maker's object is (say) a binary tree BTm of m =constn edges, then thisgreedy strategy does not seem to work. We formulate it in the following conjecture.

Conjecture. Let c be an arbitrarily large but fixed positive constant. Consider the gamewhere the board is a complete graph Kc.n of c • n points, Breaker and Maker alternatelyselect n and 1 edge(s) a move, respectively, and Maker's goal is a binary tree BTn of nedges (n = 2 \ k integer). If n> no(c), Breaker can prevent Maker from getting a copy ofBTn in n moves.

On the other hand, in linear time (i.e. in const/7 moves) Maker can obtain all trees withat most n vertices and constant size degrees.

Theorem 2. Consider the game where the board is KN with N = lOOdn. Breaker andMaker alternately select n and 1 previously unselected edges a move, respectively. Makerhas a strategy to force his graph to be tree-universal in the sense that it contains every treewith at most n vertices and maximum degree at most d.

Observe that Theorem 2 is essentially best possible (apart from the constant factor of100). Indeed, Breaker can prevent Maker from having a degree larger than 2N/b on aboard KN: if Maker selects an edge {w,u}, Breaker occupies b/2 edges from u and b/2edges from v.

In general, what can we say about Maker's largest degree, i.e., the largest star Makercan construct? For simplicity we restrict ourselves to the fair case (i.e. b = 1). This questionis apparently due to Erdos (oral communication). Szekely [12] proved, by using Lemma3 in Beck [1], that Breaker can prevent a star of size (rc/2)+const Jn log n. In the other

84 J. Beck

direction, we can prove that, for some positive constant, Maker can achieve a star of size(rc/2)+const^/n. If the complete graph Kn is replaced with the complete bipartite graphKn,n9 we obtain the following interesting 'row-column game'.

Theorem 3. Consider the game where the board is an nxn chessboard. Breaker and Makeralternately select a previously unselected cell Breaker marks his cells blue and Maker markshis cells red. Maker's object is to achieve at least (n/2) + fc (k > 1) red cells in some line(row or column). Ifk = ^Jn/32, Maker has a winning strategy.

This result is in sharp contrast with the chessboard type alternating two-coloring, wherethe discrepancy in every line is 0 or 1 depending on the parity of n.

A straightforward modification of the proof of Theorem 3 gives the above-mentionedcase where the board is Kn. We leave the details to the reader.

Finally, we study the case where Maker's goal is to build up an arbitrary prescribedgraph Gn4 of n points and maximum degree d. We prove that, for any b > 1 and d > 1there is a = c(b, d) such that Maker's graph contains all graphs Gn^ of constant degree ona board KM, N = c • n, of linear size. The quantitative version goes as follows.

Theorem/4. Consider the game where the board is KN with N = 100<P(3b)d+1 • n, andBreaker and Maker alternately select b (> 1) and 1 edge(s) a move. Maker has a strategyto force his graph to be universal, in the sense that it contains all graphs Gn,d of n pointsand maximum degree d.

Note that the exponential behaviour of c = c(b,d) is necessary: iffc = l , d = w—1 and

Gn4 = Km we just go back to Theorem A.

2. Proofs of Theorems 1-2 - 'derandomization' of the first moment method

Proof of Theorem 1. We combine the basic idea of Beck [3] with a 'truncation procedure'.Given a simple and undirected graph G, and an arbitrary subset S of the vertex-set

V(G) of G, denote by TQ{S) the set of vertices in G adjacent to at least one vertex of S.Let \S\ denote the number of elements of a set S.

The following lemma is essentially due to Posa [11] (a weaker version was earlier provedby Komlos-Szemeredi [10]). A trivial corollary of the lemma is that an expander graphhas a long path.

Lemma 1. Let G be a non-empty graph, vo G V(G), and consider a path P = {vo9v\,...,vm)of maximum length that starts from VQ. If (Vi,vm) G G (1 < i < m — 1), we say thatthe path (vo,...,Vi,vm,vm-\9...,Vi+\) arises by a Posa-deformation from P. Let end(G,P,vo)denote the set of all endpoints of paths arising by repeated Posa-deformations from P,keeping the starting point VQ fixed. Assume that for each vertex-set S c V(G) with \S\ < /c,| r G ( S ) \ S | >2 |S | . Then \end(P9G,v0)\ > k + 1.

In order to use Lemma 1, we need another result.

Lemma 2. Under the hypothesis of Theorem I, Maker can guarantee that right after Breakeroccupied (1/20) (2) edges, Maker's graph G has the following property.

Let S be a subset of V(Kn) with (l/3)e-^2OOen < \S\ < n/4. Then

\TG(S)\S\ >2\S\+e-l/2OOen.

Proof. We apply a general theorem about hypergraph games. Let Jtf* be a hypergraphwith vertex set V(Jtf) and edge set £(Jf), and let p > 1 and q > 1 be integers. A(J»f ;p, \\q)-game is a game on Jf in which two players, I and II, select p and 1 previouslyunselected vertices a move from K(jf). The game proceeds until (l/q)\V(J^)\ verticeshave been selected by I. Player II wins if he occupies at least one vertex from everyhyperedge A £ E(J^), otherwise I wins. In [3] we proved the following result: if

V ( '

then II has a winning strategy in the (J^;p, l;g)-game. (The case p = q = 1 of this resultwas proved in Erdos and Selfridge [8]. The proof is based on the method that is nowcalled 'derandomization'.)

In order to apply (1), we introduce some hypergraphs. Let m be an integer satisfying(l/3)e~1/2OOen < m < n/4, and let Jf(n;m) be the set of all complete m x (n - 3m -£-i/200en_|_ i) bipartite subgraphs of Kn. The 'vertices' of Jf(n;m) are the edges of Kn. LetJf be the union of all these J^(n;m) hypergraphs.

Now to ensure property A, in view of (1) with p = b = en and q = 20, it is enough tocheck the following inequality:

^ \mj \2m + e-i/2oo£n _ \) ~ ^ 2*

Standard calculations show that (2) holds, and Lemma 2 follows from (1) and (2). •

Now we are ready to complete the proof of Theorem 1. We show that if Maker usesthe strategy in Lemma 2, and H is Maker's graph at the end, H contains a cycle of(1 — e~{/20°E)n edges.

Let G be Maker's graph right after Breaker occupied (l/20)(!J) edges. Assume that thereexists a vertex-set Si c V(Kn) with |Si| < (l/3)e-{/2mn such that | r G (S i ) \S i | < 2|Si|.Throwing away the vertices FG(Si)U5i from G, we get a new graph G\. Again assume thatthere exists a vertex-set S2 c V(G\) with \S2\ < (l/3)e-{/2OOen such that |rGl(S2)\S2| < 2|S2|.Throwing away the vertices FG^S^ U S2 from G\, we get a new graph G2, and so on.This truncation procedure terminates (say) in t steps: Gt = Gt+\ = • • *. That is, for anyvertex-set S c V(Gt) with \S\ < (l/3)e-^2OO%

\rGt(S)\S\>2\S\. (3)

We claim

(4)

86 J. Beck

Indeed, otherwise there is an index i (< t) such that at the ith stage of the truncation, theunion S = S{ U • • • U St first satisfies (l/3)e-l/2OOen < \S\, so

e n < \ S \ < e n < n / A

and

irG(S)\s|<2|S|,

which contradicts property A in Lemma 2.It follows from (3), (4) and property A that for every set S cz V(Gt) with |S| < n/4, we

have

|rGf(S)\S|>2|S|. (5)

It immediately follows from (5) that Gt is a connected graph. We are going to show thatMaker can build up a Hamiltonian cycle on the vertex-set V(Gt).

Let P be a path in Gt of maximum length. Inequality (5) ensures that the truncatedgraph Gt satisfies the condition of Posa's lemma with k = n/4, so (see Lemma 1)\end(Gt, P9VQ)\ > w/4, where vo is one of the endpoints of P.

Let end(Gt,P,vo) = {xi,X2,...,Xfc} (k > n/4), and denote by P(x,), 1 n/4. (6)

Let

close(Gt,P) = {(xi9y) : x, G e

By (6) we have

\close(Gt9P)\ > (n/4)2/2 = n2/2>2.

Since at this moment Breaker's graph contains

-20

edges, there must exist a previously unselected edge e\ in close(Gt,P). Let ei be Maker'snext move. Then Maker's graph G\l) = Gt U { i} contains a cycle of length |P|. Moreover,G^ = Gr U {^I} is connected, thus either |P| = |K(Gf)|, and we have a Hamiltonian cyclein the truncated vertex-set, or G[l) contains a longer path (i.e. a path of length > |P| + 1).

Let Pi be a path of maximum length in G\l\ Repeating the argument above, we get that\close(G{

tl\P\)\ > n2/32. Since at this moment Breaker's graph contains (l/20)(") + en <

n1131 edges, there must exist a previously unselected edge ei in close(G^\P\). Let ei beMaker's next move. Then Maker's graph G(

t2) = Gt U {e\,ei\ contains a cycle of length

P\\. Moreover, G{2) = Gt U {e\,ei} is connected, thus either |Pi| = |F(G,)|, and we have aHamiltonian cycle in the truncated vertex-set, or G(

r2) contains a /onger path (/.e. a path of

length > |Pi| + 1). By repeated application of this procedure, in less than n moves (so therequired inequality (l/20)(") + n • en < n2/32 holds), Maker's graph will certainly containa Hamiltonian cycle in the truncated vertex-set V(Gt). Theorem 1 follows. •

Proof of Theorem 2. The main difference is that Posa's lemma is replaced with thefollowing lemma, due to Friedman and Pippenger [9].

Lemma 3. If H is a non-empty graph such that, for every set S cz V(H) with \S\ <2n — 2,we have \FH(S)\ > (d+l)|S|, then H contains every tree with n vertices and maximum degreeat most d.

We need the following analogue of Lemma 2.

Lemma 4. Maker can ensure that at the end of the game his graph satisfies:Property B: if N = lOOdn and c V(KN) satisfies 2n < \S\ < 4n, then \TG(S)\ > (d + 1)|S|,where G is Maker's graph at the end.

Proof. By imitating the proof of Lemma 2, it is enough to check the following inequality(this case is even simpler because q = 1):

/ l00dn\ flOOdn— m^ [ m J\ dm-I

m=2n v

Easy calculations show that (7) holds, and Lemma 4 follows. •

We shall now repeat the 'truncation procedure'. We show that if Maker uses the strategyin Lemma 4, then the graph G obtained by Maker at the end contains all trees with nvertices and maximum degree at most d.

Assume that there exists a set Si c= V(KN) with |Si| < 2n such that |FG(Si)| < (d+l)|Si|.Discarding the set FG(SI) U SI from G, we get a new graph G\. Again assume that thereexists a set S2 cz V(G{) with |S2| < 2n such that |FGl(S2)| < (d + l)|S2|. Discarding theset FG!(S2) U S2 from Gi, we get a new graph G2, and so on. This truncation procedureterminates, say, in t steps, Gt = Gt+\ = •••.

We claim that Gt is non-empty. Indeed, otherwise there is an index / < t such that atthe fth stage of the truncation, the union set S = Si U • • • U S,- satisfies 2n < \S\ <4n and

which contradicts Property B in Lemma 4.By definition, the non-empty graph H = Gt has the property that for every vertex-set

S cz V(H) with \S\ < 2n,

Thus, by Lemma 3, H = Gt contains all trees with n vertices and maximum degree atmost d. Since G =2 Gt = H, Theorem 2 follows. •

3. Proof of Theorem 3 - a 'fake second moment' method

Consider a play according to the rules in Theorem 3. Let xi,x2,...,X; be the blue cellsin the chessboard selected by Breaker in his first / moves, and let yuyi,...,yi-\ be the

88 J. Beck

red cells selected by Maker in his first (i — 1) moves. The question is how to find Maker'soptimal ith move j;,-. Write

Let A be a line (row or column) of the n x n chessboard, and introduce the following'weight':

- | n ,_i n 4 j

where

\ 0, otherwise.

Let y be an arbitrary unselected cell, and write

Wi(y) = wt(A) + vVj(B),

where A and £ are the row and the column containing y.Here is Maker's winning strategy: at his ith move he selects that previously unselected

cell y for which the maximum of the 'weights'

max Wi(y)y unselected

is attained.The following total sum is a sort of 'variance':

In lines A

The idea of the proof is to study the behaviour of T, as i = 1,2,3,..., and to show thatTend is 'large'.

Remark. The more natural 'symmetric' total sum

J2 (\Yi-xnA\-\XiHA\)2

2/i lines A

is of no use because it can be large if in some line Breaker overwhelmingly dominates.This is exactly the reason why we had to introduce the 'shifted and truncated weight'MA).

First we compare T, and T,+i, that is, we study the effects of the cells yt and x,-+i. Wedistinguish two cases.

Case 1: the cells yt and x,-+i determine four different lines.Case 2: the cells yt and x,-+i determine three different lines.

In Case 1, an easy analysis shows that

T,+i > T, + 1 (8)


except in the 'unlikely situation' when Wj(y,) = 0. Indeed,

so

7*1 = 71/ + 2wI(>;/) - 2wi(xi+i) + {2 or 1 or 0} > Tt + {2 or 1 or 0},

where

(2, ifwi(A)>0,wi(B)>0;{2 or 1 or 0} = I 1, if max{wi(A),Wi(B)} > 0,min{wi(A),Wi(B)} = 0;

[0, ifwi(A) = wi(B) = 0.Even if the 'unlikely situation' occurs, we have at least equality: T,+i = Tf. Because y, wasa cell of maximum weight, for x,-+i, and for every other unselected cell x, w,(x) = 0.

Similarly, in Case 2,

Ti+i > Tt + 1 (9)

except in the following 'unlikely situation': wt(B) = 0, where 4 is the line containing bothyi and x,-+i, and B is the other line containing yt. Even if this 'unlikely situation' occurs,we have at least equality: T,-+i = Tt. Because yt was a cell of maximum weight, it followsthat Wi(C) = 0, where C is the other line containing x,+i, and, similarly, for every otherunselected cell x in line A, wt(Dx) = 0, where Dx is the other line containing x.

If / is an index for which the 'unlikely situation' in Case 1 occurs, let unsel(i) denotethe set of all unselected cells after Breaker's ith move. Similarly, if i is an index for whichthe 'unlikely situation' in Case 2 occurs, let unsel(i,A) denote the set of all unselected cellsafter Breaker's (/ + l)st move in line A containing both yt and xI+i, including yt and x,+i.

If the 'unlikely situation' occurs in less than 3n2/10 moves (i.e. in less than 60% of thetotal time), we are done. Indeed, by (8) and (9),

Since Tend is a sum of In terms, we have2 ic

max (ny/204))2 > —— = —.2w lines /4 v 7 7 2 « 1 0

Equivalently, for some line A,

wn2/2G4) = 11>4 n yn 2 / 2_iI - |X n xn2/2\

where

/ \+ = i0, otherwise.

So

\A n Yn2/2_{\-\An xn2/2\ > v W I o - ^ > ^,and Theorem 3 follows.

If the 'unlikely situation' in Case 1 occurs in more than n2/l0 moves (i.e. in more than

90 J. Beck

20% of the time), then let /0 be the first time when this happens. Clearly

\unsel(io)\ > 2n2/10 = n2/5.

It follows that there are at least (n2/5)/n = n/5 distinct columns D containing (at leastone) element of unsel(io) each. So Wi(D) = 0 for at least n/5 columns Z), that is,

|D n JSST,-| —1£> n r,_,| > ^

for at least n/5 columns D. Therefore, after Breaker's i'otn move,

V {\Dnxi\-\DnYi-1\}+>l£ (10)

n columns D

Since

n columns D n columns D

by (10),

{|Dnyi_1|-|i)nx;|}+>^-.

n columns D

Since the number of terms on the left-hand side is less than n — n/5 = 4n/5, after Breaker'si'oth move, we have

Obviously Maker can keep this advantage of ^/rc/16 for the rest of the game, and againTheorem 3 follows.

Finally, we study the case when the 'unlikely situation' of Case 2 occurs for at leastn2/5 moves (i.e. for at least 40% of the time). Without loss of generality, we can assumethat there are at least n2/10 'unlikely' indices i when the line A containing both yt andx,-+i is a row. We claim that there is an 'unlikely' index *o when

\unsel(k,A)\ > n/5. (11)

Indeed, by choosing yt and x,-+i, in each 'unlikely' move the set unsel(UA) is decreasing by2, and because we have n rows, the number of 'unlikely' indices i when unsel{i,A) < n/5is altogether less than n • ^ = n2/10.

Now we can complete the proof just as before. We recall that wio(D) = 0 for thosecolumns D that contain some cell from unsel(io,A) (here A is the row containing both yio

and x/0+i). So, by (11), w,0(D) = 0 for at least n/5 columns D, that is,

for at least n/5 columns D. Therefore, after Breaker's /oth move

{\Dnx,\-\DnYt-i\}+>^. (12)

n columns D

Since

n columns D n columns D

by (12),

{|Dny,_1|-|Dnx,|}+>^-.n columns D

Since the number of terms on the left-hand side is less than n — n/5 = 4w/5, after Breaker's/oth move, we have,

Obviously Maker can keep this advantage of y/n/16 for the rest of the game, and againTheorem 3 follows and the proof is complete. •

4. Proof of Theorem 4 - a Szemeredi type embedding

The game-theoretical content of the proof is the following lemma.

Lemma 5. Maker can ensure that at the end of the game his graph satisfiesProperty C: For any two disjoint subsets U and V of V(KN) with u = \U\ > lOOMogNand v = \V\ > 100blogiV, the graph MG constructed by Maker at the end contains morethan u • v/3b edges from the u • v edges of the complete bipartite graph U x V.

Proof. Consider a play according to the rules. After Breaker's /th move, let MG;_i denotethe graph of Maker's i — 1 edges, and let J5G, be the graph of Breaker's i • b edges. Forevery U x V described in property C, let w,(C/ x V) be the 'weight'

Wi(U XV) = (1 + n^BGinUxVl~uv^-{/3b) • (1 — u\MGi

where the parameter A, 0 < X < 1, will be specified later. For every unselected edgee £ KN, let

wt(e) = Y, W^U x F>-UxV: eeUxV

Now let Maker's ith move be an unselected edge of maximum weight.Consider the total sum

Tt= Yall possible UxV in property C

We shall verify that the sequence (TJ) is decreasing:

Tt. (13)

Let e, and /;+i,i,/j+i,2,...,/i+i,f> denote Maker's ith move and Breaker's (i + l)st move,respectively. That is,

/} = MGt \ Md-u {fi+u,fi+i,2,---,fi+i,b} = BGi+l \ BG,.

92 J. Beck

Given a product set U x V in Property C, let a/((7 x K) be 1 or 0 according to whetheret € U x V or et ^ U x V; and let Pt+\(U x V) be the number of edges ft+xj, 1 < j < bcontained in 1/ x V. It is easy to see that

0 xl/)//) • (1 - Xf>{UxV) - l ) • W / ( l / x F). (14)(7 V

We need the following simple inequality. For arbitrary integers a = 0,1 and /? =0,1,...,6,

(1 -h kf/b(\ - Xf - 1 < X (X - «) •

Indeed, if a = 0, (15) is equivalent to

This inequality is immediate, since the function y = xs (0 < 5 < 1) is concave, so theslope

Iof the chord of y = x^b between x = 1 and x = 1 -f A is at most /}/fr, the slope of thetangent line at x = 1.

The case of a = 1 is even simpler. Indeed, then (15) is equivalent to

This inequality holds, since the case a = 0 implies that

((1 + Xf/b - l ) (1 - X) < ( 1 + Xf/b - 1 < X^-.

This proves (15).It follows from (14) and (15) that

X V)Wl{U X V ) - ^ U x

= Ti

and as d+x was an edge of maximum weight, we obtain (13).Now assume that during a play Breaker managed to occupy at least uv(l — I/3b) edges

from some product set U x V in Property C. Let us say that this happened after his iothmove. Then clearly 1 < T,o.

On the other hand, consider Tstart = Tu that is, the situation right after Breaker's firstmove (consisting of b edges). We clearly have

u> \00b log Nv>\00b log N

Let A = 1/2. Since b > 1, trivial calculations give

uv/3b. ^\(b-uv(\-l/3b))/b . Q _ n-uD/35 > ( -

Using this inequality, one can easily show that T\ < 1.All in all, T\ < 1 < Tio, which contradicts (13) and so proves Lemma 5. •

The following purely graph-theoretical result is essentially contained in [7].

Lemma 6. If a graph H of N = lOOd3 (3b)d+l • n vertices satisfies Property C of Lemma 5,then H contains every graph Gnj of n vertices and maximum degree d.

Proof. We closely follow the argument in [7]. Let Gn,d be a graph having n verticesxi,*2,.. . ,xn and maximum degree d. To construct a copy of Gnj in H, we will pro-ceed inductively to choose vertices yi,y2,---,yn from H so that the map x,- —• yt is anisomorphism.

Let A\ U A2 U • • • U Ad+\ be an arbitrary partition of the N-element vertex-set of Hinto disjoint subsets of almost the same size, that is, \At\ « N/(d+ 1) (i = 1,2, . . . ,d-f 1).We will choose the points yuyi,...,yn so that for each i = 1,2, ...,rc, the following twoconditions are satisfied:

(a) If 1 < s < t < i and xs is adjacent to xt in Gnj, then ys and yt come from distinctsets Aj in the partition and ys is adjacent to yt in H.

(b) If / < q < n9 V(q,i) = {yt : 1 < t < i,xt is adjacent to xq in Gn,d}, v = \V(qJ)\,1 < p < d+ I and ,4P contains no yt e V(q,i), then Ap contains a subset A*p = A*p(V(q,i))having at least \Ap\(3b)~v points so that every point in A*p is adjacent to every yt e V(qJ).

At first, condition (b) may seem hopelessly complicated to the reader. (As far as I knowit was Szemeredi who first applied conditions like (b) to prove Ramsey type theoremsin the early seventies.) However, after some thought it will be clear that this conditionis precisely what is needed to ensure that the selection of the vertices y\,y2,---,yn canproceed inductively as claimed.

Here are the details. Suppose that for some nonnegative integer i (< n) the points yt

for i < t < i have been chosen so that conditions (a) and (b) are satisfied. We show howto make a suitable choice for y,+i. (Note that this definition allows i — 0, because therule for choosing y\ is the same as for all other values of i.) First choose some 7b with1 < jo < d+ 1 so that Aj0 does not contain a point from V(i+ 1,/) (see (b)), i.e. we choosea set in the partition that does not contain yt with 1 < t < i for which xt is adjacent tox/+i. This is possible because x,-+i has at most d neighbours. Then let A*o = A*o(V(i+ 1,/))be the subset of Aj0 consisting of those points adjacent to every yt € V(i+ 1,0- Bycondition (b) we know that \A*Q\ > |^0|(3fo)~v, where v = |V(i+ 1,01- Since v < d, weobtain \A*o\ > (N/(d + l))(3b)~d > n. With the choice of any previously unselected pointfrom A*o as yI+i, we would satisfy condition (a). However, some care must be taken toensure that condition (b) is satisfied. It is clear that we need only be concerned with those

94 J. Beck

values q > i + 1 in which x,-+i is adjacent to xq. There are at most d such values. Chooseone of these, say q. Then choose j e {1,2,...,d + 1} with j =/= j0 so that Aj does notcontain any yt e V(qJ), and let [i = \V(q,i+ 1)| = 1 + \V(qJ)\. We already know that Ajcontains a subset A* = A*(V(qJ)) with at least \Aj\ • (3b)~^+l points so that every pointin A* is adjacent to every point yt G V(qJ) (note that \V(qJ)\ = fi — 1). We now applyProperty C. Note that

N\A*\ > \Aj\ • (36)^+ 1 > - — 7 ( 3 b r M > 100blogN.

J - A I 1

It follows from Property C that less than 1006logN points of A*jo = A*k(V(i+ 1,0) areadjacent to less than 1/36 of the points in A*. Fixing q and proceeding through all valuesof j different from jo, we would then eliminate at most d • 1006 log N of the points in A*Q

as candidates for yi+\. If we then range over all possible values for q, we would eliminateat most d2 • 1006 log N of the points in A*Q. In addition, we obviously cannot select any ofthe points in A*o that have been selected previously. This eliminates less than n additionalpoints. Therefore, in order to ensure that the point y,+i can successfully be chosen fromA*Q (i.e. in order to ensure both conditions (a) and (b)), we require that

1^1 > T^T(3 b)" r f ^d2' 1006 log AT+ n.

This inequality is satisfied if N = 100d3(3b)d+l • n, and the proof of Lemma 6 is complete.

•Finally, Theorem 4 immediately follows from Lemmas 5 and 6. •

References

[I] Beck, J. (1981) Van der Waerden and Ramsey type games. Combinatorica 2 103-116.[2] Beck, J. (1982) Remarks on positional games - Part I. Acta Math. Acad. Sci. Hungarica 40

65-71.[3] Beck, J. (1985) Random graphs and positional games on the complete graph. Annals of Discrete

Math. 28 7-13.[4] Bollobas, B. (1982) Long paths in sparse random graphs. Combinatorica 2 223-228.[5] Bollobas, B. (1985) Random Graphs, Academic Press, London 447ff.[6] Chvatal, V. and Erdos, P. (1978) Biased positional games. Annals of Discrete Math. 2 221-228.[7] Chvatal, V., Rodl, V., Szemeredi, E. and Trotter, W. T. (1983) The Ramsey number of a graph

with bounded maximum degree. Journal of Combinatorial Theory Series B 34 239-243.[8] Erdos, P. and Selfridge, J. (1973) On a combinatorial game. Journal of Combinatorial Theory

Series A 14 298-301.[9] Friedman, J. and Pippenger, N. (1987) Expanding graphs contain all small trees. Combinatorica

1 71-76.[10] Komlos, J. and Szemeredi, E. (1973) Hamilton cycles in random graphs, Proc. of the Combina-

torial Colloquium in Keszthely, Hungary, 1003-1010.[II] Posa, L. (1976) Hamilton circuits in random graphs. Discrete Math. 14 359-64.[12] Szekely, L. A. (1981) On two concepts of discrepancy in a class of combinatorial games. Colloq.

Math. Soc. Jdnos Bolyai 37 "Finite and Infinite Sets" Eger, Hungary. North-Holland, 679-683.

On Oriented Embedding of the Binary Tree into theHypercube

SERGEJ L. BEZRUKOV

Fachbereich Mathematik, Freie Universitat Berlin, Arnimallee 2-6, D-14195 Berlin

We consider the oriented binary tree and the oriented hypercube. The tree edges areoriented from the root to the leaves, while the orientation of the cube edges is inducedby the direction from 0 to 1 in the coordinatewise form. The problem is to embed such atree with / levels into the oriented n-cube as an oriented subgraph, for minimal possible n.A new approach to such problems is presented, which improves the known upper boundn/l < 3/2 given by Havel [1] to n/l < 4/3 + o(l) as / -> oc.

1. Introduction

Denote by Bn the graph of the n-dimensional unit cube. The vertex set of this graph is justthe collection of all binary strings of length n, and two vertices are adjacent if and only ifthe corresponding sequences differ in one entry only. Let T be a tree. It is easily shownby induction that T is a subgraph of Bn for n sufficiently large. The general question westudy here is how to find the minimal such n, which we denote by dim(T) and call thedimension of T.

Such problems arise in computer science when dealing with multiprocessor systems [6].The exact answer depends, of course, on the structure of the tree T, rather than on itssimple numerical parameters, such as the number of vertices. If one considers trees ofbounded vertex degree, which is quite natural for practical applications, one is led toconsider the polythomic tree Tk"1. This is the rooted tree with / levels, where the root hasdegree k and all the other vertices that are not leaves have degree k + 1. The dimension ofTkJ was studied in [3] (the lower bound) and in [5] (the upper bound), where it is provedthat

/ ^ < d i m ( T ^ ) < / c - / + \ + 2 / - 2 , = 2.718.... ,1)e 2

The lower bound in (1) simply follows from the cardinalities of the sets of vertices atdistance at most / from the root, while the upper bound is constructive. Despite severalattempts, there have been no improvements of these bounds in the asymptotic sense for

96 S. L. Bezrukov

arbitrary k, I. For the binary cube it is natural to imagine that the number 2 plays animportant role. In accordance with this, let us replace one of the parameters /c, / by 2.Then it is known (see [2], [3] respectively) that

dim(Tv) = 1 + 2 and

It is interesting to notice that, although T2?/ has 2/+1 — 1 < 2/+1 vertices, the lower bounddim(T2/) > / -f 1, which follows from the cardinalities, is not attainable. Actually, in [4] itis proved that one can even find in Bl+2 two copies of T2'' joined by an edge connectingtheir roots.

Therefore, in the simplest cases when one of the parameters k, I equals 2, the problemis completely solved. Let us now consider the oriented version of this problem. We orientthe edges of Tk*1 from the root to the leaves, and the edges of Bn as follows: suppose(v,w) is an edge of Bn such that the sequences v,w differ in the iih entry, where v has 0and w has 1, then we orient this edge from v to w. Now we look for an oriented subgraphof Bn isomorphic to Tk*1. In other words, we consider embeddings of Tk"1 into Bn suchthat the /th level of TkJ is embedded into the /th level of Bf\ for i = 0,1,...,/. What is theminimal possible n now? We denote this n by d\m(Tkl).

It is easy to show that the same lower bound (1), following from the inequality

holds. Indeed there is an even better lower bound [1] for dimCT '), implied by

but it gives no improvement in the asymptotic sense. As it turns out, the upper bound(1) holds for the oriented case as well, as the construction in [5] provides an orientedembedding.

Let us again consider the case when one of the numbers /c, / equals 2. If / = 2, there isno difference between the oriented and non-oriented cases, as, without loss of generality,one may always assume that the root of Tk-2 is embedded into the origin of Bn, whichforces any embedding to be oriented. It is interesting to note that in this case the lowerbound dim(Tfc>2) > 3k/2 implied by (3) is asymptotically attained.

The goal of this paper is to study the case k = 2. So, we deal with the ordinary binarytree T2?/, which we denote Tl for brevity. For this concrete value of k one can get a betterlower bound from (2), namely

1.2938... < limdimCr')//. (4)

It is easily seen that the trivial upper bound dim(T/)// < 2 equals that given by (1). Thebest known published upper bound [1] is

\imd\m(Tl)/l<3/2. (5)/

On Oriented Embedding of the Binary Tree into the Hypercube 97

The method of [1] was to find dim(Tl) for / = 1,...,6, and in particular to prove thatT6 is embeddable into B9 (here 9/6=3/2). Following this idea, one could try to find aclever embedding of Tl° into Bn° for some lo.no, which would imply the upper bound

' < wo/fo- Here we give a table of n = n(l) = dimCT*) for small values of

/ : 1 2 3 4 5 6 7 8 9 10 11n: 2 4 5 7 8 9 11 12 13 15 16

The entries of this table for / = 1,..., 7 and / = 10 are known from [1], while the other threefollow from a more detailed analysis, and we give them here without proof. The valuesfor / = 9 and / = 11 give us an improvement on (5) as 1.444... = 13/9 < 16/11 < 3/2. Wesuspect that it is possible to embed T12 into B11 (at present we are only able to embedTn into Bls), in which case we would be able to improve (5) further to n/l < 1.416 forsufficiently large /. But to find an admissible n as / increases is very difficult, and to get agood upper bound in this way is almost hopeless.

Here we present a new approach for obtaining good bounds for the oriented embedding.Our best result is

Theorem 1. l im,^dim(Tl)/ l < 4/3 = 1.333...

If we consider this result in the light of the old techniques from [1], it becomes apparentthat, to prove Theorem 1 using the old approach, one would have to prove that T3r canbe embedded into B4r for some r > 13. To demonstrate this, we computed the functionn(l) defined by (3) for / = 1,...,39 and found that the ratio n(l)/l reaches 4/3 for the firsttime just when / = 39.

Let us mention again that, for / = 1,..., 11, dim(T/) equals the lower bound given by(3). Moreover, dim(T/c'/) for k = 1 or / = 1 is also equal to the lower bound implied bythe cardinalities. So as yet, there are no examples where dim(T/) is not determined by thebound (3).

Conjecture 1. dim(T') is determined asymptotically by the inequality (2) as I —> oo.

Conjecture 2. d\m(Tkl) ~ ^ as k,l -» oo.

2. The new approach

Denote by Tj (i = 0,..., /) the ilh level of the tree Tl (i.e., the collection of all its verticesat distance / from the root) and by B" (i = 0,..., n) the ith level of Bn (i.e., the collection ofall vertices corresponding to sequences with exactly i ones).

We have the trivial upper bound dim(T/)// < 2, and thus we may, and shall, assumethroughout that Tj is embedded above the middle level of Bn. Starting from an embeddingof Tl into Bn, let us try to embed T/J/ using as few additional dimensions as possible. Itis clear that we can always succeed using two additional dimensions. The problem is totry to use just one, as we believe in the following conjecture.

98 S. L. Bezrukov

Conjecture 3. dim(T/+1) > &\m{Tl) for all I > 1.

It is possible to use only one additional dimension for Tl+l if there exists a matchingbetween the image of Tj in B" (which we also denote by Tj) and Bf+{. For example T2

may be embedded into B4 with the required matching, which implies dim(T3) = 5. Nowdim(T4) > 7 simply follows from the cardinalities, and our knowledge about dim(r3)proves dim(T4) = 7 immediately. When can one guarantee the existence of such amatching?

Let A c Bg9 and x be a given integer. Define an x-partition of A to be a partition of Ainto s parts At with \At\ < x (i = l,...,s) such that there is a set MX(A) = {at : i = l,...,s}of distinct vertices of B£+i with a, adjacent to all vertices of At (/ = l,...,s). Call such aset MX(A) a covering set for the x-partition. In particular, if x = 1, a covering set for a1-partition defines a matching between A and ££+1.

If there is an embedding of the tree T* into Bn in such a way that Tj has an x-partition,we write Tl ^>x B

n. The arguments above lead us to the following result.

Proposition 1. //' Tl ->i Bn then T/+1 ->2 £'I+1.

Proof. Embed Tl into the subcube xn+i = 0 in such a way that it has a 1-partition withcovering set A = Mi(T/). Set 5 = 7r(T/) and C = 7r(X), where ;r is the projection onto thesubcube xn+i = 1. Now embed T/^1 into A U B in the obvious way. It is clear that 7/+/has a 2-partition with covering set C. •

Unfortunately there are examples showing that it is impossible to guarantee that Tjhas a 1-partition in general, even if \Tj\ < \B£+{\. So, the matchings approach does notpromise too much, but it is the first step towards more general constructions.

Proposition 2.

(a) If Tl 2 Bn^ and Tk 2 Bn\ then TM ^{ Bn'+n\

(b) If Tl ->! Bn' and Tk >{ Bn\ then Tl+M ^2 Bn'+n\

Proof.(a) First we build an embedding of Tl into the subcube B\ of Bm+n2 based on the first

n\ coordinates, such that Tj has a 2-partition. Now for each vertex vt e Tj we considerthe subcube B\ based on the last n2 coordinates, and embed Tk in each such subcube sothat Tj{ has a 2-partition. Thus we get an embedding of Tl+k into Bni+n2. Here we meanthat the various embeddings of Tk are isomorphic.

To see that TJ^ has a 1-partition, we refer to Figure 1. In this picture we represent by

a,b and c,d vertices of Tl+k in the subcubes B[ and B{ respectively, such that

(1) these pairs are in the same parts of the second 2-partition, and(2) the vertex of Tl that is the root of the tree containing a and b is in the same part of

the first 2-partition as the vertex corresponding to c and d.

Thus from the embedding of Tk into B\ and BJ2, we deduce that there are vertices

e e B\ and / e BJ2 that cover the vertices a,b and c,d, respectively. Similarly, there

exists at least one vertex g and h that covers a, c and b, d, respectively. More exactly, theedges (a,g), (b,/z), (c,g), (d,/z) have directions of edges of the subcube B\9 while the edges(a,e), (b,e) are in B\ and (c,/), (d,/) are in B{. The required matching between T}+£ and

^ is depicted by the thicker lines.

(b) The proof is similar. •

Corollary 1. / / Tl° ^>2 Bn° then dim(Tl)/l < no/(lo + 1/3) + o(l) as I -> oc.

Proof. We have T2l° ^>\ B2n° by Proposition 2a. Now we apply Proposition 2b withk = I = 2/o and get T4/o+1 -^2 #4n°- Therefore each time the cube dimension is multipliedby 4, the height of the tree we can embed increases by a multiple of slightly more than4. More precisely, if the sequences /, and n, are defined by /, = 4/,_i + 1 and n, = 4M/_I(i = 1,...), then we have Tu ^2 Bn' for each i, and nt = no(/,- + l/3)/(/0 + 1/3), so

Wl.//£. = no/(/o + 1/3) + o(l) as i - • oo. The result follows. •

A more detailed analysis of the proof that dim(T6) = 9 shows that T6 ^ 2 #9> whichgives the upper bound limi^^dimiT^/l < 9/(6 + 1/3) « 1.421, but some work is stillrequired. Now we present, as the second elementary application of our approach, a simpleproof of the bound (5).

Proposition 3. / / Tl ->2 Bn then T/+2 ^ 2 Bn+\

Proof. First embed the tree Tl into Bn for some w such that Tj has a 2-partition withcovering set MiiTj). This is possible by Proposition 1. Now we use this embedding, andits associated 2-partition and covering set, to embed the two extra levels of the binarytree using only three extra dimensions. So, we build the 3-cube growing from each vertexof our n-cube, in particular from each vertex of Tj. For each set {ui9Vi} in our 2-partition,let w, e Bf+l be the corresponding vertex in the covering set M2(Tj). This situation isdepicted in Figure 2a, where the rectangles represent the 3-cubes growing from verticesM/,1;/. The corresponding vertices of these 3-cubes are connected, as shown in Figure 2.

100 S. L. Bezrukov

Bi B2

a.

Figure 2

We now embed two copies of T2 rooted in w,, vt into this structure, which will provide anembedding of T/ + 2 into Bn+3.

Our embedding scheme is shown in Figure 2b, where we draw the edges of the treesonly. Incomplete lines indicate a covering scheme, demonstrating that the embedding hasa 2-partition. •

Now the upper bound (5) follows immediately. We start with an embedding of T3 intoB5 with a 2-partition, the existence of which was mentioned earlier, and apply Proposition3. On the fth step of this process, we obtain an embedding of T3+2i into B5+3i, whichimplies the upper bound lim/^oodimCT')// < lim,--x(3i + 5)/(2i + 3) = 3/2.

What is important in our approach is that, given an embedding of Tl into Bn, weconstruct an embedding of Tl+€ into Bn+S, even though dim(Te) > S, which gives thebound lim/^oodim(T/)/^ ^ <V - We achieve this by using some additional informationabout the initial embedding, in this case the existence of a 2-partition.

The second important thing is that, in the proof of Proposition 3, it makes no differencewhich initial embedding we start with. The only thing one needs is a 2-partition, whichis in fact easy to guarantee. Indeed, embed Tl into any admissible Bn. Now increase thedimension of the hypercube by 1, adding the subcube xM+i = 1. Then each vertex of Tfhas a neighbour in this subcube, so Tf has a 1-partition in Bn+l.

Let us finally mention that the proof of the upper bound (5) may be further simplifiedby using the following result.

Proposition 4. / / Tl - M Bn, then Tl+2 ^{ BM


/

Bi

•s>

B,

/

Be

Bs

\

B7

v3 v4

a.

Figure 3

To prove this, one has, in fact, to show that T2 may be embedded into B4 so that T22 has

a 1-partition, which fact we have already mentioned above.

3. Towards better upper bounds

Our general aim is to get a rational upper bound for lim/^oodim(r/)/ '5 necessarilyexceeding 1.29, using constructions involving low-dimensional cubes only. It seems to beimpossible to obtain fully satisfactory results using just x-partitions, with the same xbefore and after the addition of extra levels. So we need some deeper insight.

For A c Bf, t > 2 and a sequence (xi,...,xr) of positive integers, we say that A canbe (xi,...,xt)-partitioned if there are sets Mo,Mi,...,Mt such that Mo = A, M, c Bf+i foreach /, and, for each i, there is an xrpartition of M,_i with covering set M,-. If thereexists an embedding of Tl into Bn such that Tj can be (xi,...,xr)-partitioned, we writeTl -*,,...,*, Bn.

n+4Proposition 5. / / Tl ^2,i Bn then TM ^2,3 B

Proof. Now we have to embed four copies of T3, rooted in vertices v\,...,v^ into thestructure depicted in Figure 3a, where each box represents the 4-cube. The graph of the4-cube is as shown in Figure 3b; for convenience we shall normally use the restrictedimage of it shown in Figure 3c. In this image we show just the vertices of B4, in the sameorder from left to right as they are shown in Figure 3b.The embedding we use here is shown in Figure 4, where, for simplicity, only the subcubesB\,B2,B?, and £4 are shown, without the edges connecting them. The two copies of T3

have their roots in vertices v\,V2.Now the top vertices of B\ and £2, the vertices of the 3-d level of £3 and the vertices

of the second level of £4 shown by larger solid circles in Figure 4 form a covering set

102 S. L. Bezrukov

Figure 4

Mi for a 2-partition of this piece of T/+33, and the vertices labelled by asterisks form a

covering set for a 3-partition of Mi. This covering scheme is also presented in Figure 4.Let us look further at the subcube BA. In Figure 4 only two vertices a, b of T/+3 n £4

are depicted. .They correspond to the vertices (0010) and (0100) of BA respectively (thecommas in vectors are omitted). The vertices of BA that cover them correspond to (0110)and (1100) respectively. But we also have to embed four vertices coming from the subcubeB5. In order for all these eight vertices to be distinct, we first embed the two other copiesof T3 into B5,B6,B7, and after that use the isometric transformation of these subcubesdefined by the permutation (3412) of coordinates. This permutation transforms the fourmentioned vertices into (1000), (0001), (0011), and (1001) respectively, which guaranteesthe correct embedding.

For future reference, note that there are two vertices in the second level of BA, namely(0101) and (1010), that are not in M\. They are shown as empty circles in Figure 4. TheHamming distance between these two vertices is 4. •

Using similar techniques one could prove the following properties.

Proposition 6.

(a) If Tl -^2,3 Bn then T / + 2 2 , 2(b) If Tl ->2,2,3 Bn then T / + 3 ->2,2,4(c) If Tl ^2,2,4 Bn then Tl+3 ->2,3,3(d) If Tl -^2,3,3 Bn then Tl+2 ->2,2,3

n+3 .

s2

c5

c2 c3 C4

S12

Figure 5

We do not use these properties for the proof of Theorem 1, but believe that they areuseful for further research. One could combine them with some others to get new upperbounds. In particular, Propositions 5, 6a and 6b - 6d, respectively, imply the followingresults.

Corollary

(a) lim/_(b) lim/_

2.-•

>QQ Qll l l^ .

»oQ Ql l l l i .

Tl)/l < 7/5 = 1Tl)/l< 11/8 =

.4;1.375

Our main result, Theorem 1, is an immediate consequence of the following result.

Proposition 7. / / Tl ->2,2,3,4 Bn then TM ~>2,2,3,4 Bn+\

Proof. Now we deal with the structure depicted in Figure 5, where C\,...,Cs are 4-cubesand Si,...,Si2 are the structures, consisting of seven 4-cubes, depicted in Figure 3a. Each4-cube C2, C3,C4 is connected with three structures S, (/ = 4,..., 12), as shown in Figure 5for the cube C\. We use the image of B4 shown in Figure 3b, and again reduce it to thatshown in Figure 3c.

We start with an embedding of Tl into Bn such that T\ can be (2,2,3,4)-partitioned,and now embed the three extra levels of our tree by embedding T3 into each structure 5,,as described in the proof of Proposition 5. The role of the remaining 4-cubes in Figure 5is to guarantee that T/ 3

3 can be (2,2,3,4)-partitioned. It was mentioned above that twovertices at distance 4 are free in the subcube B4 in each structure Si. Using isometrictransformations of the structures S,-, we can establish these free vertices to be just (0011)and (1100) in the structure St with i = 0(mod 3), and the vertices (0101),(1010) and(0110),(1001) in the structures St with i = 1 (mod 3) and i = 2 (mod 3), respectively. Thefree vertices are shown as empty circles in the bottom 4-cubes in Figure 6. These bottom4-cubes correspond to the subcubes £4 of the structures St (cf. Figure 3a).

104 S. L. Bezrukov

• • • • • • • •

A a.• • • •c, /

o • • • • o • o •

1— <

. 0 .

1 •

>

^ ^

• • 00• •

c

0 • • *

\ /

• 0

^ ^

• 0 • » o • • • o o » •

Figure 6

To prove that our embedding of TJ^ c a n be (2,2,3,4)-partitioned, we need to constructsets Mi,M2,M3 and M4 such that Mi is a covering set for a 2-partition of this section ofT/+3

3, M2 is a covering set for a 2-partition of Mi, M3 is a covering set for a 3-partitionof M2, and finally M4 is a covering set for a 4-partition of M3.

Mi: We use just the same construction for the set Mi as in the proof of Proposition 5.This set is shown in Figure 4 by the large solid circles.

M2: We take the top vertices of the subcube £3 to cover the top vertices of the subcubes2?i,2?2 (cf Figure 4) in each structure S,-, and use all the four vertices in the 3-d levelof the subcube B4 to cover the vertices of the 3-d levels of subcubes £3, £5. Now allthat remains to be done is to cover the four solid vertices in the second level of thesubcube £4 in each structure S,- (see Figure 4) by the six vertices in the second levelof the subcubes C, (i = 1,...,4) (cf Figure 5). The covering scheme is explained inFigure 7a. In this figure we represent the six vertices of C\ by the top block and thesecond levels of B4 in S\,S2,ST, by the three bottom blocks (for other C, and 5,-, theprinciple is the same). Each vertex of the top block is incident (in Bn) to the threecorresponding vertices of the bottom blocks, but we have to choose only two edgesto cover all the solid vertices. Now remove the edges shown in Figure 7a. Then theremaining edges between the top and bottom blocks form the required covering.

M3: As constructed above, M2 consists of the top vertices of the subcubes £3, the secondlevels of the subcubes £4 (cf Figure 4) and the second levels of the subcubes C\,..., C4.Now we have to cover all these vertices by the top vertices of the subcubes £4, thethird levels of Ci,...,C4 and the second level of C5, in such a way that no vertex ismatched to more than three from M2.

Now consider the subcube C\ and the subcubes B4 in the structures Si,S2,S3. We


(a) (b)

Figure 7

(c)

cover the top vertices of the subcubes £3, £5 from the top vertex of the subcubes £4

in each St (cf. Figure 4) and use the third edge to cover one of the three verticesin the 3-d levels of #4, as shown in Figure 6 (these three vertices are depicted bysmall circles). In order to cover the remaining three vertices of B4's (depicted by largecircles in Figure 6), we use the 3-d level of C\ and the covering scheme as shown inFigure 7b. In this picture the leftmost (large) vertex has degree three, while all theother vertices have degree two. We use the remaining three edges incident to thesevertices to cover some three vertices in the second level of C\, as shown in Figure 6.

Therefore, the vertex of each subcube B4 represented by the largest circle (seeFigure 6) in the structures St (i = 1,2,3) plays a particular role. In other structures,we use a similar principle, and the corresponding vertices of the B4S are representedin Figure 6 by large circles. Of course, one then has to correct the covering schemein the subcubes C2, C3,C4, which we do in accordance with Figure 6.

Now consider the 4-cubes Ci,...,C4 in Figure 6, and notice that the two rightmostvertices in their second levels are already covered from the 3-d levels, and just oneof the other four vertices is also covered. In order to cover the remaining threevertices in each subcube Q,...,C4, we use the leftmost four vertices of C5 and thecovering graph depicted in Figure 7c. Each vertex of the top block is incident to thecorresponding vertex in each bottom block, and removing the depicted edges we getthe required covering.

M4: Finally, we construct M4 from the top vertices of the subcubes C\,..., C4 and the thirdlevel of C5. Indeed, cover the top vertices of the subcubes B4 in each structure Sjfrom the top vertex of the corresponding subcube C,- (i = 1,...,4). Thus we have usedthree edges for each C,-. The 4th edge is used to cover the large vertices of the 3-dlevel in each Q, as shown at the top of Figure 6. The remaining (small) vertices ofQ (i = 1,...,4) are covered from the 3-d level of C5 using three edges, with a coveringscheme similar to that in Figure 7c. Now each vertex of the third level of C5 is usedto cover some three vertices of M3, and we use the 4th edge incident to each of themto cover the leftmost four vertices in the second level of C5 (see Figure 6).

•

We hope that by using similar techniques, it will be possible to operate with larger graphs,

106 5. L. Bezrukov

and construct an embedding of 10 extra levels in 13 extra dimensions and finally prove

the following.

Conjecture 4. l im/^dimfr')/ / < 13/10 = 1.30.

References

[1] Havel, I. (1982) Embedding the directed dichotomic tree into the rc-cube. Rostock. Math. Kolloq.21 39-45.

[2] Havel, I. and Liebl, P. (1972) O vnoreni dichotomickeho stromu do crychle (Czech, Englishsummary). Cas. Pest. Mat. 97 201-205.

[3] Havel, I. and Liebl, P. (1973) Embedding the polythomic tree into the rc-cube. Cas. Pest. Mat.98 307-314.

[4] Nebesky, L. (1974) On cubes and dichotomic trees. Cas. Pest. Mat. 99 164-167.[5] Olle, F. (1972) M. Sci. Thesis, Math. Inst., Prague.[6] Wagner, A. S. (1987) Embedding trees in the hypercube, Technical Report 204/87, Dept. of

Computer Science, University of Toronto.

Potential Theory on Distance-Regular Graphs

NORMAN L. BIGGS

London School of Economics, Houghton St., London WC2A 2AE

A graph may be regarded as an electrical network in which each edge has unit resistance.We obtain explicit formulae for the effective resistance of the network when a current entersat one vertex and leaves at another in the distance-regular case. A well-known link withrandom walks motivates a conjecture about the maximum effective resistance. Argumentsare given that point to the truth of the conjecture for all known distance-regular graphs.

1. Introduction

We shall be concerned with a graph G regarded as an electrical network in which eachedge has resistance 1. A well-known result due to R.M. Foster [6] (see also [3, p.41]and [9]) asserts that if G has n vertices and m edges, the effective resistance betweenadjacent vertices is r\ — (n— l)/m, provided that all edges are equivalent under the actionof the automorphism group. In this paper I shall obtain formulae for r,-, the effectiveresistance between vertices at distance i9 for i > 2, provided G is distance-transitive (DT).With hindsight, it will be clear that the same formulae hold if we assume only that G isdistance-regular (DR). The case i = 2 was also studied by Foster [7].

Another well-known fact is that the electrical problem can be regarded as a caseof solving Laplace's equation on the graph. As explained in the elegant little book byDoyle and Snell [5], this leads to significant connections with other subjects, in particularthe theory of random walks. In that context, the solution to the problem of effectiveresistances has a simple interpretation in terms of 'hitting times'. Our results can also beapplied to questions about the 'cover time', that is, the expected number of steps requiredto visit all the vertices. In particular, it appears that for all known DR graphs (except thecycle graphs), the cover time is O(n\ogn).

For the sake of completeness, we gather together here the basic notation and terminologyfor DT and DR graphs, which will be used in the rest of the paper. The author's book[2], or the standard text of Brouwer, Cohen and Neumaier [4], with its 800 references,should be consulted for details.

108 N. L. Biggs

We denote the diameter of a connected graph by d, and the distance between verticesv and vv by d(v, vv). A connected graph G is distance-transitive if, for any vertices v, vv, x, ysatisfying d(v, vv) = d(x,y), there is an automorphism of G that takes v to x and vv to y.Given integers h,i such that 0 < h, i < d, and vertices v,w, define

Shi(v,w) = \{x G KG | d(x, v) = h and d(x, vv) = i}|.

In a distance-transitive graph, the numbers Shi(v,w) depend on the distance d(v, vv), noton v and vv, so we can define the intersection numbers

Shij = Shi(v,w), where d(v,vv) = 7 , (hjje {0, l,. . . ,d}).

Consider the intersection numbers with /z = 1. For a fixed 7, SUJ is the number ofvertices x such that x is adjacent to v and d(x, vv) = i, given that d(v, vv) = 7. The triangleinequality for the distance function implies that SUJ = 0 unless i = j — l,y, or j + 1. Forthe intersection numbers SUJ that are not identically zero, we use the notation

Cj — sij-ij cij = sijji bj = sij+iji (0 < j < J),

noting that Co and bd are undefined.The numbers c;, aj, bj have the following simple interpretation. For any vertex v of G

let

Gi(v) = {x G KG I d(x, 1?) = i}, (0 < i < d).

It is clear that the sets {v} = Go(v), G\(v)..., Gd(v) form a partition of VG. Given a vertex1; and a vertex x in Gj(v), this vertex is adjacent to Cj vertices in Gj-\(v), aj vertices inGj(v), and bj vertices in Gj+\(v). These numbers are independent of v and x, providedthat d(v,x) = j .

The intersection array of a distance-transitive graph G is

C\ . . . C y . . . C

ao a\ ... ay ... a£>o fci • • • bj ...

We observe that a distance-transitive graph is vertex-transitive, and consequently reg-ular, of degree k say. Clearly, we have bo = k and ao = 0,c\ = 1. Further, since eachcolumn of the intersection array sums to fe, if we are given the first and third rows, wecan calculate the middle row. Thus it is convenient to use the alternative notation

A distance-regular graph is a graph that has the combinatorial regularity implied bydistance-transitivity, without (possibly) the prescribed automorphisms. Explicitly, it is aconnected graph such that for some positive integers d and k the following holds: thereare natural numbers bo = k,b\,...,bd-uC\ = 1,C2,...,Q, such that for each pair (v,w) ofvertices satisfying d(v,w) = j we have

1 the number of vertices in Gj-\(v) adjacent to vv is Cj (1 < j < d);2 the number of vertices in Gj+\(v) adjacent to vv is bj (0 < j < d — 1).

Clearly, a distance-transitive graph is distance-regular, but the converse is false. The

Potential Theory on Distance-Regular Graphs 109

cycle graphs are the only distance-regular graphs with degree k = 2; although they aretrivial, they are also somewhat anomalous. We exclude them by assuming that k > 3always.

In the algebraic theory of DR graphs, it is shown that there is a representation in whichthe adjacency matrix A is represented by a tridiagonal matrix B whose entries are theelements of the intersection array:

/ 0 1 0 . . . 0 0 \k a{ c2 . . . 0 00 bx a2 . . . 0 0

0 0 0 . . . ad_x cd

\ 0 0 0 . . . bd-x ad

Our main result will be formulated in terms of this matrix.

2. Calculation of potentials

In this section we shall obtain formulae for the potentials of the vertices of a DR graph,when a current enters at one vertex and leaves at an adjacent one.

It is convenient to begin with a DT graph. A particular consequence of this assumptionis that all pairs of adjacent vertices are equivalent under the action of the automorphismgroup. Choose an adjacent pair (v, w) once and for all. With respect to this pair weconstruct the distance distribution diagram, or DDD. That is, we define

Vt = {x\d(x9v) = i,d(x9w) = i+l};Wt = {x\d(x,v) = i+ld(x,xv) = i};Zt = {x\d(x,v)=d(x9w) = i}.

In terms of the intersection numbers we have

sa+n = \Vt\ = \Wt\ = \Zt\ =sin

It should be noted that the numbers of edges joining sets of vertices in the DDD arenot completely determined by the intersection array. For example, the number of edgeswith one end in Vt and one end in W\ is not determined. However, we do have someinformation.

Lemma 1. For any vertex x in F, and any set S of vertices, let S(x) denote the set of edgesjoining x to vertices in S. Let |Z/(x)| = a, | Wz(x)| = /?, |Z,-+i(x)| = y. Then we have

a + 2j8 + 7 = (bt - bM) + (ci+1 - a).

Proof. For x e Vt, we have d(v,x) = i. The number of edges joining x to vertices y such

110 N. L. Biggs

that d(v,y) = i — 1 is, by definition, c,-. The construction of the DDD ensures that allvertices adjacent to x and distance i — 1 from v are in K/_i, hence the result.

For xf e Wi, we have d(v,xf) = i + 1. The number of edges joining xr to vertices /such that d(v,y') = / + 2 is, by definition bi+\. The construction of the DDD ensures thatall vertices adjacent to x' and distant i + 2 from v are in VK,-+i, hence |W,-+i(x')| = fo/+i.Using the symmetry with respect to v and w, we conclude that |F,-+i(x)| = bi+\.

For x e V[ we have d(x, w) = i + 1 , and the vertices adjacent to x that are at distance/ from w are in K/_i, Z,, and Wy. So we get

Similarly, since d(x, t?) = /, and the vertices adjacent to x and distance / + 1 from v are inWi, Z,-+i, and K,-+i, we get

£ + 7 + ft/+i - fc/.

Adding these two equations gives the result. •

As we shall now demonstrate, the quantities evaluated in Lemma 1 are sufficient todetermine the potential of any vertex when a current J enters at v and leaves at vv, giventhat each edge has unit conductance. Describing the state of the network by means of apotential function automatically ensures that 'KirchhofTs voltage law' (that the potentialdrop around any cycle is zero) is satisfied. We assume that all vertices in V[ will be at thesame potential, and likewise for W\ and Z,-. The justification for this assumption is that itenables us to solve the equations expressing 'Kirchhoff's current law' (that the net currentat any vertex is zero). Since it is known that there is a unique solution to KirchhofTsequations (see for example [8]), any assumption producing a result must be valid. In thesame spirit, we use the symmetry with respect to v and w, and take the potentials on K;,Wi9 Zi, to be (/)/,—(/>/, 0, respectively.

Lemma 2. The potentials 0,- satisfy the equations

Ci^i-x-b^i = J-k(j)0 (1 d-\ = J — kfo.

Proof. We use the standard technique of replacing a set X of vertices that are at thesame potential by a single vertex x. Given two such sets X and Y, the edges joiningthem become parallel edges joining x to y, and can be replaced by a single edge whoseconductance is the sum of the conductances. Since we are assuming that each edge hasconductance 1, this means that the conductance of the edge xy is equal to the number ofedges joining X to Y. In particular, if each vertex in X is joined to the same number / ofvertices in Y, the conductance of xy is l\X\.

This technique enables us to regard the DDD as a network equivalent to the givengraph. A typical vertex vt in this network is obtained by identifying all vertices inVi(0 <i < d — \), and we can define w,- and z-x similarly. The formulae obtained inLemma 1 can now be interpreted as results about the conductances of edges in the

equivalent network. For example, the conductance of the edge joining vt to vt-i is C;| K;|,corresponding to the fact that |K,_i(x)| = ct.

Consider first the vertex v in G. The numbers of edges from v to V\, Z\, Wo are b\,fli, 1, respectively. Since |Fo| = 1, in the equivalent network based on the DDD, thesenumbers are the conductances of the edges joining VQ to vu zi> wo, respectively. ApplyingKirchhofTs current law at v0, we get

J = fei(0o - fa) + ai W>o ~ 0) + l((/>o - ( -

Since c\ + a\ + b\ = k and c\ = 1, this can be written as

Ci0o — b\(j)i = J — k(j)Q.

Similarly, applying KirchhofT's current law at vt, we get

cAfa-x - fa) = Oi(fa - 0) + P(fa - (-0/)) + yfa - 0) + bi+{(fa - 0I+i),

and using the formula for a + 2/? + y, this reduces to

This is valid for 1 < i < d — 2, so it follows that the value of c/0,-_i — bi<pi is constant for1 d-\ = Cd-\4>d-2 — bd-\(f)d-i = J — kfa. D

When J is given, the d equations obtained in Lemma 2 determine (/>o,</>i,...,</>d-i. Theresults come out more smoothly if we do a little reorganisation first.

As described in the Introduction, the vertices of G are arranged in disjoint subsets Gt(v)(0 < i < d), according to distance from a given vertex v. If we write /c, = |G;(i?)|, the totalnumber of vertices in G is

n = 1 +fci +fc2 + ... + fcd.

Also, by counting the edges joining Gt-\(v) to Gi(v) in two different ways, we get arecursion for the sequence (fc,-):

/co = l, diet = bi-yki-i (\<i<d).

Theorem A. / / G has n vertices and m edges, and the current is J = 2m, then the potentials

are determined recursively by the equations

0o = n — 1, bi4>i = Ci(/)i-\ — k (1 < i < d — 1).

These equations have the explicit solution

112 N. L. Biggs

Proof. This is proved by elementary algebra, starting from Lemma 2. •

Examples

(i) The dodecahedron is a distance-transitive graph with n = 20 vertices, degree k = 3,and intersection array

r 1 1 1 2

I 0 0 1 1 0 0

[ 3 2 1 1 1

Here the potentials are:

00 = 19, 01 = 8 , 02 = 5, 03 = 2 , 04 — 1.

(ii) Similarly, for the cubic DT graph with 102 vertices [4, pp.403-405] and intersectionarray {3,2,2,2,1,1,1; 1,1,1,1,1,1,3}, we get

00 = 101, 0 i = 4 9 , 02 = 23, 03 = 10, 04 = 7, 0 5 = 4 , 06 = 1.

We shall refer to this example later.(iii) Let Suz be Suzuki's simple group of order 213.37.52.7.11.13. As described in [4, pp.410-

412], the group Suz.2 is the automorphism group of a distance-transitive graph with22880 vertices and intersection array {280,243,144,10; 1,8,90,280}. The potentialsare:

0o = 22879, 0i = 93, 02 = 29/9, 03 = 1.

At this point we can extend our results to distance-regular graphs. Since the equationsfor the potentials involve only the parameters occuring in the intersection array, it is clearthat these equations determine 'potentials' in the DR case. These 'potentials' provide asolution to the network equations and, as has been pointed out, it is known that there isa unique solution. Thus we have the solution to the DR case.

It is intuitively 'obvious' that the potentials 0/ form a strictly decreasing sequence.However, we should remember that some things about the flow of electricity are only'obvious' to those who mistakenly claim to understand what electricity really is (see[5, p. 70]). In fact, the proof that 0/ > 0;+i depends upon a property of the intersectionarray which we have not yet mentioned explicitly.

Lemma 3. The intersection numbers of a DR graph satisfy

1 = c\ < ci < •.. < Cd; k = bo >b\ > ... > bd-\.

Proof. This is a standard result from the early days of DR graph theory [2, p. 135]. Italso follows from the proof of Lemma 2, specifically the equations

Ci + a + fi = Cj+i, p + y + bi+i = bt. •

Theorem B. The sequence (0/) of potentials is strictly decreasing.

Proof. In Theorem A we obtained explicit formulae for (pt, the second of which can bewritten in the form

A if * _L ^ + 1 i .bi+ibi+2...bd(pi = k [ 1 h . . . H

\ Ci+1 C;+1 Ci+2 Ci+1 Ci+2 - • • Cd

This formula for 0,- is the sum of d — i terms. Comparing the yth terms in the formulaefor (pi and 0,-+i, we have

Cj+i C/+2 •. • Ci+_/ Cj+2Cj+3 . . .

by virtue of the inequalities in Lemma 3. Since (pi contains one term more than 0,-+i, wehave the strict inequality as claimed. •

A consequence of the formulae obtained above is that, for 0 < i < d — 1,

can be written as a sum of d — i monomial terms, each of weight d — i — 1. For example,when d = 4 we get

( 1 2 3 J (po = c2c3c4 + bicic4 + bib2c4 +

( 7 ) <Pi = C3C4 + b2c4

- ' •

The rule for forming the expressions for other values of d should be clear. We shall usethem in Section 4.

3. The effective resistance vector

We defined the potentials in such a way that the potential difference between v and w is2(/>o- It follows that the equivalent resistance between vertices at distance 1 is r\ = 2(po/J,and Theorem A tells us that this is (n — l)/m. Thus we have verified the formula for r\mentioned in the Introduction in the distance-regular case.

We now show that the equations obtained in Theorem A are sufficient to determine theeffective resistances r, (2 < i < d). The trick is to use the 'method of superposition'.

Lemma 4. / / a current J enters G at v, and leaves at a vertex w, at distance i + 1 from v,the potential difference between v and w, is

Proof. Let us use the term system to denote a specific distribution of currents andpotentials on a given DR graph. We have already observed that the result is true in thesystem ZQ when a current J enters at v and leaves at w. The system Zi when a current

114 N. L. Biggs

J enters at v and leaves at w\ e W\ is the 'superposition' of Io and the system II iwhen J enters at w and leaves at wi. In So the potentials at v,w,wi are 0o, — 0o, — 0irespectively, and in III they are 0i,0o, —0o respectively. Thus, in Xi the potentials are0o + 0i>0, —(0o + 00? a n d the required potential difference is 2(0o + 0i), as claimed.

More generally, the system Z,- when a current J enters at v and leaves at vv, isthe superposition of Z,-_i and the system n,- when J enters at w/_i and leaves at w,-.Suppose we make the induction hypothesis that, in S,-_i, the potentials at t\ w/_i,w; are(7,-_i, —(jj-i, 0o — c,- respectively, where 07 = 0o + . . . + 0/. Now 11/ is obtained from Zo by atranslation that takes v, w to w,-_i,w,-, and a vertex vi to i;, so the corresponding potentialsin 11/ are 0/, 0O, —0o- Adding the two sets of potentials we have the result for £,-, and thegeneral result follows by induction. •

Let us denote by r the row-vector (ro,ri,...,r</) whose ith component r, is the effectiveresistance between a pair of vertices at distance i. The results obtained above tell us howto calculate the vector r. For example, for the dodecahedron and the 102-graph (Examples(i) and (ii)) we have

r = ^(0,19,27,32,34,35) and r =1

153(0,101,150,173,183,190,194,195).

For the general case we have the following Theorem, in which the matrix B is theintersection matrix of a DR graph, and u, v are the (d + l)-dimensional row-vectors

u = (1,1,1,...,1), v = (n,0,0,...,0).

Explicitly B — /cl is the matrix

k a\ — k

0 bx

0\ 0

00

-k . .

00

000

000

cid-\ —k

bd-\ -k )

Theorem C. The effective resistance vector r is the unique solution with r$ = 0 of theequation

r(B-/cI) = -(v-u).n

Proof. We observe that k is a simple eigenvalue of B with eigenvector u, so it follows that0 is a simple eigenvalue of B — /cl with the same eigenvector. Thus the rank of this matrixis (d + 1) — 1 = d, and the general solution of the equation is of the form r = a + /u. Itfollows that there is a unique solution with ro = 0.

We can obtain a recursion for r,- as follows. By Lemma 4,

n = -(0o (1 < i < d),

so, taking ro = 0, we can write

(t>i=J-{ri+l-ri\ (0<i<d-l).

Substituting in the equations from Theorem A, where J = 2m, we get

bim(ri+i - rt) = cMrt - r/_i) + fc, (1 < i < d - 1).

Since k = 2m/n and a, = k — bt — ci9 it follows that

c/rf_i + (at - k)r{ + biri+i = -2/n (1 < i < d - 1).

These are the components of the given matrix equation, except for the equations derivedfrom the first and last columns of B — fcl, which can be verified separately. •

Using Theorem C, it is possible to obtain a formula for r in terms of the eigenvalues andeigenvectors of B, but we do not need it here. We can also use the equation r,-+i =and Theorem A. For example,

r2 = r{ + — = n + -—(n -1-k).m b\m

If the graph has no triangles, b\ = k — 1, and this reduces to

11 n(k~iy

a result obtained under slightly different assumptions by Foster [7].

4. Applications to random walks

The analogy between a flow of electric current and a random walk is a reflection ofthe fact that both can be modelled using Laplace's equation. The random walk processrelevant here is the one in which each step consists of a move from a vertex to an adjacentone, with each possible move being equiprobable. In our case the graph is /c-regular andso the probability of a given move is l//c.

Given two vertices x and v, let qxv denote the expected number of steps required toreach v starting from x, and, in the case of a distance-regular graph, let qt be the valueof qxv when d(v9 x) = i. In terms of the parameters c,-, a*, bu the probability that the firststep from x is a move to a vertex at distance i— 1, i,i' + 1 from v is, respectively

~k9 T9 I 'Since the expected number of steps to reach v is one more than the expected number ofsteps after the first step has been taken, we have the following equation for qt:

1

Multiplying by k/m and rearranging, this gives

) ( • ) ( ) ' ( )

116 N. L. Biggs

Now k/m = 2/n, so we see that qt/m satisfies the same recursion as the effective resistance.The initial conditions are the same (ro and qo are both zero), so we conclude that qt = rar,,for i = 1,2,..., d. Of course, this is a particular case of a relationship that holds for graphsin general [5].

In matrix form, we have the equation

q(B - fcl) = fc(v - u)

for the vector q = (0,qi,...,qd), and in terms of the potentials calculated in Theorem A,

qt = 0o + 0i + . . . + 0i-i.

A related quantity of some interest in the theory of random walks on graphs is thecover time CQ of a graph G. This is the expected number of steps required to visit allthe vertices, starting from a given vertex. For our purposes all the vertices are equivalent,so the starting vertex is irrelevant. A survey of work on the cover time has been givenby Aldous: in particular, we note the result [1, p.88] that if the maximum of qxv is O(n),the cover time is O(n\ogn). In the case of a DR graph of diameter d it is clear that themaximum is qj = mrd. Thus the problem of the cover time leads us to consider upperbounds for r<*.

The formula obtained in Lemma 4, together with the fact that the sequence (</>,) isstrictly decreasing, establishes that rd is less than dr\. However, it seems that a muchstronger result may hold.

Conjecture 1. For any DR graph rd < Ar\, where A is a constant independent of the diameterd.

It is even possible that A = 2. By virtue of the results quoted above, the conjecturewould imply that for any DR graph the cover time is O(n\ogn), and that the maximumhitting time qd is not greater than 2(n — 1).

The conjecture is partly based on failure to find a counter-example among the largenumber of graphs and families of graphs listed in [4]. The 'worst' example seems to bethe cubic graph with 102 vertices, Example (ii), for which d = 7 and r-j = (195/101)n.This graph is very exceptional, being one of only three known distance-regular graphs forwhich the diameter exceeds twice the degree. The partial proof of the conjecture givenbelow points very strongly to the fact that a counter-example would have to be even moreexceptional than the 102-graph.

The following is a sketch of how it could be verified that rd < 2r\ for all known DRgraphs. This is instructive as far as it goes, but it could by no stretch of the imagination bedescribed as elegant. Even if it were proved that the list of DR graphs in [4] is essentiallycomplete (for d > 6 would be enough), this proof would certainly not find a place in TheErdos Book of ideal proofs.

We begin by proving that r < 2r\ for d < 5. The cases d = 1 and d = 2 are trivial, andd = 3 is easy using the technique described below. In view of the relationship betweenthe effective resistances and the potentials, it is sufficient to prove that

00 > 01 + 0 2 + ... + <t>d-\-

We shall use the formulae for potentials obtained in Section 2, and some of the knownparameter restrictions for DR graphs.

Lemma 5. The parameters of a DR graph with k > 2 satisfy

1 C[ < bj whenever i + j < d,

2 b{>2ifd>3.

Proof. See [4, p. 133 and p. 172]. •

Theorem D. For any distance-regular graph with diameter 4, 0o > 0i + 02 + 03-

Proof. Using the formulae displayed at the end of Section 2 (and remembering thatc\ = 1 always), we have

2,3 4 (0o - 0 i - 02 ~ h) = c2c3c4 + b{c3c4 + bxb2c4 + b{b2b3

—c3c4 — b2c4 — b2b3 — c2c4 — c2b3 — c2c3.

The terms can be collected as follows:

{(61 - l)b2 - c2}(b3 + c4) + c2c3{c4 - 1) + (b{ - l)c3c4.

Using both parts of Lemma 5, we have (b\ — \)b2 — c2 > b2 — c2 > 0 . Since c4 > 1 andb\ > 1, it follows that the entire expression is strictly positive. •

Theorem E. For any distance-regular graph with diameter 5, </>o > </>i + </>2 + 03 + 04-

Proof. Here we have

( J ((/>o — 01 - 02 - 03 - 04)

= C2C3C4C5 + bic3c4cs + b\b2c4cs + b\b2b3cs 4- b\b2b3b4—C3C4C5 — b2c4cs —

—c2c4c$ —

-c2c3c5 - c2c3b4 - c2c3c4.

Collecting up the terms on the same lines as in the previous proof we get

{(fti - \)b2 - c2}(c4c5 + b3c5 + b3b4)+c3(c2c4c5 + fric4c5 - C4C5 - c2c5 - c2b4 - c2c4).

If b\ > 3, then (fri — \)b2 — c2> b2. Using this result in the first line, and the monotonicityconditions (Lemma 4), we can find a positive term to dominate each negative one, withone positive term to spare.

If b\ = 2, we have c2 < b2 < bu so there are three cases: {b2,c2) = (2,2),(2,1),(1,1).Using the methods indicated above, we find just two 'intersection arrays' for which theresult does not hold:

{fc, 2,1,1,1; 1,1,1,1,1}, {fc,2,2,2,2; 1,2,2,2,2}.

Elementary arguments show that these arrays cannot be realised. In the first case, pick

118 N. L. Biggs

vertices v,x,y such that d(v,x) = 2,d(v,y) = 3, and d(x,y) = 1. Because fe2 = C3 = 1, xand y can have no common neighbours. This means that a\ = 0, so k = c\ + a\ + fri = 3.But the list of DR graphs with k = 3 and d = 5 is known to be complete, and it containsno such graph. Similar arguments work for the other array. •

It is clear that a little more could be squeezed out of these proofs. Probably one couldshow by similar methods that among all DR graphs with d < 7 the 102-graph has themaximum value of r^/ri. If this were done, the enterprise could be completed as follows.

The known DR graphs with d > 8 comprise just two 'sporadic' graphs (for which theconjecture can be verified immediately), and a number of infinite families. The familiescan be arranged into three (not mutually exclusive) classes:

— families with 'classical parameters';— partition graphs;— regular near-polygons.

All these families are characterised by the form of their intersection array, and can bedealt with by using the following lemma, or a variant of it.

Lemma 6. If G is a DR graph with

c2(d-2) <b2(b{ - 1),

then 0O > 0i + 02 + • • • + 4>d-\-

Proof. The explicit formulae for 0o, 0i and 02 show that

, _ n - l - / c 0O _ c2(n— 1 -k-k2) c20ob\ b\ b\b2 b\b2

By Theorem B, 0/ < 02 when i > 2, so

01 + 0 2 + ... + 0</-l < 01 +(d

b\b2

If c2(d — 2) < b2(b\ — 1), the coefficient of 0O is less than 1, so we have the result. •

For example, the Johnson graphs are the graphs whose vertices are the s-subsets of anr-set, two vertices being adjacent when the subsets have s — 1 common members. Whenr > 2s we have a family of DR graphs with 'classical parameters'

bj = ( s - j)(r - s - j)9 cj = j \ ( 0 < j < d)9

where the diameter is d = s. It is easy to check that the condition in Lemma 6 is satisfiedin this case. Similarly the doubled odd graphs form a family of regular near-polygons withdiameter d = 2k — 1 and b2 — b\ = k — 1, c2 = 1, so the condition is satisfied here too.Indeed it appears that the condition is satisfied for all the families listed in Chapter 6 of[4].

Of course, The Erdos Book would have a proof of the conjecture for all DR graphswith k > 3, independent of any classification, if such a proof exists.


References

[1] Aldous, D. (1989) An introduction to covering problems for random walks on graphs. J.Theoretical Probability 2 87-120.

[2] Biggs, N. L. (1974) Algebraic Graph Theory, Cambridge University Press. (Revised edition tobe published in 1993.)

[3] Bollobas, B. (1979) Graph Theory: An Introductory Course, Springer-Verlag, Berlin.[4] Brouwer, A. E., Cohen, A. M. and Neumaier, A. (1989) Distance-Regular Graphs, Springer-

Verlag, Berlin.[5] Doyle, P. G. and Snell, J. L. (1984) Random Walks and Electrical Networks, Math. Assoc. of

America.[6] Foster, R. M. (1949) The average impedance of an electrical network. In: Reissner Anniversary

Volume - Contributions to Applied Mechanics, J. W. Edwards, Ann Arbor, Michigan 333-340.[7] Foster, R. M. (1961) An extension of a network theorem. IRE Trans. Circuit Theory CT-8

75-76.[8] Nerode, A. and Shank, H. (1961) An algebraic proof of Kirchhoff's network theorem. Amer.

Math. Monthly 68 244-247.[9] Thomassen, C. (1990) Resistances and currents in infinite electrical networks. J. Comb. Theory

B 49 87-102.

On the Length of the Longest IncreasingSubsequence in a Random Permutation

BELA BOLLOBAS1 and SVANTE JANSON2

department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane,Cambridge CB2 1SB, England

email B.Bollobaspmms.cam.ac.uk2Department of Mathematics, Uppsala University, PO Box 480, S-751 06 Uppsala, Sweden

email svante.jansonmath.uu.se

Complementing the results claiming that the maximal length Ln of an increasing subse-quence in a random permutation of {1,2,...,«} is highly concentrated, we show that Ln isnot concentrated in a short interval: sup/ P(/ ^ Ln ^ / + n1/16 log~3//8 n) —• 0 as n —• oo.

1. Introduction

Ulam [8] proposed the study of Ln, the maximal length of an increasing subsequence of arandom permutation of the set [n] = {1,2,...,«}. Hammersley [4], Logan and Shepp [7],and Versik and Kerov [9] proved that ELn ~ l^jn and

Ln/yjn —>p 2 as n —• oo. (1.1)

Frieze [3] showed that the distribution of Ln is sharply concentrated about its mean; hisresult was improved by Bollobas and Brightwell [2], who in particular proved that

Var(Ln) = O(n1/2(logn/loglogn)2). (1.2)

Somewhat surprisingly, it is not known that the distribution of Ln is not much moreconcentrated than claimed by (1.2). In fact, it has not previously been ruled out that ifw(n) —• oo then P(\Ln — ELn\ < w(n)) —> 0 as n —• oo. Our aim in this paper is to ruleout this possibility for a fairly fast-growing function w(n), and to give a lower bound forVar(Ln), complementing (1.2).

Theorem 1.

P(\Ln - ELn\ ^ n1/16 log~3 / 8 n) -* 0 as n -^ oo.

More generally, if an and bn are any numbers such that

inf ?(an ^Ln^ bn) > 0, then (bn - an)/nl/l6 log~3 /8 n -> oo.

122 B. Bollobds and S. Janson

In particular, for sufficiently large n,

VarLn ^ n1/8 log"3/4 n.

There is still a wide gap between the upper and lower bounds, and there is no reasonto believe that the bounds given here are the best possible. In fact, a boot-strap argumentsuggests that the range of variation is at least about rc1/10, see Theorem 2 below, andit is quite possible that the upper bound in (1.2) is sharp up to logarithmic factors, asconjectured in [2].

It is well-known that Ln also can be defined as the height of the random partialorder which is in itself defined as follows. Consider the unit square Q = [0,1]2 with thecoordinate order. Thus for (x,y), (x'\y') £ Q, set (x,y) ^ (x',yf) if and only if x ^ x' andy ^ / , let (£;)£i be independent, uniformly distributed random points in Q and considerthe induced partial order on the set (6)?=i-

Let \i > 0 be a constant and let m be the Lebesgue measure in Q. Let us regard aPoisson process with intensity \idm in Q as a random subset of Q. Equivalently, let N beindependent of (6)f, with distribution Po(fi), and take the set {£,• : 1 ^ i < N}. Write HM

for the height of the induced partial order on this set.The proof of (1.2) by Bollobas and Brightwell in [2] was based on a study of Hn. In

particular they proved that

'Q^)» (1.3)

for some constant K\, every n ^ 3 and every / with 1 < k < n1/4/loglogn. For larger /,their proof yields

P(|Hn - EHn| > K2)}\ogX) < e~}}. (1.4)

These inequalities hold for non-integer n as well: also if n ^ 3 and 1 ^ A n1//4/loglogn,then for every \i < n, we have

1/4 1° g n ) ;2 (1.5)1 ) < e .

log log n /It is rather curious that our proof of a lower bound will use these results, together withthe following estimate from [2]:

0 < 2n1/2 - EHn ^ K4nl/4 log3/2 n/ log log n. (1.6)

Remark. It is shown in [2] that (1.3) holds for Ln as well. (The same is true for (1.4) and(1.5).) Similarly, Theorem 1 holds for Hn too; this follows from the proof of Theorem 1below, with a few simplifications.

The variables Ln and Hn may be defined, more generally, for random subsets ofthe d-dimensional cube [0, l]d. The results in [2] include this generalization, and itwould be interesting to find lower bounds for the variance. Unfortunately, and somewhatsurprisingly, the method used here does not work when d ^ 3. We try to explain thisfailure at the end of the paper.

On the Length of the Longest Increasing Subsequence in a Random Permutation 123

2. Proof of Theorem 1

The idea behind the proof is that Ln essentially depends only on the points in a strip ofmeasure n~a for some a > 0 (a = 1/8 if we ignore logarithmic factors). The number ofpoints in this strip is approximately Poisson distributed with expectation nl~*; hence therandom variation of this number is of order n^~a^2 and the relative variation is n~(1~a)/2.This ought to correspond to a relative variation in the height of the same order n~(1~a)/2,ignoring the further variation due to the random position of the points, which would givea variation of order at least n1//2 • n~(1-a)/2 = yf^2.

We introduce some notation. For a Borel set S a Q, let

Nn(S) = \{i^n:Zi€S}\

be the number of our n random points that lie in S, and let Ln(S) be the height ofthe partial order defined by these Nn(S) points; similarly, let Hn(S) be the height ofthe partial order defined by the restriction of our Poisson process to S. Finally, letS3 = {(x9y) £ Q : |x — y\ ^ 3} be the strip of width 23 along the diagonal. We shalldeduce our theorem from two lemmas. The first of these claims that the height onlydepends on the points in S$ for a fairly small value of 3.

Lemma 1. If K is sufficiently large, then with 3n = Kn~1^log3^4n(\og\ogn)~1^2 we have

^0 as n->oo.

Proof. We claim that K = (2K4)1/2 will do, where K4 is the constant in (1.6).In fact, we shall prove slightly more than claimed, namely that the probability that the

set {£j• : 1 ^ i ^ n} contains a point £,• ^ S$n that belongs to a maximal chain is o(l).Since the probability that a Poisson process S in Q with intensity n has exactly n pointsis at least e~ln~l/1, it suffices to show that the corresponding probability for the Poissonprocess S is o(n~1^2).

Let M be the number of points in S \ Ss that belong to a maximal chain in E. Then

M = £/(£, 3),

where

/(£,2) = /(£ ^ Ss) • /(£ belongs to a maximal chain in E).

Hence, using an easily proved formula for Poisson processes (see, e.g; [5, Lemma 2.1], and[6, Lemma 10.1 and Exercise 11.1]),

(£,3)= [Ef(z,Eu{z})ndm(z)JQ

= / P(z belongs to a maximal chain in S U {z})ndm(z). (2.1)JQ\SS

Fix z = (x,y) £ S3 and let 5 = (x + y)/2, t = (x - y)/2, Qi = [0,x] x [0,y] and

Q2 — [x, 1] x [j/, 1]. Then, writing \R\ for the area of a set R a Q, we have

I6il1/2 + \Qi\1/2 = (s2 - 1 2 ) 1 / 2 + ((l - s)2 - t 2 ) " 2

t2 , t2

< + l25 2(1-5)

= 1 ~ 25(1-5)^ 1 - 2r2

^ l~2 '

The random variables Hn(Q\) and Hn(Q2) have the same distributions as Hm and H^2,respectively, with fit = n|Q,|, i = 1,2. Setting K = ( 2 ^ ) 1 / 2 and 5 = Sn, inequality (1.6)implies that

Hence, by applying (1.5) with A = (21ogn)1/2, we find that

P(z belongs to a maximal chain in S U {z}) = P(ifw(Qi) + Hn(g2) + 1 > Hn)

< EHn - io

^ 3exp(—21ogn) = 3n~2.

Consequently, (2.1) yields EM < 3n~l, and the result follows. •

Our second lemmas states that the height is not too well concentrated.

Lemma 2. Suppose that Sn \ 0 and that P(L n ^ Ln(Ssn)) —• 0 as n —• 00. If (ocn) is any>2) then

SUpP( |L n — X| ^ 0Ln) -> 0 (35 n - ^ 00.

sequence with ctn = o(dn ) then

Proof. It is convenient to use couplings, and we begin by recalling the relevant definitions.A coupling of two random variables X and Y (possibly defined on different probabilityspaces), is a pair of random variables (X\ Y') defined on a common probability spacesuch that X' = X and Y' = Y. The notion of coupling depends only on the distributionsof X and 7 , so we may as well talk about a coupling of two distributions (which can beformulated as finding a joint distribution with given marginals).

We also define the total variation distance of two random variables X and Y (or, moreproperly, of their distributions S£(9C) and S£{^y)) as

dTV(X, Y) = sup \P(X eA)- P(7 G A)\, (2.2)A

taking the supremum over all Borel sets A. If (X\ Yr) is a coupling of X and Y then,clearly, dTV(X, Y) = dTV{X\Yr) < P(X; ^ Y'). Conversely, it is easy to construct a

coupling of X and Y such that equality holds (such couplings are known as maximalcouplings). Thus

dTv(X9Y) = minP(X'±Y')9 (2.3)

where the minimum ranges over all couplings of X and 7 . Moreover, provided theprobability space where X is defined is rich enough, there exists a maximal coupling(X\ Y') of X and Y with X' = X.

We may assume that Sn < 1 and (xn^ 8n —• oo. (All limits in the proof are taken asn —• oo.)

Let r = r(n) = \6ocnyjn] ^ l(xnyjn, a n d let \i = \i(n) = \Ssn\; t h u s

3n^ fi^ 2dn.

We use the facts that, for any /?,/?, *i,*2,

and

see e.g. [1, Theorems 2.M and l.C]. Hence

drv (Nn(SdH), Nn+r(SSn)) = dTV (Bi(w, /*), Bi(n + r, //))

AI), Po(njx)) + ^TK (Po(w/x), Po((w

Choose a maximal coupling (N'n,Nfn+r) of Nn(S$n) and Nn+r(S$n), and let (£•)£! be a

sequence of independent random points, uniformly distributed in 5^n; assume also that(£•) is independent of (Nf

n,Nfn+r). Let Lr(iV) be the height of the partial order defined by

{£ : f N}. Then (L'(JV;),L'(N;+r)) is a coupling of L ^ J and Ln+r(SSn)9 and thus

rfrF(Lw(^),Ln+r(S,j) ^ P(L'(N'n) ± L'(N'n+r))

^ P(N'n ± K+r) = dTv(Nn(S6H),Nn+r(S5n))

< UocX/2.

Furthermore, using Ln+r(Ssn+r) < Ln+r(5^) ^ Ln+r, we see that

dTV(Ln,Ln+r) < P(Ln ^ L ^ J ) + P(Ln+r ^ LB+r(S5J) + dTv{Ln(SsH),Ln+r(SsH))

^ P(Ln ± Ln(SSn)) + P(Ln+r ± Ln+r(SSnJ) + UotnSt/2 -> 0.

Hence a maximal coupling (Z/n,Z/w+r) of Ln and Ln+r satisfies P(L'n ^ ^n + r ) —• 0.We next define another coupling of Ln and Ln+r, now trying to push the variables

apart. Observe that necessarily n8n —> oo since, otherwise, for some C < oo and arbitrarilylarge n,

ELn(SSn) < ENn(SSn) = n\SSn\ ^ 2ndn ^ 2C,

which contradicts Ln/\jn — »2 and ¥{Ln ^ Ln(SsJ) —• 0. Hence r = 0(annl/2) =

o(dnl/2nl/2) = o(n).In particular, we may assume that n > 3r. Set Q\ = [0, ~]2 and Qi = (^, I]2. Then

Ln+r^Ln+r(Qx) + Ln+r{Q2). (2.4)

Moreover, Nn+r(Q\) ~ Bi(n + r,(j^)2) with an expectation of (n + r)(^)2 > ^ ^ 4a^; andit follows from Chebyshev's inequality, that

( 6 i ) ^ 2 a 2 ) ^ l . (2.5)

Since the distribution of Ln+r(Q\) conditional on Nn+r(Q\) = v equals the distribution ofLv for any v ^ 1, we obtain from (1.1) that

P(Ln + r (Q1)>2a l l ) ->l . (2.6)

Similarly, n + r — Nn+r(Q2) ~ Bi(n + r, 1 — (1 — ^) 2 ) with expectation

and thus

P(Nn+r{Q2) ^n) = ?(n + r- Nll+r(Q2) < r) - • 1. (2.7)

We define LJ,' to be the height of the partial order defined by the first n of < i, c2, • • - thatfall in Q2; obviously L", =Ln, so (L"t,Ln+r) is a coupling of Ln and Ln+r. Moreover, ifNn+r(Q2) > n, then Ln+r(Q2) > V'n, and thus (2.4), (2.6), (2.7) yield

r > L ; ; + 2 a J ^ l . (2.8)

Combining this coupling with a maximal coupling (L'n+r,L'n) of Ln+r and Ln such thatL'n+r = L,,+r, we obtain a coupling (LJ,, LJ,') of L,, with itself, i.etwo random variables LJ,and LJJ with LJ, =LJJ =Ln, such that

P(L'n > LJJ + 2a,,) > P(LB+r > LJJ + 2an) - P(Ln+r ^ Lj,) - • 1.

Finally we observe that for any real x,

P(L'n > LJJ + 2a,,) < P(L; > x + «„) + P(LJJ < x - a,,) = P(|L,, - x| > a,,)

and thus

supP( |L n -x | < a,,) < 1-P(LJ, > LjJ + a,,) -^ 0. •

Theorem 1 follows immediately from the lemmas.

3. Further remarks

Note that the proof of Theorem 1 uses the concentration results in [2], and that strongerconcentration results would imply a stronger version of Theorem 1, i.eiess concentrationthan given above. This leads to the following result, which shows that, at least for somen, the distribution of Hn is not strictly concentrated (with, say, exponentially decreasing

tails) with a variation of much less than ft"1/10. (For simplicity we consider here Hn;presumably the same result is true for Ln.)

Theorem 2. / / £ > 0 is sufficiently small, then there exist infinitely many n such that forsome m^n we have

?(\Hm-EHm\>ml/m)>n-2.

Proof. Assume, to the contrary, and somewhat more generally, that for some y, 0 < y <1/2, and all large n,

P(\Hm - EHm\ > ny) < n-2, m < n. (3.1)

The argument in the proof of [2, Theorem 9] then yields

lnl/2-EHn = O(ny) (3.2)

and Lemma 1 holds for Hn, by the argument above, with

Sn=Kny/2-l/\ (3.3)

provided K is large enough. Hence Lemma 2 (for Hn) shows that

whenever ocn = o(S^l/2), i.e; when

If y < 1/10, we may take an = ny, which then satisfies (3.5), and obtain a contradictionfrom (3.1) and (3.4). In order to obtain the slightly stronger statement in the theorem, welet y = 1/10 and note that if

P(\Hn - EHn\ > snl/l°) < n~2 < 1/2 (3.6)

for every e > 0 and n n(e), then there exists a sequence sn —> 0 such that

P( |H n -EH I I |> f i n w 1 / 1 0 )< l /2 . (3.7)

We now choose ccn = enn1/10, which satisfies (3.5), and obtain a contradiction from (3.4)

and (3.7). Hence either (3.1) or (3.6), for some £ > 0, fails for infinitely many n, whichproves the result. •

Finally, let us see what happens when we try to generalize the results to the randomd-dimensional order defined by random points in Qd = [0, l]d. Lemma 1 holds, with

5n = Kn-l/4d\og3/4n(\og\ogn)-l/2, (3.8)

by essentially the same proof; we now define Ss = {(x()d : |x,- — x7-| ^ (5, i < j}9 and notethat \Ss\ ^ Sd~{. For Lemma 2, however, we need

Kn = o(nl<d-l'2S?d-lV2), (3.9)

in which case we may take m = Knl~l/dtxn for some large K. However, (3.8) and (3.9)imply ccn = o{nP~M)^d) = o(\) for d 3, so we do not obtain any result at all. (We also

128 B. Bollobas and S. Janson

need <xn ^ 1). The method of Theorem 2 yields no result either: we obtain 3n = Kny^2~l/2d

and by (3.9) we have

y~/1/,(3—d)/4d—y(d—1)/4\ /o i A \an — O\Yl j , \5.L\J)

which again contradicts ocn ^ 1 for any y > 0 when d > 3.

We can explain this failure in terms of the heuristics at the beginning of Section 2. We

still have a relative variation of the number of points in the strip S$ of order n~^~a^2,

for some a > 0, but this translates to a variation of the height of order only w1/^-i/2+a/2^

which does not give any non-trivial result (a is rather small). Of course, this does not

preclude the possibility that there is a substantial variation of the height due to the

random position of points in the strip.

References

[1] Barbour, A. D., Hoist, L. and Janson, S. (1992) Poisson Approximation, Oxford Univ. Press,Oxford.

[2] Bollobas, B. and Brightwell, G. (1992) The height of a random partial order: concentration ofmeasure, The Annals of Applied Probability 2 1009-1018.

[3] Frieze* A. (1991) On the length of the longest monotone subsequence in a random permutation,Ann. Appl. Probab. 1 301-305.

[4] Hammersley, J. M. (1972) A few seedlings of research, Proc. 6th Berkeley Symp. Math. Stat.Prob. Univ. of California Press, 345-394.

[5] Janson, S. (1986) Random coverings in several dimensions, Acta Math. 156 83-118.[6] Kallenberg, O. (1983) Random Measures, Akademie-Verlag, Berlin.[7] Logan, B. F. and Shepp, L. A. (1977) A variational problem for Young tableaux Advances in

Mathematics 26 206-222.[8] Ulam, S. M. (1961) Monte Carlo calculations in problems of mathematical physics, Modern

Mathematics for the Engineer, E. F. Beckenbach Ed., McGraw Hill, NY.[9] Versik, A. M. and Kerov, S.V. (1977) Asymptotics of the Plancherel measure of the symmetric

group and the limiting form of Young tableaux, Dokl. Akad. Nauk. SSSR 233 1024-1028.

On Richardson's Model on the Hypercube

B. BOLLOBAS12! and Y. KOHAYAKAWA 3 |

department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane,Cambridge CB2 1SB, England

2Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA3Instituto de Matematica e Estatistica, Universidade de Sao Paulo,

Caixa Postal 20570, 01452-990 Sao Paulo, SP, Brazil

The n-cube Qn is the graph on the subsets of {1, . . . , n} where two such vertices are adjacentif and only if their symmetric difference is a singleton. Fill and Pemantle [5] startedthe study of several percolation processes on Qn and obtained many asymptotic resultsfor n —• oo. As an application of these results, they investigated the contact process withno recoveries in Qn, also known as Richardson's model for the spread of a disease. Theyobtained that, in this model, the cover time Tn of Qn starting from a single infected vertexis bounded in probability: they proved that, with probability tending to 1 as n —> oo, onehas (l/2)log(2 + >/5) + log2 + o(l) = 1.414...+ o(l) ^ Tn ^ 41og(4 + 2^3) + 6 + o(l) =14.040... + o(l). In this note we substantially improve this upper bound by showing thatone in fact has Tn < 1 + log2 + o(l) = 1.693... + o(l) in probability.

Introduction

The n-dimensional cube Qn is the graph whose vertices are the subsets of [n] = {l,. . . ,n}where two such vertices are adjacent if and only if their symmetric difference is a singleton.Here we are interested in a stochastic process on Qn, known as Richardson's model forthe spread of a disease. In this model, one vertex of Qn is at first infected and the diseaseevolves by transmission of the infection according to i.i.d. Poisson processes associated tothe edges of Qn. Every time our Poisson clock associated to an edge xy goes off, if oneof x or y is infected at that time, the other vertex becomes infected. The natural questionsconcerning this model regard the manner in which the disease spreads, and in particularthe time it takes for all the vertices of Qn to become infected. The latter question has beenrecently studied by Fill and Pemantle [5], who proved that this covering time is bounded inprobability. Our aim in this note is to give an improved upper bound for this covering time.

t Part of this work was done while this author was visiting the University of Sao Paulo, supported by FAPESPunder grant 92/3169-8.

$ Research Partially supported by FAPESP under grant 93/0603-1 and by CNPq under grant 300334/93-1.

130 B. Bollobds and Y Kohayakawa

One may study Richardson's model on Qn by studying first-passage percolation onthis graph where the passage times on the edges are i.i.d. exponential random variables.In fact, we shall study Richardson's model only looking at first-passage percolation.Let W = (We)eeE(Q") be a family of independent exponential random variables each withmean 1. If x and y are two vertices of Qn, we write d(x,y) = dg«(x,y) for the distancebetween x and y in Qn, and we set

dw(x,y) = min{/(P, W):P an x-y path in g"},

where £(P, W) = ^2eeE(P) We is the W-length of the path P. The result in [5] concerningRichardson's model on Qn is that, for a fixed xo G Qn, with probability 1 — o(l) as n —• oo,we have (1/2) log(2 + ^5) + log 2 + o(l) = 1.414... + o(l) < maxyGg* <Mx0,j>) ^ 4log(4 +2^/3) -f 6 + o(l) = 14.040... + o(l). A corollary of our main result, Theorem 5, is that thisupper bound can be improved to 1 + log2 + o(l) = 1.693... + o(l).

Indeed, we shall show that with probability 1— o(l) as n —• oo, (i) for all fixed xo G Q" wehave that m a x ^ * dw(xo>)0 ^ l+log2+o(l) , and (ii) diam(Q", W) = max{dw(x,y):x, y eQn} < 1 H- 2 log 2 -f o(l). The methods used in the proof of Theorem 5 are reminiscent ofcertain techniques first used in [3, 4].

The organisation of this note is as follows. In §1 we present some definitions andpreliminary results, and in §2 we prove the main technical lemma (Lemma 4) that isneeded in the proof of Theorem 5. This theorem is proved in §3. We close this note withsome remarks and further problems.

1. Preliminaries

Throughout this note, we let W = (We)eeE(Q») be a family of independent exponentialrandom variables each of mean 1. For this section and the next, we assume that aconstant 0 < & < 1/6 has been fixed.

One extremely simple but important idea in our analysis of (Qn, W) is that we may easilydecompose We (e e E(Qn)) into a collection of independent exponential random variables.This decomposition is then used to handle several events independently. Let X = (Xe),Y = (Ye), and Z = (Ze) be three families of independent exponential r.v.'s, where the Xe

have mean (1 — 2s)~l, and the Ye and Ze have mean s~l. Writing a A b for the minimumof a and b, if W'e = Xe A Ye AZe (e e E(Qn)), clearly W'e is an exponential r.v. with mean 1.Thus we may study (Qn, W) by analysing (gn,X), (Qn, 7) , and (Qn,Z).

All the asymptotics below refer to n —• oo. Moreover, we very often tacitly assume that nis large enough in the estimates that follow. As is usual in the theory of random graphs,we use the terms 'almost surely' and 'almost every' to mean 'with probability tending to 1as n —• oo'. For clarity, we shall denote 0 e Qn by xo- The neighbourhood of x G Qn isdenoted by T(x) = FQn(x). For graph-theoretical terminology not defined here, we referthe reader to [1].

Our first two lemmas concern (Qn,X).

Lemma 1. For all x G Qn, let Nx = Nx(e,X) c TQn(x) be defined by Nx = {y GTQn(x):Xxy ^ log2 + 3e} if x ^ XQ, and NXo = {y G TQn(xo):XXoy ^ e}. Then there is

On Richardson's Model on the Hypercube 131

a constant c\ = C\(E) > 0 such that with probability 1 — o(l) we have \NX\ ^ c\n forall x e Qn.

Proof. Fix an arbitrary vertex x ^ XQ in Qn. Then, for a fixed y G r^ (x ) ,

^ log 2 + 3E) = exp{-(log2 + 3e)(l - 2e)} <: (1 - a)/2,

for some a = a(a) > 0. From standard bounds for the tail of the binomial distribution, wesee that for a suitable constant c\ > 0 we almost surely have |NX| ^ c\n for all x ^= xo.Similarly, for a suitable constant d[ > 0, we have that \NXQ\ c'[n almost surely. Thus wemay take c\ = c\ A c'[. D

We now describe the results from [5] concerning first-passage percolation on Qn. Forall n ^ 1, let x0 = x ^ = 0 G Qn and y0 = y^ = [n] G Qn. Consider (Q\ W\ and let

To = inf{T e R:d^(xo,yo) ^ T with probability 1 - o(l)}. (1)

Equivalently, To is the smallest T (0 ^ T ^ oo) such that, for any given S > 0, wehave UmsupnlP(dw(xo,yo) ^ T + d) = 0 Thus To is the first-passage percolation timebetween two opposite vertices in Qn. Fill and Pemantle [5] have shown that To ^ 1, andan argument due to Durrett given in [5] gives that To ^ log(l + ^/2) = 0.881 • • •. Ournext lemma brings the parameter To into our problem. Before we can state this lemma weneed some definitions. A subcube of Qn is a subgraph induced in Qn by a set of the form

Qs,A = Q(S,A) = {xeQn:xnS=A},

where i c S c [n]. In the sequel we shall also denote by Q^A = 2(^-4) the subcubeinduced by QS,A, since this will not cause any confusion. The dimension dim(Qs?y4) of QS,Ais n — \S\. For an integer k ^ 0 and a set X, we write X^ for the set of all /c-elementsubsets of X.

Given x and y G Qn we define the subcube (x, y) spanned by x and y by

(x,y) = Q((x A yf,x \ (x A y)) = Q((x A yf,y\ (x A y)),

where uQ = [n]\u and as usual A denotes symmetric difference. If x c y then (x, y)equals the 'interval' [x,y] in Qn = ^([n]) regarded as a partial order under inclusion, thatis (x,y) = [x,y] = {z G Qn'.x a z c y } . I f v G ( x , y ) , let u s cal l t h e p a i r ((x,y) ;v) a rootedcube with root v. For the rest of this note, we let /co = [logft], and set

«T = {(x,y,z) eQnxQnx Qn:d(x,y) ^ 2/c0 + 1, d(x,z) = 2/c0, z e {x,y)}.

For a triple (x,y,z) e 9~ let ^ w be the family

of rooted subcubes of (x,y). In visualising ^x,y,z, it might be helpful to note that themap (py,z'.u i—• w A (y A z) gives a natural isomorphism between the subcubes (x,z)a n d ((pyiZ(x),(pyiZ(z)) = {q>y^(x\y). M o r e o v e r , n o t e t h a t t h e r o o t e d s u b c u b e s ((v,w) ;v)in J^x^z are translates of the rooted cube ((x,(py^(x)) ;x) = ((x,x A y A z ) ; x ) . In fact,we may clearly regard Qn as (x,z) x (x ,^ 2 (x)) . Let us say that ({v,w} ;v) G ^x,y,z is

132 B. Bollobds and Y. Kohayakawa

inexpensive if dx(v, w) is at most (To + E)/(1 — 2e), and expensive otherwise. In the sequelwe write mo = (2

/c/c°). Rather crudely we have nl3S < mo ^ n139.

Lemma 2. There is a constant c2 = c2(s) > 0 such that the following holds almost surely.For all (x9y9z) G ZT the number of rooted subcubes in ^x,y,z that are inexpensive is atleast c2mo = c2 (fco°).

Proof. Let (x,y,z) G y be given. Fix v, w G Qf (t > 1) with d(t;,w) = / , and let(7 = (£/e)ee£((X) be a family of independent exponential r.v.'s of mean 1. By the definitionof To, we have that for any given fixed 8 > 0 we have \im^ V(du(v9 w) ^ To + 3) = 0. Itfollows that there is an integer <f0 = ^o(e) such that JP(du(v,w) < (To + e)/(l -2e)) ^ 1/2if £ ^ ^o- Moreover, there clearly exists a constant c'2 = c2(s) > 0 such that the probabilitythat a rooted cube ((v9w) ;v) is inexpensive is at least c2 if 1 < dim(v9w) ^ /o- Let usnow set c2 = (l/2)(c2 A 1/2).

Let ((ty,Wi) ;i;,-) (1 < i ^ mo) be the mo rooted cubes in yx,y,z- Note that the (ty,w,-)are pairwise vertex-disjoint and hence the events that ((i;,-,wt) ;vt) (1 < i < mo) should beinexpensive are independent. Thus the number J of inexpensive rooted cubes in yx,y,z hasbinomial distribution Bi(mo,p) with parameters mo = (2/c

/c°) and p ^ 2c2. It now followsthat P(J < c2m0) ^ exp{—pmo/8} ^ exp{—n13} = o(\y\~*)9 as required. D

Our next result is a simple lemma concerning (Qn, Y). Recall that fco = n°8wl- ^ x>y G Qn are such that d(x,y) = 2t9 let Lxy denote the middle layer [z G {x,y) :d(x,z) =d(z,y) = /} of (x,y).

Lemma 3. Let Vo = VQ(Y) be the set of vertices z ofQn that are incident to at least (s2/3)nedges e with Ye ^ e. Then almost surely for all x, y G Qn with d(x,y) = 2/co at most n verticesin the middle layer Lxy of (x9y) fail to be in Vo.

Proof. Let x = x0 = 0 and fix y G [n]{2ko). For any edge e G E(Qn) we have P(Y, ^ e) =1 — exp(—e2) ^ e2/2, and hence for fixed z G Lxy the probability that z does not belongto Vo is at most po = exp{—^2^/36}. Thus the probability that we have |LXJ, \ Vo\ n isat most

fm\ fnlA\exp{-e2n2/36} ^ exp{-eV/36} ^ exp{-Q(n2)}.

\nj \n JThus the probability that there exist x, y G Qn with d(x,y) = 2ko such that \LXJ \ VQ\ ^ nis at most 4n exp{-Q(n2)} = o(l). D

Let Vo = V0(Y) be defined as in Lemma 3. For all z G Vo we let N'z - N'z(e, Y) = {zr GrG-(z): 7«, ^ e}. Thus |N^| ^ (e2/3)n for any z G Fo.

2. The main lemma

In this section we study (<2",Z), and prove a lemma (Lemma 4) that will be fundamentalfor the proof of our main result, Theorem 5. We stress that the argument in the proof

On Richardson's Model on the Hyper cube 133

below is quite crude, and that with a little more patience one may prove a stronger result.However, such a strengthening of this lemma would not, as far as we can see, improveTheorem 5. Before turning to our lemma, recall that 0 < e < 1/6 has been fixed.

Lemma 4. . Set K = L(£6rc/65O)1/7J, and let m2 = omn/1 ^ n2, where co = co(n) -> ooas n —> oo. Let x = xo = 0 and fix y c [n] with 1 ^ k = \y\ ^ K. Let S c TQn(x)and T a TQn(y) with \S\, \T\ ^ m be given. Then in (Qn,Z) there is an S-T path P ofZ-length tf(P,Z) = YleeE(P)Ze ot. most s with probability at least 1 - e - ° ) ' l / l o g 0 ) .

Proof. For all e G E(Qn) and 2 < j < K + 1, let Z ^ be an exponential r.v. withmean K/e, and assume that all these variables are independent. Set Z( ; ) = (Z^)ee£(Qn)for 2 < j < K + 1. Clearly Ze and A ; Z i ; ) h a v e t h e s a m e distribution for all e e E(Qn).Let t0 = s/(2K +k-2). Then for all 2 ^ j ^ K + 1 we have

>2K(2K+k-2) "

For all 2 ^ 7 ' ^ K + 1, let Hj = Hj(Zu)) c Q" be the spanning subgraph of Qn whoseedge-set is {e G E(Qn):Z{

ej) < r0}. Note that the // , (2 < 7 < K + 1) are K independent

random elements of ^(Qn,p)9 the space of random spanning subgraphs of Qn whose edgesare independently present with probability p. Let us set Gy = //2U- • -Ui/y (2 < 7 < K +1).We now claim the following.

Claim. In GK+I there is an S-T path of length at most 2K + k — 2 with probability atleast 1 — exp{—con/ log co}.

Note that a path as in the claim above has Z-length at most to(2K + k — 2) = e, andhence to prove our lemma it suffices to prove this claim. To verify this claim we start bysetting Si = {v eS:v c/i y} and T\ = {v G T:y a v}. Clearly |Si|, |Ti| ^ m — k > m/2. Weshall think of Qn as Qn~k x Q^ in the following way. For each v cz yc = [n] \ y, let

Qv = {v,v Ay) = (v,vUy) = {z cz [ n ] : z n / = t;}. (2)

Note that Qv is a cube of dimension k = |y|, and that g = (xo,yc) = {v e Qn:v a yc} is acube of dimension n — k. Also, we may naturally regard Qn as (xo,yc) x (xo^)-

Write /ti = |yc| = n — k and note that clearly n\ = (1 + o(l))n. Let us set

S2 = {z G (/) ( 2 ) :z G I>(Si) and zUy G ren(Ti)}.

A moment's thought shows that \S'2\^ (m22)- Thus, setting

S2 = {z G (/) ( 2 ) :z G rH2(Si) a n d z U j E rH2(Ti)},

quite crudely we have E(|S2|) ^ i^^P1 ^ m2p2/9. Hence, with probability at least 1 —exp{-m2p2/1800} = 1 - exp{-Q(e4/7con)}, we have \S2\ > mV/10 .

For the rest of the argument, we condition on |S2| ^ m2p2/10 ^ (l/5)(mp/n)2("21). In

the sequel, if U cz (xo^0), we write U V y for {u U y: w e U}, namely the translate of Uby the map u\-+ uU y, which is contained in the cube (y, [n]). Let us now fix 3 ^ j ^ K

134 B. Bollobas and Y. Kohayakawa

and condition on the existence of S2 <= (yc)(2\...,Sj-i <= (yc)(J-V for which the followingconditions hold for all 2 ^ i < j .

(ii) for all z G Sf there is an S\-z path of length i — 1 in G, = H2 U • • • U //;,(m) for all z G S,- there is a Ti-(z U y) path of length 1 - 1 in G,- = if2 U ••• U if,.Now let Sj = r en(S ;_i) n ( / ) ( ^ = {zr G (yc)W:z a z' for some z G Sy-_i}. Then, by thelocal LYM inequality (see for instance [2], §3), we have that

We now set Sj = {z eS-:z e rHj(Sj-i) and z U y G TH.(Sj-x Vy)}. Then very crudely wehave that

and so, with probability at least

condition (/) above holds for i = j , that is \Sj\ ^ (4/5)2"-/ (mpj^/n)2 (nj). Note that bythe definition of S7 conditions (ii) and (m) above hold for i = j . Let

SK={ZE (yc)W:dGK(S9z)9 dGK(T,zUy) ^ K - 1},

where dc* denotes the distance in the graph GK- Then, using the argument abovefor 3 < j ^ K in turn, we see that, with probability at least

we have

^ (=£!)'(••).To estimate Po, note that if 3 < j ^ K then

-

since the function /(x) = (2?/x)x (- > 0) is log-concave. However,

p2n\

and hence A ^ (1/216 + o(l))m2np4 = Q(cwn10/7). Thus, very crudely,

Po > 1 - K exp{-Q(conwn)} ^ 1 - exp{-con}. (4)

On Richardson's Model on the Hyper cube 135

Let us now condition on (3). To finish the proof of our claim, we shall show that with veryhigh probability there is an S^-(Sf^ V y) path of length k in HK+\. We shall do this usingan extremely crude argument. Consider the family $F = {Qv: v e S^} of/c-cubes as definedin (2). Recall that Qv is a translate of (xo,y) by v, and note that in particular the Qv

(v e S£) are pairwise disjoint. Consider for each u e ^ a fixed v-(v U y) path Pv c Qv oflength k, and note that the probability Pi that there should not be an S^-(S^ V y) pathof length k in HK+{ is at most (1 -p*)'5*! <$ (1 -P

K)\SK\ <C exp{-pK\S£\}. However, as (3)holds,

Kj ^ 5\pn) Uz

which is much larger than con. Thus Pi ^ fwn. This bound and inequality (4) finish theproof of the claim, and hence Lemma 4 is proved. •

3. The main result

Recall that W = (We) is a collection of independent exponential r.v.'s each with mean 1.Also, recall that To is the first-passage percolation time between two diametrically oppositevertices in Qn when the passage times are given by W (cf. (1) in Section 2).

Theorem 5. The following hold with probability 1 — o(l) as n —• oo.

(0 Writing x0 = 0 G Qn, we have maxyeQn dw(x0,y) < To + log 2 + o(l),

Proof. (0 Let 0 < e < 1/6 be fixed. We regard We as Xe AYef\Ze(ee E(Qn)), where Xe,Ye, and Ze are as in Section 2, i.e. the Xe are exponential with mean (1 — 2c)"1, the Ye

and Ze are exponential with mean e"1, and all these variables are independent. Let usnow consider the following events. (We keep the notation introduced in Section 2.)

(a) For all x e Qn we have |JVX| = \Nx(e,X)\ ^ cm,(b) for all {x,y,z) G , the number of inexpensive rooted subcubes in ^x,y,z is not smaller

(c) for all x j G 2 " with d(x,y) = 2/c0, we have |LX0, \ Vo\ = |LX>>. \ Fo(7)| < w.

Note that by Lemmas 1, 2, and 3 conditions (a), (/?), and (c) above hold almost surely. Forthe rest of the argument we condition on (Qn,X) and (Qn, Y) satisfying (a), (fc), and (c).

Let SXo = NXo = NXo(e,X) = {y e TQn(xo):XXoy < e}. For every x e g" - x0, set Sx =N'x = Nf

x(s, Y) if x G Fo = Fo(7), and Sx = Nx = N X ( E , Z ) otherwise. For a fixed pair x,j / G f i " with d(x,y) < K = L(e6n/650)1/7J, if we let S = Sx and T = Sy, Lemma 4 tellsus that dz(S, T) ^ £ with probability at least 1 — exp{—n5/4}. Thus (d) below holds withprobability 1 — o(l).

136 B. Bollobas and Y Kohayakawa

(d) For all x, y G Qn with d(x,y) ^ K we have

{ 3e if x,y G Fo U {x0}

Iog2 + 5e if x G Fo U {x0} or y G Fo U {x0}2 log 2 -f 7e otherwise.

We now condition on (Qn, W) satisfying (d), and claim that consequently we havemzx{dw(xo,y)'.y G Qn} ^ To + log2 + 3eT0 + 12e. First note that it follows from (d)that max{dw(xo,y):y G Q", d(xo,y) ^ K} is as small as claimed, and hence let y G Qn

be such that d(xo,y) > K ^ 2/co. Pick z G (xo,y) with d(xo,z) = 2&o, and note thatthen (xo,y,z) G 3~. Since we are conditioning on (b) and (c) above, we may find v G LXoJ,such that v G Fo, w = v A y A z G Fo, and ((u, w) ;v) is an inexpensive rooted cube. Butthen

dw{xo-> y) ^ d\y(xo, i ) + d]y(v, w) -f- Jp^(w, y)Theorem 5 (i) follows by letting s —> 0.(ii) Minor modifications to the above argument gives a proof of Theorem 5 (ii). •

The result above, coupled with the upper bound To ^ 1 for the first-passage percolationtime To established by Fill and Pemantle [5], gives the following corollary.

Corollary 6. The following hold with probability 1 — o(l) as n —• oo.

(i) Writing x0 = 0 G gn, we fawe m a x ^ * <Mx<),)0 < 1 + Iog2 + o(l),(ii) diam(g", W) ^ 1 + 2 log 2 + o(l). •

4. Concluding remarks and open problems

The best bounds so far for the first-passage percolation time To in Qn, defined in §1, arethe ones given in [5], namely 0.881... = log(l + ^2) ^ To ^ 1. We believe that percolationhappens at a sharply defined time.

Conjecture 7. For all S > 0 we have limn_,oo 1P{dw(xo,yo) ^ To — 3} = 0. D

We also believe that the analogous phenomenon happens in Richardson's model. As astarting point, it would be interesting to settle the following problem.

Problem 8. Do the following two assertions hold with probability 1— o(l) as n —»oo?

(i) Writing x0 = 0 G Qn, one has maxyeQn dw(xo,y) = To + log2 + o(l).

(ii) One has diam(Qn, W) = To + 2 log 2 + o(l). •

References

[1] Bollobas, B. (1979) Graph Theory - An Introductory Course, Springer-Verlag, New York,viii-h 180pp

[2] Bollobas, B. (1986) Combinatorics, Cambridge University Press, Cambridge, xii-\- 177pp

On Richardson's Model on the Hypercube 137

[3] Bollobas, B., Kohayakawa, Y. and Luczak, T. (1996a) On the diameter and radius of randomsubgraphs of the cube, Random Structures and Algorithms, to appear

[4] Bollobas, B., Kohayakawa, Y. and Luczak, T. (1996b) Connectivity properties of randomsubgraphs of the cube, Random Structures and Algorithms, to appear

[5] Fill, J. A. and Pemantle, R. (1993) Percolation, first-passage percolation, and covering times forRichardson's model on the n-cube, The Annals of Applied Probability 3 593-629.

Random Permutations: Some Group-TheoreticAspects

PETER J. CAMERON^ and WILLIAM M. KANTOR*

tSchool of Mathematical Sciences, Queen Mary and Westfield College,Mile End Road, London El 4NS, U.K.

* Department of Mathematics, University of Oregon, Eugene, OR 97403, U.S.A.

The study of asymptotics of random permutations was initiated by Erdos and Turan, in aseries of papers from 1965 to 1968, and has been much studied since. Recent developmentsin permutation group theory make it reasonable to ask questions with a more group-theoretic flavour. Two examples considered here are membership in a proper transitivesubgroup, and the intersection of a subgroup with a random conjugate. These both arisefrom other topics (quasigroups, bases for permutation groups, and design constructions).

1. Permutations lying in a transitive subgroup

Sn and An denote the symmetric and alternating groups on the set X = {l , . . . ,n}. Asubgroup G of Sn is transitive if, for all i,j e X, there exists g € G with ig = j . In apreliminary version of this paper, we asked the following question:

Question 1.1. Is it true that, for almost all permutations g € Sn, the only transitive subgroupscontaining g are Sn and (possibly) An?

Here, of course, 'almost all g e Sn have property P' means 'the proportion of elements ofSn not having property P tends to 0 as n —• oo\

An affirmative answer to this question was given by Luczak and Pyber, in [15]. We willdiscuss the motivation for this question, and speculate on the rate of convergence.

To analyse the question, we make the customary division of transitive subgroups intoimprimitive and primitive ones. A subgroup G is imprimitive if it leaves invariant somenon-trivial partition of X, and primitive otherwise. Imprimitive subgroups may be large,but the maximal ones are relatively few in number: just d(n) — 2 conjugacy classes, whered(n) is the number of divisors of n. (If the permutation g lies in an imprimitive subgroup,

140 P. J. Cameron and W. M. Kantor

then it lies in a maximal one, which is precisely the stabiliser of a partition of X into sparts of size r, where rs = n and r9s > 1.) On the other hand, primitive groups are moremysterious; but it follows from the classification of finite simple groups that

— they are scarce (for almost all n, the only primitive groups are Sn and An, see [3]);

— they are small (order at most ncloglogn with 'known' exceptions, see [1]).

In addition, many special classes of primitive groups (for example, the doubly transitivegroups), have been completely classified.

The number of permutations that lie in some primitive subgroup other than Sn or An canbe bounded, since such permutations have quite restricted cycle structure (a consequenceof minimal degree bounds, see [14] - note that these bounds are a consequence of theclassification of finite simple groups - or by more elementary means, as Luczak and Pyber[15] do). So we will concentrate on imprimitive subgroups, and, in particular, the largestimprimitive subgroups: those preserving a partition of X into two sets of size n/2, for neven.

A permutation fixing such a partition must either fix some (n/2)-set, or interchangesome (n/2)-set with its complement. Now a permutation interchanges some (n/2)-setwith its complement if and only if all its cycles have even length. The number of suchpermutations is

( (n - l ) ! ! ) 2 = ( (M-l ) (n-3) . . .3 .1) 2 ,

which is easily seen to be n\O(l/y/n). (This formula is easily proved using generatingfunction methods. A 'counting' proof is given in [2]. Curiously, it is equal to the numberof permutations with all cycles of odd length, see [7, 8, 9, 10]. We are not aware of a'counting' proof of this coincidence!)

On the other hand, a permutation fixes an (w/2)-set if and only if some subfamily of itscycle lengths has sum n/2. There seems to be no simple formula for the number of suchpermutations; but Luczak and Pyber show that their proportion is at most An~c", whereA and c are positive constants. Indeed, more generally, the proportion of permutationsfixing some fc-set tends to 0 as k —• oo (as long as n > 2k).

We turn now to the motivation for this question. A quasigroup is a set with a binarymultiplication in which left and right division are uniquely defined (equivalently, themultiplication table is a Latin square). In a quasigroup Q, left and right translations arepermutations, represented by the rows and columns of the multiplication table of Q. Themultiplication group Mlt(Q) of Q is the group generated by these permutations. This group'controls' the character theory of Q [16]. In particular, if Mlt(Q) is 2-transitive, then thecharacter theory of Q is trivial. Smith conjectured that this happens most of the time, andthis is indeed true.

Theorem 1.2. For almost all Latin squares A, the group generated by the rows of A is thesymmetric or alternating group.

This is proved in [2], but follows more directly from the affirmative answer to Question1.1, since the rows of a Latin square obviously generate a transitive permutation group, and

Random Permutations: Some Group-Theoretic Aspects 141

the first row of a random Latin square is a random permutation (that is, all permutationsoccur equally often as first rows of Latin squares).

This suggests several related questions:

1 Is it true that, for almost all Latin squares, the first two rows generate the symmetricor alternating group? (By a theorem of Dixon [4], almost all pairs of permutationsgenerate Sn or An; and a positive proportion of these (1/e, in the limit) have theproperty that the second is a derangement of the first, and hence occur as the firsttwo rows of a Latin square. But not all derangements occur equally often.) Moregenerally, study further the probability distribution on derangements induced by theirfrequency of occurrence in Latin squares. What is the ratio of the greatest to thesmallest number of completions?

2 Is it true that the multiplication groups of almost all loops are symmetric or alternat-ing? (A loop is a quasigroup with identity. Thus we are requiring that the first row andcolumn of the Latin square correspond to the identity permutation, and the deductionof the analogue of Theorem 1.2 from Question 1.1 fails.)

3 What proportion of Latin squares have the property that all the rows are evenpermutations? (If the limit is zero, the alternating group can be struck out from theconclusion to Theorem 1.2.)

4 Is the proportion of permutations that do lie in a proper transitive subgroup 0(n~1//2)?(By our remarks above, this would be best possible.)

2. Bases and intersections of conjugates

Introducing the next topic requires a fairly long detour. Let G be a permutation group ona set X. A base for G is a sequence (xi,.. . ,xr) of points of X whose pointwise stabiliseris the identity. It is irredundant if no point is fixed by the pointwise stabiliser of itspredecessors. Bases are of interest in several fields, including computational group theory.

If G has an irredundant base of size r, then 2r < \G\ < n(n — 1)... (n — r + 1), whencelogn|G| < r < log2|G|. It is easy to construct examples at or near either side of thisinequality. Nevertheless, it is thought that, for many interesting groups, the base sizeis closer to the lower bound. In particular, certain primitive groups whose order ispolynomially bounded should have bases of constant size.

To elucidate this, we look more closely at primitive groups. The 0'Nan-Scott theorem(see [1]) divides these into several classes. All but one of these classes consist of groupsthat can be 'reduced' in some way to smaller ones or studied by other means. The oneclass left over consists of groups G that are almost simple (that is, that have a non-abeliansimple normal subgroup N such that G is contained in Aut(AT)). Using the classification offinite simple groups, it is possible to make some general statements about almost simpleprimitive groups. For example, the following result holds (see [1, 12]; the latter papergives c = 8).

Theorem 2.1. There is a constant c with the following property. Let G be an almost simpleprimitive permutation group of degree n. Then either

(a) G is known (specifically, G is a symmetric or alternating group Sm or Am, acting on theset of k-sub sets o/{l, . . . ,ra} or on the set of partitions o/{l , . . . ,m} into s parts of sizer, or G is a classical group, acting on an orbit of subspaces of its natural module or onan orbit of pairs of subspaces of complementary dimension); or

(b) \G\ < nc.

(The methodological point raised by this and similar theorems is that in the study of finitepermutation groups, after the classical divisions into intransitive and transitive groups,and of transitive groups into primitive and imprimitive groups, one should also divideprimitive groups into large' and 'small' groups, the large ones being 'known' in somesense. This principle applies to both theoretical and computational analysis.)

It is conjectured that there is a constant d (perhaps d = 3) such that, if G is almostsimple and primitive and does not satisfy (a), then almost every c'-tuple of points is a basefor G.

According to the classification of finite simple groups, the simple normal subgroup Nof G is an alternating group, a group of Lie type, or one of the 26 sporadic groups. Inthe first of these three cases, we were able to prove the conjecture (with c' = 2).

Theorem 2.2. Let G be an almost simple group, not occurring under Theorem 2.1 (a). Ifthe simple normal subgroup of G is an alternating group, then almost all pairs of points arebases.

We outline the proof.The first observation is that if G is transitive and H is a point stabiliser, the proportion

of ordered pairs of points that are bases is equal to the proportion of elements g e G forwhich H n Hg = 1, where Hg is the conjugate g~lHg.

Second, primitivity of G is equivalent to maximality of the subgroup H. Moreover, ifm 7 6, then Aut(Am) = Sm, so we may assume that G = Sm or Am. Consider H (the pointstabiliser in the unknown action) acting on M = {l, . . . ,m}. If H is intransitive, it fixes a/c-subset of M for some k; by maximality, it is the stabiliser of this /c-set, and the actionof G is equivalent to that on /c-sets. Similarly, if H is transitive but imprimitive, then it isthe stabiliser of a partition, and G acts on partitions of fixed shape. Both of these casesare included under Theorem 2.1 (a). So H is primitive on M. (This is an example of the'bootstrap principle': note that m is much smaller than n.)

Thus, finally, we need a result about random permutations.

Proposition 2.3. Let H be a primitive subgroup of Sm, not Sm or Am. Then, for almost allpermutations g e Sm, we have H Pi Hg = 1.

This is true, and can be shown by a simple counting argument, except in the case of thelargest primitive groups (the automorphism groups of the line graphs of Kr or Krj, withm = Q) or r2 respectively), where some special pleading is required. In outline: counttriples (h, k, g) with h, k e / / , h, k ^ 1, g e G and hg = k. The number of such triples is notmore than \H\2c, where c is the largest order of the centralizer of a non-identity element

Random Permutations: Some Group-Theoretic Aspects 143

in H; and it is not less than the number of elements g with H C\Hg ^ 1. Now use the fact

that primitive groups are small, and their elements have relatively few fixed points (and

so relatively small centralizers).

Remark. There is an analogy between intersections of conjugates and automorphism

groups. (For example, if the group G is the automorphism group of a particular structure

S, then the intersections of pairs of conjugates of G represent those groups that can be

represented in the following way: impose two copies of the structure S on the underlying

set, and consider all those permutations which are automorphisms of both structures

simultaneously.)

Thus, Proposition 2.3 should be compared with the statement 'almost all graphs have

trivial automorphism group' [6]. As the analogue of Frucht's theorem [11], we propose

the following conjecture.

Conjecture 2.4. Let Gi ,G2, . . . be primitive groups of degrees n\,ni,..., where rc, —• oo and

Gt ^ Sni or AHi for all i. Let X be an abstract group that is embeddable in G, for infinitely

many values of i. Then, for some i, and some permutation g e SHi, we have G, n Gf = X.

This has been proved by Kantor [13] for the family of groups G, = PFL(/, q), nt =

(ql — l)/(q — 1), for a fixed prime power q. (In this case, every finite group is embeddable in

Gi for all sufficiently large i.) Kantor used this result to show that, for a fixed prime power

q, every finite group is the automorphism group of a square 2-((ql — l)/(q — l),(ql~l —

l)/(q - 1), (q1-2 - l)/(q - 1)) design for some i.

References

[I] Cameron, P. J. (1981) Finite permutation groups and finite simple groups. Bull. London Math.Soc. 13 1-22.

[2] Cameron, P. J. (1992) Almost all quasigroups have rank 2. Discrete Math. 106/107 111-115.[3] Cameron, P. J., Neumann, P. M. and Teague, D. N. (1982) On the degrees of primitive

permutation groups. Math. Z. 180 141-149.[4] Dixon, J. D. (1969) The probability of generating the symmetric group. Math. Z. 110 199-205.[5] Donnelly, P. and Grimmett, G. (to appear) On the asymptotic distribution of large prime

factors. J. London Math. Soc.[6] Erdos, P. and Renyi, A. (1963) Asymmetric graphs. Acta Math. Acad. Sci. Hungar. 14 295-315.[7] Erdos, P. and Turan, P. (1965) On some problems of a statistical group theory, I. Z. Wahrschein-

lichkeitstheorie und verw. Gebeite 4 175-186.[8] Erdos, P. and Turan, P. (1967) On some problems of a statistical group theory, II. Acta Math.

Acad. Sci. Hungar. 18 151-163.[9] Erdos, P. and Turan, P. (1967) On some problems of a statistical group theory, III. Acta Math.

Acad. Sci. Hungar. 18 309-320.[10] Erdos, P. and Turan, P. (1968) On some problems of a statistical group theory, IV. Acta Math.

Acad. Sci. Hungar. 19 413-435.[II] Frucht, R. (1938) Herstellung von Graphen mit vorgegebener abstrakter Gruppe. Compositio

Math. 6 239-250.[12] Kantor, W. M. (1988) Algorithms for Sylow p-subgroups and solvable groups. Computers in

Algebra (Proc. Conf. Chicago 1985), Dekker, New York 77-90.[13] Kantor, W. M. (to appear) Automorphisms and isomorphisms of symmetric and affine designs.

J. Algebraic Combinatorics.


[14] Liebeck, M. W. and Saxl, J. (1991) Minimal degrees of primitive permutation groups, with anapplication to monodromy groups of covers of Riemann surfaces. Proc. London Math. Soc. (2)63 266-314.

[15] Luczak, T. and Pyber, L. (to appear) Combinatorics, Probability and Computing.[16] Smith, J. D. H. (1986) Representation Theory of Infinite Groups and Finite Quasigroups, Sem.

Math. Sup., Presses Univ. Montreal, Montreal.

Ramsey Problems with Bounded Degree Spread

G. CHENt and R. H. SCHELP*

tNorth Dakota State University, Fargo, ND 58105

^Memphis State University, Memphis, TN 38152

Let k be a positive integer, k > 2. In this paper we study bipartite graphs G such that,for n sufficiently large, each two-coloring of the edges of the complete graph Kn gives amonochromatic copy of G, with some k of its vertices having the maximum degree of thesek vertices minus the minimum degree of these k vertices (in the colored Kn) at most k — 2.

1. Introduction

Ramsey's theorem assures a specified local order in the midst of global chaos. Specifically,given graphs G and // , each with no isolates, there exists a number r(G,H) such thatevery red-blue coloring of Kn with n > r(G,H) yields either a red copy of G or a bluecopy of H. What are the properties of such monochromatic copies in the global settingof the two-colored complete graphs? In this paper we will investigate the degrees in thetwo-colored Kn of vertices belonging to such monochromatic copies.

Let y: E(Kn) H-» (R,B) denote a two-coloring of the edges of the complete graph oforder n using colors red (R) and blue (£), and let (R) and (B) denote the correspondingmonochromatic graphs. For X c V(Kn), let (X)R and (X)B denote the subgraphs inducedby X in (R) and (B) respectively. Given graphs G and H, each with no isolates, writeX £ Ry(G,H) if \X\ = \V(b)\ and G <= (x)R, or \X\ = \V(H)\ and H c (x)B. By Ramsey'stheorem, there exists a number r(G,H) such that Ry(G,H) =fc 0 for every y: E(Kn) \-> (R,B)whenever n > r(G,H). We shall refer to a set in Ry(G,H) as a Ramsey host. The followingresults were obtained in [1] and [2], where the degrees of vertices in Ramsey hosts wereinvestigated.

Theorem 1. (Albertson [1]) In every two-coloring of the edges of the complete graph of order> 6, there is a monochromatic triangle KT> for which two vertices have the same degree.

Theorem 2. (Albertson and Berman [2]) For all n, there exists a red-blue coloring of the

146 G. Chen and R. H. Schelp

edges of Kn that contains no red K4 and no two vertices of equal degree joined by a blueedge.

Theorem 2 tells us that Theorem 1 is best possible in the sense that the triangle X3

cannot be replaced by Kn, n > 4. Generally, given a graph G, we say that it has theRamsey repeated degree property if for all sufficiently large n every two edge-coloringy: E(Kn) H-» (R,B) yields a Ramsey host X e Ry(G,G) in which there are vertices x, ysatisfying dy(x) = dy(y). Here dy refers to either the degree in (R) or in (B), since verticesof the same degree in (R) have the same degree in (B). Recently, Erdos, Chen, Rousseauand Schelp generalized Theorem 1 by proving the following result.

Theorem 3. (Erdos, Chen, Rousseau, and Schelp [3]) For each m> I, the complete bipartitegraph Km^m and the odd cycle Cim+x have the Ramsey repeated degree property.

In the same paper they proved the following result.

Theorem 4. (Erdos, Chen, Rousseau, and Schelp [3]) In every two-coloring of the edges ofthe complete graph of order > r(G,H), there is a Ramsey host X such that

max d (x) - min d (y) < r(G, H) - 2.xeX yex

Further, the result is best possible in the sense that for every sufficiently large n, there is atwo-coloring of the edges of Kn such that for every Ramsey host X the following inequalityholds.

max d (x) - min d (y) > r(G, H) - 2.xeX yeX

Let V\ and V2 be two subsets of V(G). We use E(V\, V2) to denote the edges with oneend vertex in V\ and the other in V2. The degree of x in the graph G will be denoted bydG(x), or simply d(x) if the identity of G is clear from the context. Also the neighborhoodof x will be denoted by NQ(X) or N(x) when G is clear. For Y <= V(G) the degree spreadof Y is defined as

AG(Y) = maxd(y) — mind(v).yeY yeY

Let k be a positive integer and G be a graph. Let n be a sufficiently large integerand y: E(Kn) 1—• (B,R) be a two-coloring of the edges of the complete graph Kn. Weare interested in a generalization of the Ramsey repeated degree property replacing twovertices by k vertices in the Ramsey hosts for G. Since there are graphs that only havetwo vertices of the same degree, we do not expect all k of these vertices to have the samedegree. Thus our interest is in finding the minimum difference among the degrees of suchvertices. To be specific, let H be a graph and Y <= V(H). The degree spread of Y isdefined as

A(7) = maxdH(y) -mindH{y).

For the case of a two-colored complete graph with edge coloring 7, the degree spread ofY is the same for H = (B) and H = (R), and we denote this common value by A- (Y). Inthis paper, we will consider

4^(G,H) d= max min min A,(Y),n 7 XeR..(G,H) \Y\=k '

Ramsey Problems with Bounded Degree Spread 147

where the final maximum is taken over all two-colorings y: E(Kn) i—• (R,B) and the initialminimum is taken over all the Y ^ X. Thus ^ ( G , / / ) measures the smallest possibledegree spread of some /c-subset of vertices that must appear as a vertex subset of eithera red G or blue H under each two coloring of the edges of Kn.

Let V = {v\, v2, - • vm}, W = {wi, n>2, • • • wm}> and E = {vtVj, vtWj : 1 k - 2. Thus,

for all k > 2.In this paper we wish to determine the graphs G and H for which *Pjj(G, H) < k — 2 for

all sufficiently large n. When G = H, we write ^ ( G ) in place of Vkn(G,G).

2. Main theorem

Let G be a graph. A vertex subset / of V(G) is called distance q-independent if d(x, y) > q+\for every pair of vertices x and y in /. Notice that / is distance 1-independent if and onlyif / is independent. Also if / is distance q-independent with q > 2, then N(x) nN(y) = 0for each x ^ y in / . Let k > 2 be a positive integer. A bipartite graph G = (Ki, K2) iscalled (q,k)-independent if for each positive integer 1 < m < k — 1, there is a distanceq-independent set / of G such that

|/ n Ki| = m a n d |/ n K2| = /c -m.

Notice the following:

— The even cycle Ct with t > 4/c is (2, /c)-independent.— If G is (q, /c)-independent and H is a spanning subgraph of G, then H is (q,/c)-

independent.— Let G be a connected bipartite graph with diameter > 4/c. Then G is (2, /c)-independent.

With the above definition we state our main theorem as follows.

Theorem 5. Let k > 2 be a positive integer and G be a (2, k)-independent bipartite graph.Then there is a positive integer N such that for every positive integer n > N, ^ ( G ) < k — 2.Thus each two-coloring of the edges of the complete graph Kn gives a monochromatic copyof G with some k of its vertices of degree spread < k — 2 in the colored Kn.

The following results follow directly from the theorem.

Corollary 1. Let k > 2 be a positive integer and G be a connected bipartite graph withdiameter at least 4/c. Then, for n large, ^ ( G ) < k — 2.

Corollary 2. Let k > 2 be a positive integer and G be one of the following graphs:

— an even cycle Ct with t > 4/c;— a path with more than 4/c vertices;

— a tree containing a path of length > 4k;— a bipartite graph with at least k components.

Then, for n large, *F*(G) < k - 2.

The following results are needed in the proof of the main theorem. The first lemmageneralizes one of the most well-known facts about graphs, namely that in every graphthere are two vertices with the same degree.

Lemma 1. (Erdos, Chen, Rousseau, and Schelp [3]) Let G be a graph of order n and kbe a positive integer less than n. Then G contains a set Y of k vertices with degree spreadA(Y) at most fc — 2.

The following result is an analogue to Ramsey's theorem for bipartite graphs.

Lemma 2. ([4]) For all positive integers p there exists an N such that for every n> N eachedge coloring of Kn,n with two colors contains a monochromatic Kp,p. The least such N willbe denoted by r2(p).

By a well-known argument [5], the following holds.

Lemma 3. Let e be a positive number and p be a positive integer. There is a positive integerN such that for every graph G of order n> N, if the vertex subset

C = {v : dG(v) > P—nl-llp)

has more than en vertices, then G contains a copy of Kp^p with a vertex part in C.

3. Proof of the Main Theorem

Let G = (Fi,F2) be a (2,/c)-independent bipartite graph. Let max{|Ki|, \V2\) = p. The'sufficiently large' nature of n will be assumed throughout the argument, and no attemptwill be made to accurately estimate a threshold value of n at which the desired propertyfirst appears. Let y: E(Kn) i—• (R,B) be a given two-coloring of the edges of Kn. FromLemma 1, we start with k vertices {xi, x2, • • •, x/J for which |dy(x,-) — dy(xj)\ < k — 2. Inthe remainder of the proof we let m = n — k.

Partition the vertex set V(Kn) — {x\, x2, • • •, Xk] into 2k cells, (A\, A2,- • -,Ak), whereA[ G {B,R} such that a vertex v € (A\, A2,---, Ak) if the edge vx\ is colored with thecolor A\9 VX2 is colored with the color A2 , . . . , vxk is colored with the color Ak. Two cellsA = (Au A2,...,Ak) and A* = (A\, A*2,...,A*k) are conjugate if {At} U {A*} = {B,R} foreach i = 1, 2, . . . , fc: that is, the cell A* can be obtained from A by changing the bluecolors to red colors and the red colors to blue colors.

First we assume that either \(R, R, . . . , R)\ or |(£, B, . . . , B)\ > m/2k. Without loss ofgenerality, we assume that \(R, R, . . . , R)\ > m/2k. In fact, in this case we will prove thatthere is a monochromatic KPiP for which there are k vertices whose degree spread is atmost fc — 2. Let

C = {ve(R,R,...,R) : dR(v)> 2k+xp^nx-"}.

and D = V(Kn) - C - {xu x2, . . . , xk}. If \C\ > m/2k+1, then, by Lemma 3, there is a red

Kp,p-k in the red graph induced by V(Kn) — {xi, X2, . . . , x/J with the p — k vertex part

set in C (since n is large). Combine this with {xi, X2, . . . , x/J, giving us a red XP)P with k

vertices with degree spread at most k — 2. Thus |D| > m/2fe+1. Notice that each D G D has

ds(v) >n — 2k+lp~pnl~~p — 1. Clearly, for H large, D contains k vertices y\ y^ . . . , yk of the

same blue degree. Further, these k vertices have a large common blue neighborhood and

can be easily enlarged to a blue Kp,p containing yu yi, •••> yk- Thus, we may assume that

\(R,R,..., R)\ < m/2k and |(B, B, . . . , B)\ < m/2k.Since there are 2k cells, one of the other cells, say 4, must contain at least m/2k vertices.

In this case, we let ro = p and r\ = r2(p). For every i > 1, let ri+\ = r2(rt). We willshow that there is a monochromatic copy of G containing the vertex set {xi, X2, . . . , Xk}whenever m > 22krki. Hence the theorem will hold.

Without loss of generality, assume that

A = (R,R9 . . . , £ , B , B, . . . , B),Y VS t

where 5 and t are positive integers. For every pair of numbers i and j with 1 \A\ > m/2k and dB(xj+s) > \A\ > m/2k. Recall that \dR(xi)-dR(xj+s)\ <k — 2, and m = n — k is sufficiently large. Thus there exists a cell

AiJ = ( ^ i 9 ^ 2 ' ' ' ' » ^k )

with v4)J = B and X^ s = R such that

Since m/4k > r^/4 > rsr, there are A(l, 1) £ A and AJ j c A\^ such that

and all edges in E(A(1, l),A\ {) are colored with the same color.Since |A(1,1)| > rst-i and |J4I,2| > rst > rst_i, there are two vertex subsets A(l,2) c

v4(l, 1) and A\2 c ^1,2 such that

\A(l2)\>rst-2, |^ 2 |>r s f _ 2 ,

and all edges in E(A(1,2),A*12) are colored with the same color.Continuing in the same manner, we can show that there are

4(1,1) 3 4(1,2) => ••• 3 4(l,r) and

A\x c ,4 U , 4J>2 c Xi,2, . . . , 4J>f c ^i, t

such that

|4(l,i)| > rsf_/ > r(s_i)f, 14^1 > r(s_i)r

for each i = 1, 2, . . . , f, and all edges in £(4(1,0,4*,) are the same color.For the moment, assume for all i = 1, 2, ... t that the edges in E(A(l,i),A\i) are red.

Recall that G = (V\, V2) is (2,/c)-independent. Thus there are vertices y\, yi, •.., ys ^ V\and ys+\, ys+2, "', yk € K2 such that d(yt,yj) > 3 for each 1 ro = p, we can embed G in the red graph (R) such that V2 ^ 4(1, t) and

N(ys+i) c A*ul, N(ys+2) <= A\^ . . . , N(yk.{) c 4^_ 1 ? J/j - (Jfl} N(ys+i) <= 4 j r Replacingy, by the vertex x,- in the embedded graph for i = 1, 2, . . . , fc, we obtain a red graph Gwith k vertices xi, x2, * * * *& for which the degree spread is at most k — 2. Thus we mayassume that there is an i\ such that all edges in E(A(l,i\),A*li) are blue.

Note that |4(l,z'i)| > r(s_i)f and |42,i| > rst > r(s_i)f. There are 4(2,1) ^ ^4(1,i"i) andA*2 j ^ A^\ such that

14(2,1)| > r(s_i)f_i, 14^1 > r(s_i)r_i,

and all the edges in £(4(2, l ) , ^ ^ ) are the same color.Since \A(2,1)| > r(s_i)r_i and \A2^\ > rst > r(s_i)f_i, there are 4(2,2) ^ 4(2,1) and

A*22 ^ 42,2 such that

|4(2,2)| > r(,_1)r_2, 14 21 > r(s_i)f_2,

and the edges in £(4(2,2), 4^ 2) are the same color.Continuing in the same manner, there are

4(2,1) 3 4(2,2) 3 ••• 3 4(2,t)

such that

for each / = 1, 2, . . . , t, and all edges in £(4(2, i), 4^) are the same color.Notice that rt(S-2) > ro = p. If for every i = 1, 2, . . . , /c, the edges in £(4(2,0,4^) are

red, in the same manner as argued above, we can show that there is a red copy of Gthat contains xi, X2, . . . , x& whose degree spread at most k — 2. Thus we may assume thatthere is an i2 such that the edges in £(4(2,i2),A*2h) are blue. Notice that |4(2,/^)| > r(S-2)tand \A*2h\ > r{s_2)t.

Continuing in the same manner, we may assume that there are 2s vertex subsets

4 3 4(1, ii) 3 4(2, i2) 3 • • • 3 4(5, is), and

such that

and 14 (5, is)\ > ro = p. Further, for all j = 1, 2, . . . , 5, the edges in E(A(j, ij), 4*,) are blue.Since G = (V\, V2) is (2,^-independent, there are vertex sets

{zu z2, . . . , zs} c vu and {zs+u z,+2, . . . , zk} c K2

such that for each pair of vertices z,- and zy, d(zj,Zj) > 3 whenever 1 < / < 7 < /c. Then Gcan be embedded in the blue graph (B) such that V\ ^ 4(s,is) and

s-\

N(Zl) c ^ ^ , jV(Z2) c= 4 ^ , ... N(zs_0 c 4_U a_i 9 F2 - |JiV(zy-) c ^ . .7=1

Replacing each z, by x, for i = 1, 2, ... fe, we obtain a blue copy of G containingxi, X2, ... Xfc whose degree spread at most k — 2. This completes our proof.


References

[1] Albertson, M. O. (preprint) People who know people.[2] Albertson, M. O. and Berman, D. M. (preprint) Ramsey graphs without repeated degrees.[3] Erdos, P., Chen, G., Rousseau, C. C. and Schelp, R. H. (1993) Ramsey problems involving

degrees in edge-colored complete graphs of vertices belonging to monochromatic subgraphs.Europ. J. Combinatorics 14 183-189.

[4] Graham, R. L., Rothschild, B. R. and Spencer, J. H. (1990) Ramsey Theory (2nd Edition), JohnWiley & Sons, New York.

[5] Kovari, T., Sos, V. T. and Turan, P. (1954) On a problem of Zarankiewicz. Colloq. Math. 350-57.

Hamilton Cycles in Random Regular Digraphs

COLIN COOPER^ ALAN FRIEZE*§ and MICHAEL MOLLOY*

^School of Mathematical Sciences,University of North London,

London, U.K.* Department of Mathematics, Carnegie-Mellon University,

Pittsburgh PA15213, U.S.A.

We prove that almost every r-regular digraph is Hamiltonian for all fixed r > 3.

1. Introduction

In two recent papers Robinson and Wormald [8, 9] solved one of the major open problemsin the theory of random graphs. They proved the following result.

Theorem 1. For every fixed r > 3 almost all r-regular graphs are hamiltonian.

For earlier attempts at this question see Bollobas [2], Fenner and Frieze [5] and Frieze[6], who established the result for r > r$.

In [8] (r=3) a clever variation on the second moment method was used, and in [9] (forr > 4) this idea plus a sort of monotonicity argument was used.

In this paper we will study the directed version of the problem. Thus, let Qnr = Qdenote the set of digraphs with vertex set [n] = {l,2,...,w} such that each vertex hasindegree and outdegree r. Let Dnr = D be chosen uniformly at random from Qnr.

Theorem 2.( 0 r = 2

lim PHD is Hamiltonian) = < ,n->oo [ 1 r > 3.

Supported by NSF grant CCR-9024935

154 C. Cooper, A. Frieze and M. Molloy

The case r = 2 follows directly from the fact that the expected number of Hamiltoncycles in Dni tends to zero.

Our method of proof for r > 3 is quite different from [8, 9] although we will use theidea that for r > 3, a random r-regular bipartite graph is close, in some probabilisticsense, to a random (r — l)-regular bipartite graph plus a random matching.

Our strategy is close to that of Cooper and Frieze [4], who prove that almost every3-in, 3-out digraph is Hamiltonian.

2. Random digraphs and random bipartite graphs

Given Dnr = ([n],A\ we can associate it with a bipartite graph B = Bnr = (j){Dnr) =([n], [n],E) in a standard way. Here B contains an edge {x,y} iff D contains the directededge (x,y). The mapping 0 is a bijection between r-regular digraphs and r-regular bipartitegraphs, so B is uniform on the latter space, which we denote by Q^r.

For r > 3 we wish to replace Bnr by Bnr-\ plus an independently chosen randomperfect matching M of [n] to [n]. This is equivalent to replacing D by no U D, where Iloand D are independent and

(i) Ilo is the digraph of a random permutation,b

Of course n is the union of vertex disjoint cycles. We call such a digraph a permutationdigraph. Its cycle count is the number of cycles.

The arguments of [9] allow us to make the above replacement. A brief sketch of whythis is so would certainly be in order.

Let XM denote the number of perfect matchings in Bnr. Arguments in [9] demonstratethe existence of e(b) > 0 such that for b > 0 fixed,

lim Pr(XM > E(XM)/b) > 1 - e{b\n—KX)

where e(b) —• 0 as b —• oo.Now consider a bipartite graph & = (Q^r_1?Q^r,<f). There is an edge from G e ^ r _ i

to G e Q^r iff G = GUM, where M is a perfect matching. Now choose (G,G) randomlyfrom S. Let A denote some event defined on O*r and A = {(G,G) eS \G e A}. Then,since the maximum and minimum degrees of the Q%r_x vertices of gft are asymptoticallyequal to n\e~^r~1^ [1],

Pro(^) = (1 + o(l))Pri(.4),

where o(l) refers to n —• oo, Pr0 refers to the space $ with the uniform measure, andPr\ refers to (randomly chosen) G = Bnr-\ plus a randomly chosen M, disjoint from

On the other hand, if Pr refers to Bnr,

ProU) =

Hamilton Cycles in Random Regular Digraphs 155

> (Pr(A)-e(b))/b.

Thus

Pr(A) < e(b) + (b + o(l))Pri(A).

Thus, if A is {<f>~x(Bnj-\ UM) is non-Hamiltonian} (M disjoint from Bnjr-\ here), we canshow that Pr(A) —> 0 (as n —• oo) by proving that Pri(4) —• 0 (as n —> oo), since b can bearbitrarily large.

Finally, if Pr2 refers to Bnr-\ plus a randomly chosen M (not necessarily disjoint fromB«,r-i), then Pr2(A) —• 0 (as n —> oo), implies Pr\(A) —• 0 (as n —• oo) since the probabilitythat M is disjoint from J5w,r_i in this case tends to the constant e~^~l) > 0.

We have thus reduced the proof of Theorem 1 to showing that

lim Pr(n0 U D is Hamiltonian) = 1.n—•oo

In fact we have only to prove the result for r = 3 and apply induction. Thus assume r = 3from now on.

We will use a two phase method as outlined below.Phase 0. As Ilo is a random permutation digraph, it is almost always of cycle count at

most 21ogn, see, for example [3].Phase 1. Using D, we increase the minimum cycle size in the permutation digraph to at

least n0 = [lOOn/logn].Phase 2. Using Z), we convert the Phase 1 permutation digraph to a Hamilton cycle.In what follows inequalities are only claimed to hold for n sufficiently large. The term

whp is short for with high probability i.e. probability 1 — o(l) as n —• oo.

3. Phase 1: Removing small cycles

We partition the cycles of the permutation digraph Flo into sets SMALL and LARGE,containing cycles C of size \C\ < no and \C\ > no, respectively. We define a NearPermutation Digraph (NPD) to be a digraph obtained from a permutation digraph byremoving one edge. Thus an NPD T consists of a path P(F) plus a permutation digraphPD(T) that covers [n] \ V(P(T)).

We now give an informal description of a process that removes a small cycle C from acurrent permutation digraph IT. We start by choosing an (arbitrary) edge (VQ9 UQ) of C anddelete it to obtain an NPD To with Po = P(T0) e &(uo9vo)9 where &(x,y) denotes the setof paths from x to y in D. The aim of the process is to produce a large set S of NPDssuch that for each F e 5,

(i) P(F) has a least no edges, and(ii) the small cycles of PD(F) are a subset of the small cycles of IT.We will show that whp the endpoints of one of the P(F)s can be joined by an edge to

create a permutation digraph with (at least) one less small cycle.The basic step in an Out-Phase of this process is to take an NPD F with P(F) e ^(uo,v)

and examine the edges of D leaving v. Let w be the terminal vertex of such an edge, andassume that F contains an edge (x, w). Then V = F U {(v, w)} \ {(x, w)} is also an NPD.

V is acceptable if(i) P(F') contains at least no edges, and(ii) any new cycle created (i.e. in V and not F) also has at least no edges.If F contains no edge (x, w), then w = uo. We accept the edge if P(T) has at least no

edges. This would (prematurely) end an iteration, although it is unlikely to occur.We do not want to look at very many edges of D in this construction and we build a

tree To of NPDs in a natural breadth-first fashion, where each non-leaf vertex F gives riseto NPD children V as described above. The construction of To ends when we first havev = \yjnlogn\ leaves. The construction of To constitutes an Out-Phase of our procedureto eliminate small cycles. Having constructed To we need to do a further In-Phase, whichis similar to a set of Out-Phases.

Then whp we close at least one of the paths P(F) to a cycle of length at least no. If\C\ > 2 and this process fails, we try again with a different edge of C in place of (UO,VQ).

We now increase the formality of our description. We start Phase 1 with a permutationdigraph IIo and a general iteration of Phase 1 starts with a permutation digraph FI whosesmall cycles are a subset of those in Flo. Iterations continue until there are no more smallcycles. At the start of an iteration, we choose some small cycle C of II. There then followsan Out-Phase, in which we construct a tree To = To(II, C) of NPDs as follows: the rootof To is Fo, which is obtained by deleting an edge (VQ,UO) of C.

We grow To to a depth at most |"1.51ogw~|. The set of nodes at depth t is denoted bySt. Let F e St and P = P(F) e ^(uo,v). The potential children V of F, at depth t + 1 aredefined as follows, with w the terminal vertex of an edge directed from v in D.

Case 1: w is a vertex of a cycle C € PD(T) with edge (x, w) € C. Let V = F U {(v, w)} \{(x,w)}.

Case 2: w is a vertex of P(F). Either w = MO, or (x, w) is an edge of P. In the formercase, F u {(u,w)} is a permutation digraph IT, and in the latter case, we let V =Fu{(t;,w)}\{(x,w)}.

In fact we only admit to St+\ those V that satisfy the following conditions:

0(i), the new cycle formed (Case 2 only), must have at least no vertices, and the pathformed must either be empty or have at least no vertices. When the path formed isempty we close the iteration and if necessary start the next with IT.Now define W+, W- as follows: initially W+ = W- = 0. A vertex x is added to W+whenever we learn any of its out-neighbours in D, and to W- whenever we learn anyof its in-neighbours. W = W+ U W-. We never allow \W\ to exceed n9/l0.The only information we learn about D is that certain specific arcs are present. Theproperty we need of the random graph D is that if x ^ W+ and S is any set of vertices,disjoint from W,

These approximations are intended to hold conditional on any past history of thealgorithm such that \W\ < n9/10. Furthermore, if x e W+, but only one neighbour y is

known then, where y # S,

Similar remarks are true for A/_(x). Thus, since W remains small, N+(v) are usually(near) random pairs in W.

C(ii) x<£W.An edge (v, w) satisfying the above conditions is described as acceptable. In order toremove any ambiguity, the vertices of St are examined in the order of their construction.

Lemma 3. Let C e SMALL. Then

Pr(3t < |"log3/2 v] such that \St\ > v) = 1 - O((loglogn/logn)2).

Proof. We assume we stop construction of To, in mid-phase if necessary, when \St\ = v,and show inductively that whp (3/2)' <\St\<21, for t > 3. Let t* denote the value of twhen we stop. Thus the overall contribution to | W\ from this part of the algorithm is atmost \SMALL\ x 2 r + 1 < n0M.

In general, let Xt be the number of unacceptable edges found when constructing St+1,(t = 1,2, ...,£*). The event of a particular edge (v, w) being unacceptable is stochasticallydominated by a Bernouilli trial with probability of success p < log log n/n. (in general,inequalities are only claimed for sufficiently large n). To see this, observe that there is aprobability of at most 201/ log n that in Case 2 we create a small cycle or a short path.There is an O(n~1/10) probability that x e W. Finally there is the probability that w lies ina small cycle. Now in a random permutation the expected number of vertices in cycles ofsize at most k is precisely k/n. Thus whp Ilo contains at most rcloglogrc/(21ogn) verticeson small cycles, so, given this, the probability that w lies on a small cycle is at mostloglogn/(21ogn).

For t < c, constant, the probability of 2 or more unacceptable edges in layers t < cis 0 (22c(loglogn)2/(k)gn)2), and thus |St+i| > 2|Sf| - 1 > (3/2)r for 3 < t < c withprobability 1 — O((log log n/ log n)2).

In order to see this, note that in the case where there is only one acceptable edge at thefirst iteration, subsequent layers expand by a power of 2, and |Si| =2 otherwise.

For t > c,c large, the expected number of unacceptable edges at iteration t is at mostH — 2p\St\, and thus, by standard bounds on tails of the Binomial distribution,

This upper bound is easily good enough to complete the proof of the lemma. •

Now, To has leaves Fj, for / = 1,..., v, each with a path of length at least no (unless wehave already successfully made a cycle). We now execute an In-Phase. This involves theconstruction of trees TtJ = 1,2,...v. Assume that P(F,) e ^(uo,Vi). We start with F, and2u and build Tt in a similar way to To, except that here all paths generated end with vt.This is done as follows: if a current NPD F has P(T) € ^(u,Vi), we consider adding an

edge (w,w) e D and deleting an edge (w,x) e F (as opposed to (x, w) in an Out-Phase).Thus our trees are grown by considering edges directed into the start vertex of each P(F),rather than directed out of the end vertex. Some technical changes are necessary, however.

We consider the construction of our v trees in two iterations. First of all we grow thetrees only enforcing condition C(ii) of success, and thus allow the formation of smallcycles. We try to grow them to depth k = n°g3/2 vl- We also consider the growth of the vtrees simultaneously. Let 7,-/ denote the set of start vertices of the paths associated withthe nodes at depth { of the ith tree, i = 1,2..., v, t = 0 ,1 , . . . , fc. Thus Tu0 = {u0} for all i.We prove inductively that Tt/ = T\/ for all ij. In fact, if T,-/ = T\/, the acceptable Dedges have the same set of initial vertices, and, since all of the deleted edges are Flo-edges(enforced by C(ii)), we have T",y+i = T1/+1.

The probability that we succeed in constructing v trees Ti,T2,. . .Tv, say, is, by theanalysis of Lemma 3, 1 — O((log log n/ log n)2). Note that the number of nodes in each treeis at most 2/c+1 < n87, so the overall contribution to \W\ from this part of the algorithmis O(w87logn).

We now consider the fact that in some of the trees some of the leaves may havebeen constructed in violation of C(i). We imagine that we prune the trees Ti ,T2 , . . . r v

by disallowing any node that was constructed in violation of C(i). Let a tree be BADif after pruning it has less than v leaves. Now an individual pruned tree has essentiallybeen constructed in the same manner as the tree To obtained in the Out-Phase. (We havechosen k large enough so that we can obtain v leaves at the slowest growth rate of 3/2per node.) Thus

and

v ( l 0 , g l ° g " )

and

Pr(3 > v/2 BAD trees) =

Thus

Pr(3 < v/2 GOOD trees after pruning)< Pr(failure to construct TUT2,... Tv) -f Pr(3 > v/2 BAD trees)

Thus with probability 1-O((loglogn/lognj1), we end up with v/2 sets of v paths, eachof length at least lOOn/logn, where the /th set of paths have Vt, say, as their set of startvertices, and Vt as a final vertex. At this stage each vt $. W+, and each V\ C\ W_ = 0. Hence

2v /2

n

= 0{n~x).

Pr(no II edge closes one of these paths) < 1 1 + 0 , ,,,„

Consequently the probability that we fail to eliminate a particular small cycle isO((log log n/ log n)2) and we have the following.

Lemma 4. The probability that Phase 1 fails to produce a permutation digraph with minimalcycle length at least no is o(l).

At this stage we have shown that TloUD almost always contains a permutation digraphn* in which the minimum cycle size is at least no. We shall refer to IT as the Phase 1permutation digraph.

4. Phase 2: Patching the Phase 1 permutation digraph to a Hamilton cycle

Let Ci,C2,...,C/c be the cycles of FT, and let ct = \Q \ W\, c\ < c2 < • • • < ck, andc\ > no — rc3/4 > 99\ogn/n. If k = 1, we can skip this phase, otherwise let a = n/\ogn. Foreach C,-, we consider selecting a set of m, = 2[ct/a\ + 1 vertices v e Ct\ W, and deletingthe edge (v,u) in n*. Let m = Y^!i=\ m,-, and relabel (temporarily) the broken edges as(vt,Ui)J e [m] as follows: in cycle C,, identify the lowest numbered vertex x, that loses acycle edge directed out of it. Put v\ = x\, and then go round C\ defining V2,v^...vmx inorder. Then let vmi+\ = X2, and so on. We thus have m path sections Pj e ^(u^j^Vj) inIT* for some permutation <\>. We see that <j> is an even permutation as all the cycles of 0are of odd length.

There is a chance that we can rejoin these path sections of IT* to make a Hamiltoncycle using D. Suppose we can. This defines a permutation p, where p(i) = j if Pt is joinedto Pj by (Vi,u<t>(j)), where p e Hm the set of cyclic permutations on [m]. We will use thesecond moment method to show that a suitable p exists whp. Unfortunately a technicalproblem forces a restriction on our choices for p.

Given p, define y = (pp. In our analysis we will restrict our attention to p G R^ ={p e Hm : (\>p = y,y e Hm}. If p G R^, we have not only constructed a Hamilton cycle inIT* UD, but also in the auxiliary digraph A, whose edges are (Uy(i)).

Lemma 5. (m - 2)! < |/fy| < (m - 1)!

Proof. We grow a path I,y(l),y2(l),...,yfc(l) in A, maintaining feasibility in the way wejoin the path sections of n* at the same time.

We note that the edge (i,y(i)) of A corresponds in D to the edge (v^u^)). In choosingy(l) we must avoid not only 1, but also 0(1), since y(l) = 1 implies p(l) = 1. Thus thereare m — 2 choices for y(l), since 0(1) ^ 1.

In general, having chosen y(l),y2(l),...,y*(l), 1 < k < m — 3, our choice for y/c+1(l) isrestricted to be different from these choices, and also 1 and /, where U{ is the initial vertexof the path terminating at i^ i ) made by joining path sections of II*. Thus there are eitherm — (k + 1) or m — (k + 2) choices for yfc+1(l), depending on whether or not i = 1.

Hence, when k = m — 3, there may be only one choice for ym~2(l), the vertex h say.After adding this edge, let the remaining isolated vertex of A be w. We now need to showthat we can complete y, p so that y,p € Hm.

Which vertices are missing edges in A at this stage? Vertices l,w are missing in-edges,

and h, w out-edges. Hence the path sections of IT* are joined so that either

u\ —> Vh, uw —• vw o r u\ —• vw, uw —• Vh.

The first case can be (uniquely) feasibly completed in both A and D by setting y(h) =w,y(xv) = 1. Completing the second case to a cycle in IT* means that

y = (l,y(l), ...,ym~2(l))(w), (1)

and thus y ^ Hm. We show that this case cannot arise.When, y = 4>p and 0 is even, y and p have the same parity. On the other hand, p G Hm

has a different parity to y in (1), which is a contradiction.Thus there is a (unique) completion of the path in A. •

Let H stand for the union of the permutation digraph IT and D. We finish our proofby proving the following.

Lemma 6. Pr(H does not contain a Hamilton cycle) = o(l).

Proof. Let X be the number of Hamilton cycles in H resulting from rearranging the pathsections generated by 0 according to those p e Rx. We will use the inequality

Em\ mHere probabilities are now with respect to the D choices for edges incident with verticesnot in W, and on the choices of the m cut vertices.

Now the definition of the m, gives

In . In .k <m< hfc,

a a

so

(1.99)logn < m < (2.01)logn.

Also

k < m/199,m/ > 199 and — > -^— \<i<k.mt 2.01

Let Q denote the set of possible cycle re-arrangements. Then co G Q is a success if Dcontains the edges needed for the asssociated Hamilton cycle. Thus, where e = O(l/n1/10),

E(X) = S^ Pr((D is a success)

>- ( ^ + ) ) < w

1-0(1) (2m

>

"

en) ^\{\m)mi

- o{\))(2n)-mim (2m\ra^yra

(1-O(1))(2TT)-W/3 9 8 / 3.98 \

mjm" V 2.0502/

(lm\ / ea y\en) V2.01 x 1.02/

(3)

Let M, Mr be two sets of selected edges that have been deleted in J and whose pathsections have been rearranged into Hamilton cycles according to p, p' respectively. LetN, Nf be the corresponding sets of edges that have been added to make the Hamiltoncycles. What is the interaction between these two Hamilton cycles?

Let s = \M n M'\ and t = \N n Nf\. Now t < s, since if (v,u) e N n N\ there must bea unique (v, u) e M n M' that is the unique ./-edge into u. We claim that t = s impliest = s = m and (M,p) = (M\pf). (This is why we have restricted our attention to p e #</>.)Suppose, then, that f = s and (t;,-,M,-) G M n Mr. Now the edge (UJ,M7(J)) G N, and sincet = s, this edge must also be in N'. But this implies that (vy^Uy^) G M\ and hence inM Pi Mr. Repeating the argument, we see that (vyk^,uyk^) G M n Mr for all /c > 0. But yis cyclic so our claim follows.

We adopt the following notation. Let t = 0 denote the event that no common edgesoccur, and (5, t) denote \M n Mf\ = s and |N n N'\ = t. Thus

EEE5=2 t=\ «

= E(X) + Ex + £2 say.

Clearly,

(4)

(5)

For given p, how many p' satisfy the condition (s, t)l Previously | ^ | > (m — 2)!, and now\R(j)(s, t)\ < (m — t — 1)! (consider fixing t edges of F').Thus

m s—1(m — t - 1)! /n

Now,

- oil / Vm m,- - O

J

where the o(l) term is O((\ogn)3/n). Also

> 7T-

2 2

> — for (7i H- • • • dfc = s,

and

Hence

171 S— 1

s2-Ol \ sms f s2

^ W V2w7

= 0(1).

To verify that the right-hand side of (7) is o(l), we can split the summation into

L ^ J / (101)nexp{-s/2m}y 1

and

S2= y f(2.0l)nexp{-s/2m}2a J s\

Ignoring the term exp{—s/2m}, we see that

[ ( 5 0^O g"J< ^ ^ ( ( 1 . 0 0 5 ) l o g n ) -^ ?.

since this latter sum is dominated by its last term.

Finally, using exp{—s/2m} < e~1/8 for s > m/4, we see that

S2 < n(L005^m < n9/10.

The result follows from (2) to (7). •

Acknowledgement

We thank the referee for his/her comments.

References

[1] Bender, E. A. and Canfield, E. R. (1978) The asymptotic number of labelled graphs with givendegree sequences, Journal of Combinatorial Theory (A) 24 296-307.

[2] Bollobas, B. (1983) Almost all regular graphs are Hamiltonian. European Journal of Combina-torics 4 97-106.

[3] Bollobas, B. (1983) Random graphs, Academic Press.[4] Cooper, C. and Frieze, A. M. (to appear) Hamilton cycles in a class of random digraphs.[5] Fenner, T I. and Frieze, A. M. (1984) Hamiltonian cycles in random regular graphs, Journal

of Combinatorial Theory B 37 103-112.[6] Frieze, A. M. (1988) Finding hamilton cycles in sparse random graphs, Journal of Combinatorial

Theory B 44 230-250.[7] Kolchin, V. F. (1986) Random mappings, Optimization Software Inc., New York.[8] Robinson, R. W. and Wormald, N. C. (1992) Almost all cubic graphs are Hamiltonian, Random

Structures and Algorithms 3 117-126.[9] Robinson, R. W. and Wormald, N. C. (to appear) Almost all regular graphs are Hamiltonian,

Random Structures and Algorithms.

On Triangle Contact Graphst

HUBERT de FRAYSSEIX, PATRICE OSSONA de MENDEZand PIERRE ROSENSTIEHL

CNRS, EHESS, 54 Boulevard Raspail, 75006, Paris, France

It is proved that any plane graph may be represented by a triangle contact system, that is acollection of triangular disks which are disjoint except at contact points, each contact pointbeing a node of exactly one triangle. Representations using contacts of T- or Y-shapedobjects follow. Moreover, there is a one-to-one mapping between all the triangular contactrepresentations of a maximal plane graph and all its partitions into three Schnyder trees.

1. Introduction: on graph drawing

An old problem of geometry consists of representing a simple plane graph G by means ofa collection of disks in one-to-one correspondence with the vertices of G. These disks mayonly intersect pairwise in at most one point, the corresponding contacts representing theedges of G. The case of disks with no prescribed shape is solved by merely drawing foreach vertex v a closed curve around v and cutting the edges half way. The difficulty ariseswhen the disks have to be of a specified shape. The famous case of circular disks, solvedby the Andreev-Thurston circle packing theorem [1], involves questions of numericalanalysis: the coordinates of the centers and radii are not rational, and are computedby means of convergent series. This problem is still up to date, and considered in manyresearch works. In the present paper we will consider triangular disks. A contact point(A,B) is a node of the triangle A and belongs to the side of the triangle B (but is not anode of B). The asymmetry of the pair (A,B) defines an orientation of the correspondingedge. Such an arrangement is called a triangle contact system (see Figure 1). It is obviousthat any triangle contact system S defines an oriented simple plane graph G(5). Our resultis that any simple plane graph may be represented by a triangle contact system, and thatthese representations for a maximal plane graph are in one-to-one correspondence withthe Schnyder partitions (see definition below).

+ This work was partially supported by the ESPRIT Basic Research Action Nr. 7141 (ALCOM II).

166 H. de Fraysseix, P. Ossona de Mendez and P. Rosenstiehl

(a)

toFigure 1 (a) A triangle contact system (b) A common angle representation of the system la

(c) An isosceles representation of the system la

On Triangle Contact Graphs 167

The main tools we shall make use of are the so-called canonical or shelling order ofthe vertices of a maximal plane graph G introduced in [6], and a partition of the interioredges of G into three trees, due to Schnyder [9].

2. Triangle contact systems and Schnyder partitions

We consider planar objects defined as closed 2-cells. The instances of planar objectsappearing below are closed triangles, segments. We also consider T-shaped objects, orY-shaped objects as limit cases.

Definition. A contact system is a finite family of planar objects such that two objects ofthe family intersect in at most one point, and that three objects have no common point.

A consideration of the tubular neighborhoods of the objects shows that a contact systemS defines a unique simple plane graph G(S). Each bounded face of G(S) corresponds in Sto a bounded hole of the representation, the unbounded face of G(S) corresponds in S tothe unbounded hole. The system S is biconnected if G(S) is biconnected.

Definition. A triangle contact system S is a contact system such that every object is aclosed triangle, and such that each contact point is a node of exactly one triangle. Asubsystem of S is a family of triangles of S. A free node is a node of a triangle which isnot a contact point.

A triangle contact system S is maximal if and only if each hole of S is delimited byexactly three triangle sides.

Notice that the graph G(S) defined by a maximal triangle contact system S is a maximalplanar graphT.

Two triangle contact systems S and Sf are isomorphic if there exists an isomorphismmapping of the sides of the triangles of S into the sides of the triangles of Sf.

Definition. A canonical order, or shelling order, of the vertices of a maximal plane graphG with external face M, V, W is a labelling of the vertices v\ = u, vi = v, v^,..., vn = w meetingthe following requirements for every 4 < k < n:

— the subgraph Gk-\ <= G induced by v\,V2,...,Vk-\ is 2-connected, and the boundary ofits exterior face is a cycle Q_i containing the edge uv;

— the vertex Vk belongs to the exterior face of G/c, and its neighbors in Gk-\ form asubinterval of the path Q_i — uv, with at least two-elements.

It is proved in [6] that such a labelling is always possible and can be computed in lineartime by packing the vertices one by one.

' The converse is not true, as pointed out by the referee: K?> may be represented by a non-maximal trianglecontact system


Given a biconnected triangle contact system S and a bounded hole H of 5, let tu bethe number of triangles adjacent to H and let pn be the number of free nodes belongingto the boundary of H.

Theorem 2.1. A biconnected triangle contact system S is isomorphic to a subsystem of amaximal triangle contact system if and only if

tH-PH = 3 (1)

for any bounded hole H of S.

Actually, one can prove by simple counting that, if (1) holds for any bounded hole, wehave on the unbounded hole H^:

tHoo-pHoo=-2 (2)

In order to prove necessity, we need the following lemma.

Lemma 2.1. Let G be a maximal planar graph and G be a biconnected subgraph of G.Then there exists a sequence of k biconnected subgraphs of G: G\ = G,..., G,,..., G = G'obtained at each step by deleting exactly one vertex.

Proof. This lemma is a straightforward consequence of the shelling packing order appliedto G while starting from G. •

This lemma implies that a subsystem Sf of a maximal triangle contact system S can beobtained by deleting triangles Tt one by one, keeping the subsystems St biconnected.

Proof of theorem 2.1. The proof of necessity is by induction on the number of removedtriangles. The property holds for a maximal triangle contact system, since, for a hole //,tH = 3 and pn = 0. Assume the property holds when the first i — 1 triangles are removed,and consider the removal of the /th triangle Tt from S,-_i and the corresponding hole H.As 5,-_i is biconnected, each triangle adjacent to Tt in 5,_i is adjacent to exactly two holesadjacent to Tt. Therefore, the number d of holes adjacent to Tt equals the number oftriangles adjacent to Tt, that is the degree of T,-. Let H\,...,Hj,...,Hd denote these holes.We have

tH ~PH =

that is tH — PH = 3. Thus, (1) holds for H.The condition is sufficient. Let S be a biconnected triangle contact system satisfying

condition (1), H a bounded hole of S, and let tH denote the number of triangles adjacentto H and having i free nodes in the boundary of H. We shall prove that condition (1)


allows us to fill up H with triangles. There exists a triangle To adjacent to H and withoutfree nodes in H. If the hole H is not yet filled up, To can be chosen in such a way that itis adjacent to a triangle T\ having at least one free node in H. We consider three cases:

— T\ has three free nodes in H. Add a triangle T in contact with To and Tu in theposition displayed in Figure 2a; the number t3

H decreases.— T\ has two free nodes in H. Add a triangle T in contact with To and T\ in the position

displayed in Figure 2b; t2H decreases and t3

H remains unchanged.— T\ has one free node in H. Add a curve-sided triangle T in contact with To and T\ in

the position displayed in Figure 2c; £# decreases, while t2H and t3

H remain unchanged.

At each step the contact system remains biconnected, and as t2H + t3

H + tu decreases, theiteration stops with tu = 3. Equation (2) allows us to fill up the unbounded hole Hoo by asimilar process. The final system obtained can be redrawn as a maximal triangle system(see Section 3), as required. •

One more definition given in [9]:

Definition. A Schnyder realizer of a maximal plane graph is a partition of the interioredges of G in three sets Yr, Yg, Yb of directed edges such that for each interior vertex v

— v has indegree one in each of 7r, Yg, Y ,— the counterclockwise order of the edges incident on v is: entering in Tr, leaving in Tb,

entering in Tg, leaving in Tr, entering in Tb, leaving in Tg.

The first condition of the definition implies that Yr, Yg and Yb are three trees orientedfrom their roots. Schnyder proved in [9] that any maximal plane graph has a realizer.

Any Schnyder realizer may be extended into a Schnyder partition by assigning the edgesof the exterior face to the three trees in such a way that all the edges of the graph arepartitioned into three trees.

In the following, Or, Og and Ob will denote the partial orders on V(G) induced by thethree oriented trees of a Schnyder realizer, and Or, Og and Ob will denote the reversedpartial orders.

Now we relate triangle contact systems and Schnyder partitions.

Theorem 2.2. A maximal triangle contact system S defines a Schnyder partition of G(S).

For the proof of this theorem we need a lemma. Given a maximal triangle contactsystem S, let v be a non-free node of a triangle T, belonging to the side of a triangle T',and let / be the mapping that associates with v the node of T' opposite to v (see Figure3).

Lemma 2.2. The mapping f is acyclic.

Proof. We prove a stronger result: there exists no elementary cycle of nodes Ni,...,Nk

such that JVj+i = /(iV,-) and such that N\ and Nk belong to a common triangle. Such a

cycle of k nodes, if it exists, defines a cycle of k triangles. The deletion of the triangles of

170 H. de Fraysseix, P. Ossona de Mendez and P. Rosensdehl

H

(a)

H

(b)

(c)Figure 2 How to fill up the hole H


Figure 3 The contact graph edges in three colours


S inside the cycle produces a subsystem of S that is still biconnected. For the hole H sodefined, tH = k, and pu >tu — 2, because each triangle except the first and the last haveexactly one free node in H. According to Theorem 2.1, the hole H cannot be filled up,and this contradicts the maximality of S. •

Proof of theorem 2.2. Given an internal triangle T, consider two different chains definedby both starting from T. By the argument of the lemma, these two chains cannot end atone and the same triangle. Then the three chains starting at T lead to the three externaltriangles of S; call them Tr, Tg and 7&, taken in the circular order. The chains endingat a node of Tr (respectively Tg, Tb) constitute an acyclic connected subgraph of G(S),which we call the red (respectively green, blue) tree, and which we orient away fromTr (respectively Tg, Tb). Thus, the edges of G(S) are partitioned into a red, a green anda blue tree, all three oriented away from their respective roots. The nodes belongingto one and the same side all have the same image under / , and hence belong to thesame tree. It follows that the sides of each triangle are colored red, green and blue. Nowconsider three adjacent triangles, and note that the side colors are all in the same circularorder. Therefore, all the triangles are colored in the same circular order. The nodes of atriangle represent the three incoming edges; the other contacts, sorted by colors, representoutgoing edges. So, this coloration and this orientation define a Schnyder partition ofG(S). •

Remark. Two non-isomorphic triangle contact systems representing the same maximalplane graph G define two distinct Schnyder partitions of G.

Now, given a Schnyder partition of a maximal planar graph, the way to generate atriangle contact system that defines it is the purpose of the next section.

3. Construction of a triangle contact system

In this section we prove the main result of the paper. Let us first recall how a shellingorder defines a Schnyder realizer (using the above notation). When at step k the vertexVk is packed, color red (respectively green) the leftmost (respectively the rightmost) edgeincident to Vk and Ck-u and orient it toward Vk\ color blue all the other edges incidentto Vk and Ck-u and orient them away from Vk. Each vertex of G gets a red and a greenfather when packed and a blue father corresponding to its last neighbor packed. It is easyto check that this partition into three colored trees is a Schnyder realizer. If we extendthe coloration to the outer face in such a way that each edge is assigned to one of thetwo trees to which it is incident, we get a Schnyder partition.

We shall first prove a representation result for maximal plane graphs. The general case(Theorem 3.2) will follow immediately.

Theorem 3.1. Any maximal plane graph G has a triangle contact representation.

Proof. Consider a shelling order and its corresponding Schnyder realizer. In the following,

Tk will denote the triangle representing the vertex Vk and (j)r(k) (respectively 0g(fcthe shelling label of the red (respectively green and blue) father of Vk in the shelling order.

We start with a maximal triangle contact system formed by the triangles Tu T2 and Tw,these triangles having their bases parallel to the x-axis, at ordinates 1,2 and n, respectively.

We construct iteratively the representations of the graphs Gk (2 < k < n). Each triangleTk (2 < k < n) gets its blue base parallel to the x-axis at ordinate k and its opposite nodeat ordinate (j>b(k).

At each step, the triangles bounding the representation of Gk will correspond, in thesame circular order, to the vertices of Q , and the intersection of the unbounded hole withthe half-plane y > k is y-convex (the intersectection with any vertical line is connected).

Assume that the representation of Gk-\ has been completed according to the previousconstraints. By definition of shelling orders, the neighbors Vkl9...,Vkp of Vk in Gk-\ forman interval of Q_i. As Vk is the blue father of the vertices ^,.(1 < i < p), the free nodes ofthe corresponding triangles have ordinate k. The vertices Vkx and Vkp are, respectively, thered and green fathers of v^ As noticed above, their blue fathers are packed after Vk, andhave a label greater than k. Let xr (respectively xg) denote the abscissa of the point ofthe right (respectively left) side of triangle T^ (respectively Tkp) at ordinate k. Accordingto the y-convexity of the intersection of the unbounded hole and the half plane y > k — 1,the region defined by y > k and xr < x < xg is empty. The triangle Tk is placed (crossingfree) with coordinates (xr,fc),(x&,fc),(a^xr + (1 — (Xk)xb,^b{k)) with a G [0,1]. The twoconditions on the representation are obviously preserved for Gk-

The representation of Gn is obtained by adding the already defined triangle Tn to therepresentation of Gn-\. •

Remark. We may require the triangles to be isosceles (or right-angle) by a proper choiceof T\, T2 and Tn and the assignment of the value \ (respectively 0) to all the oik coefficients.It is easy to check that there are graphs that cannot be represented by a contact systemof equilateral triangles.

Theorem 3.2. Any plane graph G has a triangle contact representation.

Proof. The graph G can be augmented into a maximal plane graph G by adding verticesand edges incident to the added vertices. Then a representation of G follows from arepresentation of G by deleting the triangles corresponding to the added vertices. •

Proposition 3.1. A triangle contact representation of a plane graph G can be computed inO(n2+e) time, with any given e > 0.

Proof. The graph G can be augmented in linear time into a maximal plane graph G byadding at most 2 vertices per face. The graph G and a shelling order of its vertices can becomputed in linear time. By choosing a right-angle or isosceles triangle representation, eachcoordinate needs linear precision. As shown by D. Knuth, each intersection computationmay then be achieved in O(n{+€) time. •


In a representation, the ratio between the largest and the smallest triangle may beexponential, and for some graphs this is unavoidable. In the following, we shall describeanother type of contact representation on an n x n grid.

4. Other representations

From the triangle contact representation of a maximal plane graph G, say the isoscelesone, we now deduce another representation.

We construct a T-contact system (contact system of T-shaped objects): for each isoscelestriangle T draw the perpendicular height corresponding to its horizontal base and, if Tis neither T\, T2 nor Tn, extend the base of T on both sides, until a contact with theperpendicular height of its red and green fathers is reached (see Figure 4a).

This representation does not require a triangle contact representation to be achieved.Actually, any x-coordinates of the vertical segments compatible with the shelling orderand any y-coordinates of the horizontal segments compatible with Or n Og lead to aT-contact representation (see Figure 4b). As a linear extension of a partial order may becomputed in linear time, we have the following theorem.

Theorem 4.1. Any plane graph may be represented by a T-contact system on a n x n gridin linear time. •

From an isosceles triangle representation of a maximal plane graph G, one can deducea representation of G by a contact system of Y-shaped objects (see Figure 5a). Such arepresentation can be performed directly by using a procedure similar to the one describedfor triangles, and we can require that the Y-shaped objects be composed of segmentsbelonging to three fixed directions, as shown in Figure 5b.

From an isosceles triangle representation of a maximal plane graph, one can also derivea tessellation and a rectilinear representation of G (see Figure 6).

5. Final remarks

We have shown that two non-isomorphic triangle contact systems representing the samemaximal plane graph G define two distinct Schnyder partitions of G. Moreover, we haveshown that a Schnyder realizer obtained by a shelling order allows one to constructa maximal triangle contact system with which the realizer is associated. Actually, anySchnyder realizer may be defined by a shelling order. The following theorem is proved in[3].

Theorem 5.1. Given a Schnyder realizer (Yr,Yg,Yb) of a maximal plane graph G, a totalorder of V(G) is a shelling order defining (Yr, Yg, Yb) if and only if, it is a linear extension

Because of the different possible constructions of the triangles T\,Ti and Tn in thepreviously described algorithm, we have the following theorem.


(a)

(b)

Figure 4 (a) T-contact (b) T-contact system on the nxn grid


(a)

(b)

Figure 5 {a) Y-contact system representation (b) Y-contact system with 3 fixed directions on the grid


(a)

(b)

Figure 6 (a) Tessellation representation (horizontal segments are vertices, vertical segments are faces

and rectangles are edges), (b) Rectilinear representation (horizontal segments are vertices, vertical

segments are edges)

Theorem 5.2. The non-isomorphic triangle contact systems representing a maximal plane

graph G are in one-to-one correspondence with the Schnyder partitions of G.

Acknowledgments

We thank the referee and A. Machi for their great attention and many useful suggestions.

References

[1] Andreev, E. M. (1970) On convex polyhedra in Lobacevskii spaces. Mat. Sb. 81 445-478.[2] Di Battista, G., Eades, P., Tamassia, R. and Tollis, I. G. (1989) Algorithms for drawing planar

graphs: an annotated bibliography. Tech. Rep. No. CS-89-09, Brown University, 1989.[3] de Fraysseix, H. and de Mendez, P. O. (In preparation) On tree decompositions and angle

marking of planar graphs.[4] de Fraysseix, H., de Mendez, P. O. and Pach, J. (submitted) A streamlined depth-first search

algorithm revisited.


[5] de Fraysseix, H., de Mendez, P. O. and Pach, J. (1993) Representation of planar graphs bysegments. Intuitive Geometry (to appear).

[6] de Fraysseix, H., Pach, J. and Pollack, R. (1990) Small sets supporting Fary embeddings ofplanar graphs. Combinatorica 10 41-51.

[7] B. Mohar (To appear) Circle packings of maps in polynomial time.[8] Rosenstiehl, P., and Tarjan, R. E. (1986) Rectilinear planar layout and bipolar orientation of

planar graphs. Discrete and Computational Geometry 1 343-353.[9] W. Schnyder (1990) Embedding planar graphs on the grid. In: Proc. AC MS I AM Symp. on

Discrete Algorithms 138-148.[10] Tamassia, R. and Tollis, I. G. (1989) Tessellation representation of planar graphs. In: Proc.

Twenty-Seventh Annual Allerton Conference on Communication, Control, and Computing 48-57.

A Combinatorial Approach to Complexity Theoryvia Ordinal Hierarchies

WALTER A. DEUBER and WOLFGANG THUMSER

University of Bielefeld, Faukultat Mathematik, Postfach 10 01 31 33501 Bielefeld 1, Germany

Long regressive sequences in well-quasi-ordered sets contain ascending subsequences oflength n. The complexity of the corresponding function H(n) is studied in theGrzegorczyk-Wainer hierarchy. An extension" to regressive canonical colourings is indicated.

1. Introduction

For many mathematicians the most noble activity lies in proving theorems. It must havecome as a blow for them when Godel [7] showed that there are unprovable theorems. Atthe beginning they still could find some consolation in hoping that such culprits might onlyoccur in Peano arithmetics through esoteric diagonalization arguments. Nowadays there isa wealth of the most natural valid theorems that can be stated in the language of finitecombinatorics but are not provable within that system.

Mathematicians understand to a certain extent how to find unprovable theorems andhow to prove their unprovability within a formal system. In that sense we are relying onthe classical work by Gentzen [5], Kreisel [15] and Wainer [31]. Moreover, we shall applytheir beautiful ideas to something that seems to be well understood, viz to well-quasi-orderings. This is an old concept found in Gordan [6], and Kruskal [16] correctly pointedout that it was 'a frequently discovered concept'. That is why we are not reinventing it andare well aware that any sequence (st) of specialists starting with the author must contain anarbitrary long subsequence of experts knowing more than s0, a fact, which gives a nicetheme for this paper. Leeb was one of the first to deal with structural problems of wqo's,which are related to this paper [18]. Some beautiful ideas of P. Erdos are valuable for theanalysis of such phenomena occurring in all well-quasi-orders. For related combinatorialquestions we would also like to draw the reader's attention to the beautiful paper ofNesetfil and Loebl [19].

180 W. A. Deuber and W. Thumser

2. How to use complexity theory

We are interested in first order statements Vx3yA(x,y) in the language of Peano arithmeticswhere A is primitive recursive. Let g(x) be the smallest y satisfying A(x, y). We are interestedin the question of whether g is defined for every x. Let us anticipate the answer, which hasbeen known for a long time: if g grows fast enough, the statement 'g is defined everywhere'is not provable within Peano arithmetics.

In order to specify growth rates in complexity theory, we define a hierarchy of referencefunctions, There are various hierarchies available and, depending on the combinatorialproblems and personal taste, one can make a choice. Here we concentrate on theWainer-Grzegorcyzk hierarchy, cf. [8] and [31].

The first few functions are defined as follows:

fi+i(ri) =fi o ... of(n), where the iteration is n fold, and finally

fjji) =fn(n) is the Ackermann function defined by diagonalization.

The first few levels are well known :/0 grows like the identity,/^ linearly, f2 exponentially,/ 3 is the tower function;/4 is sometimes called the 'wow'-function [9], ....

Using the Cantor Normal form to define fundamental sequences representing ordinalsthere is no difficulty extending the hierarchy up to f, for instance

L+i+i(n) :=L+i ° • • • °L+i(n) "-times

L+Sn) : = / L + » diagonalization

f^in) '-=flon{n) diagonalization

f (n) \=f^"{n) diagonalization with the w-tower of height n.

For details, see [31].One can measure complexity with respect to these reference functions by defining

g > h iff lim ^ = 0(*) J —g(n)

. and g ~fx iff a is the smallest ordinal with/a+1 > g.

One should be aware that this complexity measure is fairly insensitive to small changesbut, as we shall see, it will allow rather clean-cut statements on combinatorial complexity.

Theorem (Kreisel). Let A(x,y) be a primitive recursive formula in the language of Peanoarithmetics andg{x) be the smallest witness y for A(x, y). If g > f or g ~ fe then 'g is definedfor all x* is not provable in Peano arithmetic.

This theorem demonstrates that it might be useful to understand complexity theory withrespect to such hierarchies. From the point of view of nonprovability in Peano arithmetic,only certain reference functions such as f are of interest, but we shall see that the otherlevels of complexity occur in rather natural contexts too. Here we concentrate on surveying

A Combinatorial Approach to Complexity Theory via Ordinal Hierarchies 181

some of these results and give examples for combinatorial problems which correspond tovarious levels.

3. Regressive sequences in wqos

Recall that a well-quasi-ordering is a poset (A, ^ ) that contains no infinite antichains andno infinite strictly descending sequence; thus any infinite sequence of elements of A mustcontain an infinite weakly ascending subsequence.

Let (A, ^ ) be a wqo-set with an obvious ranking r defined by successively taking minimalelements. Call a sequence (ao,a^ ...) regressive iff r(at) ^ / for every ieoj.

Theorem 3.1. Let (A, ^ ) be wqo. Then there exists a function H{A o :w->w such that everyregressive sequence (a0, .",aH(n)) contains a weakly ascending subsequence with n terms.

Harzheim proved this for (N, ^ ) [10] and (f Jd, ^ ) [11]. The general version might befolklore. The following proof should be known to all specialists. It came to the authorsmind when teaching on fixed point theorems in compact spaces.

Proof. Consider the space S of regressive w-sequences over A. Finite sequences should befilled up with minimal elements. Thus with

Rt = {xeA\ r(x) ^ /} one has 5 = IIW i^.

As a product of the finite sets Rt, the space 5 is compact in the Tychonoff topology, andas a metric space it is also sequentially compact.

Assuming that the theorem fails, pick a wqo set (A, ^ ) and an«ew such that for everyheoj there exists a regressive 'bad' sequence a(h) = (a(

oh\ ...,a{^), i.e., an //-term sequence

not containing any n term ascending subsequence. Thus the sequence (a(h))heoj has anaccumulation point a e S. As A is wqo, it follows that a must contain an infinite weaklyascending subsequence, so it contains a weakly ascending subsequence a' with n terms. Ofcourse a' is contained in an initial segment of a, the accumulation point. Thus it is containedin an initial segment of some a{h\ yielding the desired contradiction. •

Of course one could also use Konig's infinity lemma for a proof. We do not know whetherthe theorem can be generalized. To start with, finite sets Rf, in order to have a compact 5,does not seem to be the most general idea [21].

In this paper we are going to explore the complexity of H{A ^ for various posets (A, ^ ) .For some of the most natural and commonly occurring wqo's the Wainer-Grzegorczykhierarchy seems to be quite adequate for neat results.

4. Low complexity levels, product of chains

Harzheim [10] established the following

Theorem 4.1. H{K o(/i) = 2n'\ Thus H(N, o ~ fv

Proof. In order to establish the result, we proceed in the framework of complexity theoryand showi the upper bound H(n) ^ 2n~l

ii the lower bound H(n) > 2n~\For (i) we make use of some beautiful ideas from [4].Let (at) i= \,...,H(n) be a regressive sequence of positive integers. So far, H(n) is

unknown and we want to show 2n~1 ^ H(n). Define a mapping

f:{l,...,2n-1}^{\,...,2n-1} by

/>( \ — p e n g th of a weakly ascending sequence[of maximal length with first element in av

Case a. There is an i with *f(/) ^ n.Obviously this shows that an ascending subsequence of length n exists.

Case fl. /(/) < n for all /.Thus £ may be viewed as a colouring of {1, ...,2*~1} with at most n—\ colours.

Definition. A subset X of IJ is called large iff \X\ > minX

By the pigeon-hole principle, there is alarge subset X = {iv ...,// +1} ^ {1, ...,2"~1} that ismonochromatic for a certain £. By definition, each of the elements

is the starting point for a weakly ascending subsequence of maximal length / . Therefore (*)has to be a strongly descending sequence. (In order to see a{ > ait, suppose that, on thecontrary at ^ ai. Then a longest sequence starting at at could be extended by a{ yieldinga longest ascending sequence of length / + 1). The length of (*) is i1 + 1 and its first elementhas rank ^ /19 which gives a contradiction.

It remains to show that 2n~l is such that it allows the application of the pigeon-holeprinciple. Colour {1,...,«— 1} with n—\ colours in such a way that no large subset occursmonochromatically. Observe that / :{l , ...,7}->{l,2,3}, defined by

1 2 3 4 5 6 7\

h h /3 h h f)is a colouring such that every extension to 8 = 23 would yield either a large set or need anew colour. It is easy to see that any colouring of {1,. . . , 7} in which the colours do not occursuccessively either already contains a large set or can be rearranged to the above example,showing that the greedy strategy yields a = 2n~x as an upper bound for H(n).

As for the lower bound (ii), Figure 1 gives an explicit regressive sequence a of length2n~l — \ without weakly ascending subsequence of length n:a = (0103210). •

Another possibility for establishing upper bounds, which turned out to be useful in morecomplicated situations, employs a tree argument: a beautiful idea occurring in [1].

a4 a5

Figure 1

Given a regressive sequence a defined on the first few, say, a, integers, we recursivelyconstruct a sequence of binary trees T19 ..., Tp ..., Ta, in which— the internal nodes are labelled by 1,... J— the leaves are unlabelled— the pendant edges (those going into leaves) are labelled by 0 , . . . , /The construction is initiated by Figure 2.

(**)

Figure 2

Given 7J with internal nodes 1, ...J and pendant edges labelled 0, ...,y, (cf. (**)), defineTj+1 as follows: as a is regressive, we know that a(j+ 1) ^ 7 .

Thus there is a pendant edge labelled with a(j+ 1). The corresponding leaf in T} nowbecomes an internal node of Tj+1 labelled j+ 1. Moreover, two new pendant edges areattached to it and labelled by a(y + 1), a(j+ 1)+ 1. The other pendant edges of Tj are keptas such in Tj+1, those with labels < a(j+ 1) going unchanged and for those with labels >a(j+ 1) the labelling being increased by one. It is immediate from the construction that thefunction a is always increasing along the paths of internal nodes. If the size of the treebecomes larger than 2n~l — 1, we cannot help avoiding increasing subsequences of size atleast n, which proves the upper bound. The example a = (0103210) of Figure 1 leads toFigure 3.

The complete binary tree of depth n — 1 may be obtained from the example for the lowerbound, which shows that H(n) ^ 2n~l.

Needless to say, in the simple situation of Harzheim's result, our efforts for provingupper and lower bounds by rather sophisticated looking methods may give an overloadedimpression. To us it seems to be the simplest approach (cf [29]), and moreover, it isgeneralizable to more tricky situations (see Section 6). The general case for products ofchains was given by [11].

Theorem 4.2. HNd ~ fd_xfor d^2.


1 2 0 1 2 3 0 1 2

3 4

1 2 3 4 5 6 0 1 2 3 4 5 6 72 3 4 5

5. The intermediate levels, Higman's theorem for finite alphabets

One of the classical results in wqo theory is Higman's theorem:If(X^) is wqo, then Hig{X, ^ ) , the set of finite words over X endowed with embeddabilityinto subwords, is wqo.

We shall indicate the complexity of the corresponding //-functions. In doing so, weobserved a proof for Higman's theorem, which is as constructive as possible and,astonishingly, avoids minimal bad sequences. The proof makes use of some earlyobservations of [13] on finite sets, but apart from that, should be folklore to the specialists.

The crucial phenomenon is best observed by taking t = {1 ^ , . . . , ^ /} to be a finite,linearly ordered alphabet. Let a,beHigt. If 5 ^ H i g b, then b has a certain structureimposed by a. In order to appreciate the idea, let t = \0, a = (2,10,7) and b = (ft1? ...,bn).Case 1. ax = 2 ^ bt for all / (the embedding of a fails already in the first place). Then

beHigia.-l).C a s e 2 . L e t ix b e m i n i m a l w i t h b t 2 . T h u s b = ( b 1 . . . bt -^ * b t * ( b i i + 1 . . . b n ) w i t h

(b1...bh_1)e H i g ^ - l ) , and (bh+1...bn)eHigt is such that

i

By iteration one obtains the following general result.

Definition. Let I b e a poset and aeX. Then [a) is the principal filter {z \ aby a, and X\[a) is the complement of the principal filter [a).

z} generated

Fact. Let (X ^ ) be a poset, a = (a1... an), b = {bi... bm) e Hig (X) and a ^ Hig b. Then there

exist f < n, boeHig(A^i)),..., b/+1 eHig(X\[a/+1)) and elements b* e[a^, . . . , /?* e[a,) with

b = box b* xbxx b* ... x bM.

Basically, the fact says: if a ^ Hig b, then b is contained in a product whose factors are ofthe form Hig(X\[^)) or principal filters. Moreover, the length of all these products has anupper bound depending on the length of a.

Theorem 5.1. (Higman) If(X^) is wqo, then Hig(X, ^ ) is wqo.

Proof. The theorem holds for X=0. So assume that the theorem holds for allcomplements of principal filters X\[a) of some X. We will show that it holds for X. As suchan induction works for wqo's, the theorem follows.

So, let 50,51,52... be a sequence of elements of Hig X. Assume a0 ^ at for all / (i.e. thegreedy approach to show that the sequence is 'good' fails), then each at, i = 1,2,... has astructure as given by the fact. Thus there exists an / such that for infinitely many /

/+\

ai e Y\ (complement of principal ultrafilter) x Y\ X.

The induction hypothesis and the product lemma [12] imply that there is a weaklyincreasing subsequence of afs. Thus X is wqo. •

The proof looks quite constructive at first sight. A careful analysis reveals that for ageneral poset X the proof is not at all constructive. Nevertheless, for special Xs it is strongenough that with some additional work [30] one can obtain the following theorem.

Theorem 5.2. Let t < (±>. Then HHig(t) —/,/- i .

Remark. In the framework of regressive sequences the problem asking for Hmgi(o) seems tobe ill posed, as the rank function according to our definition (r(a) = lgth(a)) does not makesense, it is imaginable that for adequate definitions reasonable results could be obtained forHmg{(0) and beyond. For well-posed but somewhat artificial modifications of this problem[24].

6. The upper levels, the w-towers

Kanamori-McAloon [14] gave a model theoretic proof for the unprovability of a theoremon regressive colourings of /c-element sets. Here we shall analyze the correspondingcomplexity questions. By doing so we shall explain how 'canonical Ramsey theory', iargesets' in the sense of Paris and Harrington [22] and 'tree arguments' can be applied in orderto obtain sharp complexity results. These are related to the results of [2].

Before generalizing the concept of regressive sequences, we indicate as a combinatorialtool the Erdos-Rado canonization lemma [3]. We need

Definition. Let n,keco U {to}. ( l ' ' ," ' I = I I denotes the set of all k element (respectively\ k 1 \k

infinite) subsets of n. Let X = {x0, ...,xk_1}<, Y = { j 0 , . . . , j A ,_ 1 } < be ^-element subsets of M,

and / b e a subset of {0, ...,(&—1)}. Let X\I— {xieX/ieI}<. Thus we have

X:I= Y.I iff x1=yi for all iel.

The countable case of the canonical version of Ramsey's theorem can be stated as follows.

Lemma. [3] Let keoj be fixed and A: -> w be a colouring into the natural numbers. Then

there exist I <= {0,..., (k— 1)} and an infinite subset Me[ of natural numbers such that for

w •all X,Yel I the relation

\k J

X:I=Y:I holds iff A(X) = A(Y).

Example. In the special case where k = 2 the theorem assures the existence of an infiniteset such that the restricted colouring is— constant

A(X) = A( Y) for all X, YeM (I = 0 ) ,

— or injective

A(X) = A(Y) iff X= Y (/={0,1}),

— or depends only on minimum elements

— or depends only on maximum elements

A(X) = A(Y) iff max

In order to generalize regressive sequences we make use of the following definition

Definition. Fix n and k as above. A colouring A: \^to is called min-regressive if\kj

A(x) < minXfor all Xe I. For M c n we call a colouring A: U w min-homogeneous\kj \k J

if(M

min X = min Y implies A(X) = A( Y) for all X, Ye

Note that for k = 1 we recover the notion of a regressive sequence.A classical example is the van der Waerden colouring, which assigns to every arithmetic

progression of length k in {1,...,«} its first element diminished by one, and which assigns0 to the other A>tuples.

The following theorem is obvious to all those familiar with canonical Ramsey theory.

Theorem 6.1. Let kea) be fixed. For every m there exists a smallest n = Hk(m) with thefollowing property:

Let A: I I -> OJ be mm-regressive. Then there exists an m-element subset Mofn such that the\k)

restriction AM is min-homogeneous.\ k I

Proof. Work with the countable version of the Erdos-Rado canonization lemma. As thecolouring A is min-regressive the pertinent canonical cases must be min-homogeneous.Finally, apply compactness to obtain the existence of Hk. •

A natural problem is the analysis of the complexity of Hk. Here we rely on [25], [24] and[29].

Theorem 6.2. Let k^2. Then

H ~w f11k ' Joy tower of height k-V

Here we will give an explicit description of the arguments showing H2 -fr theAckermann function. As in Section 4, the proof consists in giving(i) a lower bound ~ f0

(ii) an upper bound ~ fi0.For the lower bound we need the following lemma.

Lemma 6.3. H2(Ram(2,m + 3,k)) ^fk(m), where Ram(2,m-\-3,k) is the ordinary Ramseynumber, arrowing (m + 3)2

k.

Proof. Given m, k, let m* = Ram(2,m + 3,k), n* = //2(m*). Observe that for .v < v < tothere exist unique 0 ^ k* < k and 1 { < x satisfying

fjp(x) :=fk* o ... o/fc* (x) < y <fk*o... o/A.* (x).

/-times / + l-times

This is well defined as

fk*(x) <fk*+i(x) =fkl}(x).

Now, define a regressive mapping by

/ otherwise.

n*\Let M*e[' ) be such that A is min-homogeneous on M*. We define a ^-colouring

* otherwise.

M*( M \ (M\Let Mel ^ t>e s u c n t n a t A* M ? Ms a constant colouring and let x < y < z be the three

largest elements of M. Then m ^ x and, as the function/^ is increasing, it suffices to showthat fk(x) ^ z. Assume to the contrary that fk(x) > z > y. Hence fk(y) > z also, asA W ^fk(y). Say, A({x,j}) = A({JC,Z}) = t and A({x,y}) = A({x,z}) = A({x,z}) = k*. ThenfUx) ^ V < z </^+ 1W. Apply /,.. to this inequality. Then z <fkt\x) </fc*( v). But thiscontradicts fk*(y) ^ z.

Corollary 6.4. The function H2(m) is not primitive recursive.

Proof. As Ram(2,m + 3,k) ^ /:(w+3)-k\ it is primitive recursive. But H2(Ram(2,m + 3,m))^fm{m) =fM{m) by the lemma. As the primitive recursive functions are closed undercomposition, the assertion follows. •

For the upper bounds we have the following theorem.

Theorem 6.5. H2(m) <f(0(m)for all m ^ 3.

Proof. In order to prove this theorem, we consider trees as partially ordered sets, thesmallest element being the root. As in Section 4, we use a tree argument: For a given

regressive mapping A: U n, define a tree (7^, ^ T) on {2,..., n — 1} by

15

/ < T m iff A({&, /}) = A({k, m}) for all k with k < T f.

For example, the tree depicted in Figure 4 corresponds to regressive mappings A: I \'" I -> 15

such that

A(2,3) - A(2,4) = A(2,5) = A(2,6),

A(3,4) = A(3,5) = A(3,6),

A(2,7) = A(2,8) = ••• = A(2,14),

A(7,8) = ••• = A(7,14).

Nothing is asserted about the remaining pairs.

9 10 11 12 13 14Figure 4

Mills [20] called a tree small branching if the successor-degree of each node / is atmost /.

The following observation is trivial but useful

Observation 1. Let A »n be a regressive mapping and (TA, ^ T ) be the associated tree.

Then:(i) k < T / implies that k < £

(ii) TA is small branching(iii) every chain is min-homogeneous.

For estimating H2(m) from the above, we ask how large n must at least be such that everysmall branching tree TA contains a chain of length m. Denote by M(m) the smallest such n.

Figure 4 shows that M(4) > 14, and it is easy to see that, in fact, M(4) — 15.Figure 5 indicates that M(5) > 239.41 — 2, and again it is not difficult to see that

M{5) = 239.41 — 1. The idea behind Figures 4 and 5 is fairly obvious. To build a large smallbranching tree without chains of length m, one fills in the branches from left to right byplacing smaller numbers as far down on the tree as possible to save vertices higher up forlarger numbers. These larger numbers then allow more immediate successors, thus makingthe tree as big as possible. Such trees are well known in computer science as balancedpreordered trees.

39

5 . . .

238x 41 -

2 3 9 x41 - 2

Lemma 5.6. Let n = M(m)—\, and let T be a small branching tree defined on {2, ...,/?!•

without any m-element chains. Then T is a balanced preordered tree.

The somewhat technical proof was given in [20] and [29], and a beautifully illustratedversion may be found in the highly recommended forthcoming book ' Aspects of RamseyTheory' [26].

Let Mm(k) be the smallest positive integer n such that every small branching balancedpreordered tree on [k,n] contains an m-element chain (thus M(m)+ 1 = Mm{2)).

Observation 2.(1) M2(k) = k+l,(2) Mm+1(k) < Mh

m(k+\), (k-fold iteration).

Proof. (1) is obvious and (2) follows immediately from the construction of small branchingtrees, cf. Figure 5. •

By boolean combination of definitions, we obtain the following observation.

Observation 3. Mm(k) ^//w_j(A:+ 1 ) - 1 for all k,m^ 1.

In order to obtain the upper bound/,, for H2 and conclude the proof of Theorem 6.4, itsuffices to combine Observations 1-3.

Remark 1. It is possible to extend these arguments and obtain a proof of Theorem 6.2 ingeneral. For details see [29].

Remark 2. Erdos and Mills [2] gave upper bounds for the Paris-Harrington function forcolouring pairs with a fixed number of colours; the Ramsey case [27]. The above resultscover the canonical min-homogeneous case for pairs and ^-tuples in general.

7. Outlook and problems

In this paper we have concentrated on the levels of the Grzegorczyk-Wainer hierarchy upto e0. Of course we could, and did, go beyond. [28] gives an account of the finiteminiaturization of Kruskal's theorem for trees, another classic in wqo theory. For the caseof binary trees, [30] shows that for regressive sequences of binary trees HBin ~ / , whereas[28] indicates that the general case for regressive sequences of arbitrary trees is far beyondfv . Finally, we would like to mention Leeb's jungles [18], which unfortunately have notreally been penetrable for us so far.

As a general problem and idea, we suggest searching for other 'natural' combinatorialfeatures that may be extended by compactness arguments and lead to fast growingfunctions and unprovability results.

Closer to the extension of HarzheinVs result (cf. Theorem 4.1, it would be interesting tofind orders related to each level of the hierarchy. When stating Higman's theorem, weassumed the alphabet to be an antichain. Of course such alphabets may be partially ortotally ordered. How does this order affect the growth of the corresponding //-functions?

References

[1] Erdos, P., Hainal, A., Mate, A. and Rado, R. (1984) Combinatorial set theory: Partitionrelations for cardinals. Studies in Logic and the Foundations of Mathematics 106. North-Holland.

[2] Erdos, P. and Mills, G. (1981) Some Bounds for the Ramsey-Harrington Numbers. J. of Comb.Theorw Ser. A 30, 53-70.


[3] Erdos, R. and Rado, R. (1950) A combinatorial Theorem. Journal of the London MathematicalSociety 25, 249-255.

[4] Erdos, P. and Szekeres, G. (1935) A combinatorial problem in geometry. Composite Math. 2.464-470.

[5] Gentzen, G. (1936) Die Widerspruchsfreiheit der reinen Zahlentheorie. Mathematische Annalen112, 493-565.

[6] Gordan's, P. (1885) Vorlesungen uber Invariantentheorie, Hrsg. v. Geo. Kerschensteiner. 1. Bd.Determinanten (XI, 20IS.). Teubner, Leibzig.

[7] Godel, K. (1931) Uber formal unentscheidbare Satze der Principia Mathematica und verwandterSysteme, I. Monatschefte fur Mathematik und Physik 38, 173-198.

[8] Grzegorczyk, A. (1933) Some classes of recursive functions. Rozprawy matematiczne 4. InstytutMatematyczny Polskiej Akademie Nauk, Warsaw.

[9] Graham, R., Rothschild, B. and Spencer, J. (1990) Ramsey theory, Wiley, New York.[10] Harzheim, E. (1967) Eine kombinatorische Frage zahlentheoretischer Art. Publicationes

Mathematicae Debrecen 14, 45-51.[11] Harzheim, E. (1982) Combinatorial theorems on contractive mappings in power sets. Discrete

Math. 40, 193-201.[12] Higman, G. (1952) Ordering by divisibility in abstract algebras. Proc. London Math. Soc. 2,

326-336.[13] Jullien, P. (1968) Analyse combinatoire - Sur un theoreme d'extension dans la theorie des mots.

CR. Acad. Sci. Paris, Ser. A 266, 851-854.[14] Kanamori, A. and McAloon, K. (1987) On Godel incompleteness and finite combinatorics.

Annals of Pure and Applied Logic 33, 23-41.[15] Kreisel, G. (1952) On the interpretation of nonfinitistic proofs. Journal of Symbolic Logic 17. II.

43-58.[16] Kruskal, J. B. (1972) The Theory of Well-Quasi-Ordering: A Frequently Discovered Concept.

Journal of Combinatorial Theory (A) 13, 297-305.[17] Leeb, K. (1973) Vorlesungen uber Pascaltheorie. Arbeitsbericht des Instituts fur mathematische

Maschinen und Datenverarbeitung, Friedrich Alexander Universitat Erlangen Niirnberg, Bd. 6Nr. 7.

[18] Leeb, K. Personal communications.[19] Loebl, M. and Nessetfil, J. (1991) Unprovable combinatorial statements. In: KeedwelL A. D.

(ed.) Surveys in Combinatorics.[20] Mills, G. (1980) A tree analysis of unprovable combinatorial statements. Model theory of

Algebra and Arithmetic. Springer-Verlag Lecture Notes in Mathematics 834, 248-311.[21] Nessetfil, J. and Rodl, V. (1990) Mathematics of Ramsey Theory, Springer-Verlag. Berlin.

Heidelberg.[22] Paris, J. and Harrington, L. (1977) A mathematical incompleteness in Peano Arithmetic.

Handbook of Mathematical Logic. In: Barwise, J. (ed.) North-Holland Publishing Company.1133-1142.

[23] Promel, H. J., Thumser, W. and Voigt, B. (1989) Fast growing functions based on Ramseytheorems. Forschungsinstitut fur Diskrete Mathematik, Bonn (preprint).

[24] Promel, H. J., Thumser, W. and Voigt, B. (1991) Fast growing functions based on Ramseytheorems. Discrete Mathematics 95, 341-358.

[25] Promel, H. J. and Voigt, B. (1989) Aspects of Ramsey Theory I: Sets, Report number 87495-OR, Forschungsinstitut fur Diskrete Mathematik. Universitat Bonn, Germany.

[26] Promel, H. J. and Voigt, B. (1993) Aspects of Ramsey Theory, Springer Verlag, Berlin.[27] Ramsey, F. P. (1930) On a problem of formal logic. Proceedings of the London Mathematical

Society 30, 264-286.[28] Simpson, S. G. (1987) Unprovable theorems and fast growing functions. In: Simpson. S. G.

(ed.) Logic and Combinatorics. Contemporary Mathematics 65, 359-394.


[29] Thumser, W. (1989) On upper Bounds for Kanamori McAloon Function, preprint 89-10,Sonderforschungsbereich 343 "Diskrete Strukturen in der Mathematik", Universitat Bielefeld.

[30] Thumser, W. (1992) On the well-order type of certain combinatorial structures, Bielefeld(manuscript, submitted).

[31] Wainer, S. S. (1972) Ordinal recursion and a refinement of the extended Grzegorczyk hierarchy.Journal of Symbolic Logic 37 281-292.

Lattice Points of Cut Cones

MICHEL DEZAf and VIATCHESLAV GRISHUKHIN*

fCNRS-LIENS, Ecole Normale Superieure, Paris

^Central Economic and Mathematical Institute of Russian Academy of Sciences (CEMI RAN), Moscow.

Let R+(JTA ]) ,Z(jrn) ,Z+(Jfn) be, respectively, the cone over 1R, the lattice and the coneover Z, generated by all cuts of the complete graph on n nodes. For / > 0, let Al

n := {d e1R+(JTW) nZ{Jfn) : d has exactly / realizations in Z+{3fn)}. We show that A'n is infinite,except for the undecided case A® ^ 0 and empty A'n for / = 0, n < 5 and for i > 2, n < 3.The set A\ contains 0, l,oc nonsimplicial points for n < 4, n = 5, n > 6, respectively. Onthe other hand, there exists a finite number t{n) such that t(n)d e Z+(Jfn) for any d e ^Jj;we also estimate such scales for classes of points. We construct families of points of A®and Z+{Jfn), especially on a 0-lifting of a simplicial facet, and points d e R + ( J T , , ) withdin = t for 1 < i < n — 1.

1. Introduction

In this paper we study integral points of cones. Suppose there is a cone C in IR" that isgenerated by its extreme rays e\,e2,...,em, all e,- e Zn.

Let d be a linear combination,

d= Y. k&- *1}

l</<mWe call the expression a K-realization of d if/.,- G X, 1 < f < m, and K is either of R + ,

If A/ > 0 for all U then d G C, and (1) is an #+-realization of d. If /,,• is an integer forall i, then d e L where L is a lattice generated by the integral vectors <?,, 1 < / < m, and(1) is a Z-realization of d. Obviously L ^ Zn. If // > 0 and is integral for all /, we callthe point d an h-point of C. Hence h-points are the points having a Z+-realization. A

* This work was done during the second author's visit to Laboratoire d'Informatique de TEcole NormaleSuperieure, Paris

194 M. Deza and V. Grishukhin

point d e C n L is called a quasi-h-point if it is not an h-point. In other words, d is aquasi-h-point if it has IR+- and ^-realizations but no Z+-realization.

We consider cut cones, i.e. those where e, are cut vectors. Let Jf'„ be the set of allnonzero cut vectors of a complete graph on n vertices. Then IR+(jf „) is the cut cone.The members of the cut cone R+(Jfw) are exactly semimetrics, which are isometricallyembedded into some Zi-space, i.e. into Rn with the metric \x — y\t . Between them, themembers of integer cut cone Z+(JTn) are exactly semimetric subspaces of some hypercube{0, l}m equipped with the Hamming metric. In particular, the graphic metric d(G) belongsto Z+(Jfn), (l/2)Z+(Jf*n) if and only if G is an isometric subgraph of a cube or of a halvedcube, respectively. The above equivalences explain the interest of the cut cones, such asR+(jf"w) and Z+(3fn). See [12] for a detailed survey of applications of cut polyhedra. Asexamples, we recall applications for binary addressing in telecomunication networks, themax-cut problem in Combinatorial Optimization, and the feasibility of multicommodityflows. More specifically, the integer cut cone Z+(Jf w) provides some tools for DesignTheory (see, for example, [9] and Section 8 below) and for the large subject of embeddinggraphs in hypercubes.

In fact, those problems are related to feasibility problems of the integer program

{AX = d,Xe VD, (2)

where A is the n x m matrix whose columns are the vectors ex.In this paper we attack the integer programming aspects of the cut cones, the main

general problem of which is to give a criterion of membership in Z+(Jf), X ^ Jfm formetrics of given class. Examples of possible approaches to it are as follows.

1 Criteria in terms of inequalities and comparisions, as in [3]: Jf = JTn, (n < 5); [10],[13]: Jf is a simplex, i.e. cuts of Jf are linearly independent, X = OddXn.

2 Criteria in terms of enumeration, as in [1] for (l,2)-valued d, or in [15] for d = d(G),where G is a distance-regular graph.

3 A polynomial criterion as in [14] for graphic d = d(G) and other of d.

But in this paper we use other concepts (quasi-h-points and scales), which come fromthe basic concept of the Hilbert base; see Sections 3 and 4, and 8 and 9 below, respectively.

Finally, we also address adjacent problems on cut lattices (characterization and somearithmetic properties), and on the number of representations of a metric in Z+(Xn).

2. Definitions and notation

Set Vn = {l,...,n}, En = {(ij) : 1 < i < j < n}, then Kn = (Vn,En) denotes the completegraph on n points. Denote by P(/1,/2,...,/A) = Pk the path in Kn going through the vertices

ii, f2, ...,*"*•

For S c vn, S(S) c En denote the cut defined by S, with (ij) e d(S) if and only if| S n {ij} \= 1. Since S(S) = d(Vn — S), we take S such that n £ S. The incidence vectorof the cut 8(S) is called a cut vector and, by abuse of language, is also denoted by S(S).Besides, S(S) determines a distance function (in fact, a semimetric) ds(S) on points of Vn

as follows: ds(s)(Uj) = 1 if (Uj) G <5(S), otherwise the distance between / and j is equal to0. For the sake of simplicity, we set 5({iJ,k,...}) = S(i,j,k,...).

Lattice Points of Cut Cones 195

We use Jfn to denote the family of all nonzero cuts d(S), S ^ Vn. For any familyJf c jfm define the cone C(Jf) := R + p f ) as the conic hull of cuts in jf. So, bydefinition, X is the set of extreme rays of the cone C(JT). The cone C(Jf) lies in thespace R( j f ) spanned by the set Jf\ We set Cw := C(Jfn).

So, each point d G C(Jf) has a representation rf = X!<5(S)ejf- 5 (5). Since As > 0, therepresentation is called the ^^-realization of d. The number 5Z<5(S)ejf ^s is called r/ze s/z£of the 1R+ -realization.

The lattice L(Jf) := Z(Jf) is the set of all integral linear combinations of cuts in Jf.Let Ln = L(Jfn). The lattice Ln is easily characterized: d G Ln if and only if d satisfies thefollowing condition of evenness

dij + dik + d^ = 0 (mod 2), for all 1 < i < j < k < n. (3)

So, 2Zn{n~l)/2 c L n c Zn{n~l)/2.The points of L(Jf) with nonnegative coefficients, i.e., the points of Z + ( J T ) , are called

h-points. We denote the set of h-points of the cone C(Jf) by hC(Jf). For d G Z+(Jf),any decomposition of d as a nonnegative integer sum of cuts is called a ^-realizationof d. An h-point of C,, is (seen as a semimetric) exactly isometrically embeddable into ahypercube (or h-embeddable) semimetric. This explains the name of an h-point.

For d G Cn, define

s(d) := minimum size of R+-realizations of d,

z(d) \— minimum size of Z+ -realizations of d if any.

Let d(G) be the shortest path metric of a graph G. We set

zln := z

For this special case, G = KtU s(d) = s(2td(Kn)) is equal to aln := ^ " ^ .

A point d G C p f ) is called a quasi-h-point of C(JT) if d belongs to L(JT) but has noZ+-realization. We set

A(jf) i= C(Jf) 0 L(X') - Z+(Jf).

Recall (see [18]) that a Hilbert basis is a set of vectors <?i,...,^ with the property thateach vector lying in both the lattice and the cone generated by e\,...,ek is a nonnegativeintegral combination of these vectors. A(X) = 0 would mean that Jf is a Hilbert basis ofC(Jf). Actually, Jf would be the minimal Hilbert basis of C(Jf) if it is a Hilbert basis,since JT is the set of extreme rays of C(Jf) (see [4]).

Define

A((jf) := {d G C(Jf) n L(Jf) : d has exactly i Z+-realizations},

A\x := A\Xn\

So, the above defined set A(jf) is ^°(JT). Define

n\d) := min{t G Z+ : rd has > i ^-realizations}

= minjr G Z+ : td (£ Ak(Jf) for all 0 < k < /}.

A cone C = JR+(Jf) is said to be simplicial if the set Jf is linearly independent; a pointd G C is said to be simplicial if d lies on a simplicial face of C, i.e., if d admits a unique]R+-realization.

Let dim Jf be the dimension of the space spanned by Jf. Call e(Jf) := | Jf | — dim jf,the excess of Jf\ Set

\Xn :\S\=l o r n-\S\ = / } .

For even n we also set

EvenJfn = {5(S) G Jfn : | S | , n - | S | = 0 (mod 2)},

OddJfn = {S(S) G Jfn : |S|,w — \S\ = 1 ( m o d 2)}.

For a subset T ^ Vn denote

EvenTjfn = {S(S) EJfn:\SnT\=0 (mod 2)},

OddTjfn = {S(S) eJfn:\SnT\ = l (mod 2)}.

So EvenJfn = EvenTJfn, OddJfn = OddTjfn for T = Vn, n even.Remark that Jf%m = {<5(S) G Jf^m : 1 S} = {3(S) G Jf^w : 1 G 5}.

Denote by Jf^.Jff, J f f^ m o d ^ the families of <5(S) G JTn with |S| G {ij,n-i,n-j},\S\ $. {i,n — i}, min{|iS|,H— |5|} ^ /(mod a), respectively.

We write C£ for C(JTg), where a and b are indices or sets of indices.

3. Families of cuts Jf with A(Jf) = 0

Of course A(jf) = 0 if e(JT) = 0, i.e. if the cone C(Jf) is simplicial. It is easy to see thatC(Jfl

n) is simplicial if and only if either / = 1, or / = 2, or (/, w) = (3,6). Also e(Jf 3) = 0,what is a special case of the formula

e(Jfn) = 2n~{ - 1 -

Some examples of JT with a positive excess but with A(Jf) = 0 are:

(a) Jf*4, Jf^ with excess 1 and 5, respectively. The first proof was given in [3]; for detailsof the proof see [10], where, for any d G Cn n Ln, n = 4,5, the explicit Z+-realizationof d is given.

(b) OddJfe with the excess 1. For the proof see [10].(c) (See the case n = 5 of Theorem 6.2 below.) The family of cuts (with excess 5) on a

facet of C(Jfe) that is a 0-lifting of a simplicial pentagonal facet of C{X^).

But Jf^'2 with excess n has A(Jt) ^ 0 for n > 6. Below we give some examples of JTwith A($C) ^ 0, which are, in a way, close to the above examples of JT with A(Jf) = 0.

We denote by Q(b) the linear form J2\<i<j<nbibjxij f° r b e %"• ^ YH=\bi = 1> ^ e

inequality g(fc) < 0 is called a hypermetric inequality. We call d G JR"^-1)/2 a hypermetricif it satisfies all the hypermetric inequalities. We denote the hypermetric inequality byHypn(b). It is easy to verify that S(S) satisfies all hypermetric inequalities. Moreover, for

large classes of parameters b (see [4], [6]) Hypn(b) is a facet of C(3fn). The only knowncase when a hypermetric face is simplicial is (up to permutation) Hypn(\

2,— ln~3,n — 4),n > 3, and (its 'switching' in terms of [6]) Hypn(—1, T" 2 , - (n - 4)). Call the facetHypn(l

2,-ln-\n-4) the mam n-/aa?f. Call the facet Hypn(l2,0k,-ln-k-\n - k - 4) the

/c-/o/d 0-lifting of the main (n-Zc)-facet. It is a facet of C(Jf „), because every /c-fold O-liftingof a facet of Cn-k is a facet of Cn (see [4]). We call 1-fold 0-lifting simply O-lifting. Welist, up to a permutation, all facets of C(Jfn) for 3 < n < 6:

— The unique type of facets of C{C/fi) is the main 3-facet (triangle inequality);— The unique type of facets of C(Jf*4) is the main 4-facet (which is the 0-lifting

Hyp4(—1,12,O) of a main 3-facet);— All facets of C(Jf$) are 2-fold O-liftings of a main 3-facet (i.e. 0-lifting of a main

4-facet), and the main 5-facet Hyps(l3,— I2), called the pentagonal facet;— All facets of C{J^e) are: 2-fold O-liftings of a main 4-facet, 0-lifting of a main 5-facet,

the main 6-facet Hyp6(2,1,1,-13) and its 'switching' Hyp6(-2,-l, I4).

Lemma 3.1. If X is a family of cuts S(S), \S\ < (n/2), lying on a face F of Cn, the family

X1 = X U {3({n + 1})} U {S(S U {n + 1}) : d(S) e J f}

is the family of cuts lying on a 0-lifting of the face F. If, for the above Jf\ C(Jf) is asimplicial facet of Cn, we obtain, for n > 4,

f) = n(n - 3)/2.

Proof. If C(Jf) is a simplicial facet of Cn, then dim Jf = \Jf\ = Q) - 1. Obviously,| J T | = 2|Jf| + 1. Since JT is a simplicial facet of Cn+U we have, dim X' = ("+1) - 1also. Hence

= n(n - 3)/2.

D

Recall that 4(jf) = 0 for Jf = tf5,X\,C/f\,C/f\,Xx£ = OddJf6, and for the family ofany (except triangle) facet of X^, since 3f\ is simplicial for i = 1,2,3, and Jf 5, OddJfsare examples given at the beginning of this section.

4. Antipodal extension

A fruitful method of obtaining quasi-h-points is the antipodal extension operation at thepoint n. For d G IR"^-1)/2 we define antj e ]R"^+1)/2 by

(antad)ij = dtj for 1 < i < j < n,(antad)n,n+i = a,

=0L — djn for l<j <n—l.

For Jf c jfn, define

antJf = {ant{3(S) : 8(S) G Jf} U {<5(n + 1)}.

Note that

antid(S) = S{S) if w G S, a n d antx8(S) = 8(S u{n+ 1}) if n £ S.

Hence

arcfJf = {<5(S) :<5(S) G Jf ,n G S} U {(5(S U {n + 1}) : <5(S) e JT ,n£

Observe that if d G C(Jf) and d = ^ ( S ) G j r /-s<5(S), then

+1). (4)5{S)eJT S

Also, if

~~ 1),

then a = ^2S Xs + Ac and d = J2s(S)e)f ^s^(S) is the projection of ant^(d) on JR/1^-1*/2.So antad G IR(anf Jf) if and only if d G 1R( Jf).Note that the cone IR(anf JT) is the intersection of the triangle facets Hypn+\(l2,—lj, 0"~2),

where bn = bn+\ = 1, b7 = —1 and b\ = 0 for i j= 7, 1 s(d),(iii) antad G hCn+\ if and only if d G nCn and a > z(d),(iv) antad /s a simplicial point of Cn+\ if and only if d is a simplicial point of Cn and

a > s(d). •

Clearly, s{ant^d) = a if antad G Cn+\ and z(ant0Ld) = a if ant^d G hCn+\. Also, anrad G A\for i > 0 if and only if d G A\v a G Z + , a > z(d).

Proposition 4.1 obviously implies the following important corollary.

Corollary 4.2. Let d G hCn, and a be an integer such that s(d) < a < z(d). Then ant^d Gtad is a quasi-h-point in Cn+\.

5. Spherical ^-extension and gate extension

Let d G Cw+i. We write d = (d0^1), where

d° = {di7 : 1 < 1 < 7 < n}9 d1 = {d?>+1 : 1 < / < n).

A point d G Cn+i is called the spherical t-extension, or simply t-extension, of the pointd° G Cn if d = (d°,dl) and djn+l = t for all i e Vn. We denote the spherical f-extension ofd° by exttd°.

Let jn be the n-vector whose components are all equal to 1. Then for the f-extension(d°,dx), we have dx = tjn.

Proposition 5.1. exttd is a hypermetric if and only if

(i) d is a hypermetric,(ii) t>(ZbibjdiJ)/I.{'L-l)

for all integers b\,...,bn with X := Y^\ / > 1 and g.c.d. b\ = 1.

Proof. If exttd is hypermetric, then ^2bibj{exttd)ij < 0 for any b\,...,bn, bn+\ G Z+ with

bibjdij+\<i<j<n \<i<n

Since bn+\ = 1 — Z, the second term is equal to —rZ(Z — 1). We obtain (i) if fen+i = 0 or1; otherwise Z(I - 1) ^ 0, and we get (ii). •

Corollary 5.2. exttd is a semimetric if and only if d is a semimetric and t > (1/2) max(/7) d\j.In fact, apply (ii) above to the case bx = b}•• = 1, bn+\ = — 1 and bk =0 for other ft's.As with Proposition 5.1, one can check that anttd is a hypermetric (a semimetric) if and

only if d is a hypermetric (a semimetric, respectively) and

for any integers b\,...,bn with X := J2" bt > 1 and g.c.d. ft,- = 1

(f > -maxi</<_,-<„_i(d,7 + 4 , + djn), respectively).

There is another operation, similar to antipodal extension operation. We call itthe gate extension operation at the point n (called the gate). For d G IR"^-1)/2, definegatj e R '^-1) / 2 by

(gatxd)jj = djj for I < i < j < n,)w,n+i = a,

)i,n+i = a + din for 1 < / < n - 1.

The following identity shows that gafad is, in a sense, a complement of ant^d:

antad + gatit-nd = 2exttd. (5)

Recall that we take 5 in 5(S) such that w ^ S. Hence, for Jf c jfn, we have

g ^ JT = JfU{S(n+ 1)}.

Actually, anr Jf n = OddTjfn+u gat Jfn = {S(n + 1)} U EvenTJfu+u for T = {n,n + 1}.

Note that the cone R+(ga£ Jf) is the intersection of the triangle facets Hypn+$ l j ,0w-2 , - l , lB+i), where b{ = bn+{ = 1, bn = - 1 , bj = 0 for ; ^ i, l<j<n- 1.

It is clear that any IR+ -realization of gata^ (if it belongs to Cn+$ has the formY^s^s^(S) + a<5(n + 1) where n + 1 ^ S, and where the above realization is any ]R+-realization of d. So, ga£ad G L ^ ^ + b / z C n + i , ^ ^ , respectively) if and only if d GLn(CmhCmAl

n, respectively) and a G Z(R + ,Z+ ,Z , respectively).Also, gatad is a hypermetric (a metric) if and only if a G 1R+ and d is a hypermetric (a

metric, respectively).Hence if a G Z+, we have

In particular, gata J is a quasi-h-point if and only if d is.The following facts are obvious.

1 If di is the ^-extension of d?, i = 1,2, then di + d2 is the (ti + £2)-extension of d? + d .2 If d° lies in a facet of the cut cone, the ^-extension of d° lies in the 0-lifting of the

facet.

We call a point d e Cn even if all distances d/; are even.Let d = J2S ^sd(S) be a Z+ -realization of an h-point d. We call the realization (0,1)-

realization (27L^-realization) if all ^ are equal to 0 or 1 (are even, respectively). Wehave

Fact. Let d be an h-point. Then d = d\ -\-di, where d\ has a (0,1)-realization, and di has a2Z^-realization.

Obviously, if d has a 2Z+-realization, d is even. But if d is even, it can have no2Z+-realizations.

The following Proposition is an analog of Proposition 4.1.

Proposition 5.3.

(i) exttd e Ln+l if and only if d e 2Zn{n~l)/2 and t e Z,(ii) exttd G Cn+\ if d G Cn and It > s(d),(hi) suppose that d has 2Z^-realizations, and let zeven(d) denote their minimal size; then

exttd G hCn+\ if d G hCn and 2t > zeven(d).

Proof, (i) is implied by the trivial equality d,-,w+i + dy,n+i -f dtj = 2t + d^, 1 < i < j < n.From (5) we have exttd = (l/2)(anfad + gaf2f-ad). Taking a = s(d) and applying (ii) of

Proposition 4.1 we get (ii).Taking a = zeven(d\ applying (iii) of Proposition 4.1 and using ant++=mn{d),gat2t-=n.en(d)d e

2Z+(Jfn+1), we get (hi). D

Define extfd = extt(ext™~ld), where ext\d = exttd.

Proposition 5.4. Iflt > s(d), then ext™d G Cn+m for any m G Z+, and

t™-{dl It - T ^ T T ) < s(ext™d) <2t- 2~m(2t -\m/2]

Proof. From Proposition 5.3(ii) we get

s(exttd) < -s(ants{d)d + gat2t-S(d)d) = t + -s{d) < It.

By induction on m, we obtain ext™d G Cn+m for all m G Z+, and the upper bound fors(extfd).

The lower bound is implied by the fact that the restriction of extfd on m extensionpoints is td(Km). Since s(td(Km)) = (\/2)at

m (see Section 2), we have

D

Remark. So, if s(d) < 2t, then limm^oo s{extfd) = 2t.Probably, there exist mo = mo(t,d) such that s(ext™d) = It for m > mo.We conjecture that ext™d & Cn+m for m > mi if s(d) > It. For example, if t = 1 and

d = d(G) (d(G) is the shortest path metric of the graph G), then it can be proved thatmi = 2.

If the conjecture is true,

s(d) = 2min{r : ext™d G Cn+m for all m G Z+}.

Recall, that Proposition 4.1(ii) implies

s(d) = min{a : antad G Cn+\}.

In terms of ext™d we also have analogs of (i) and (hi) of Proposition 4.1.

Proposition 5.5.

(i) ext™d G Ln+m for all meZ+ if and only if d G 2Zn("~1)/2 and t is even.

(ii) ext™d G hCn+mfor all m G Z + if and only if t is an even positive integer, and ext\pd G

hCn+l.

Proof. The evenness of t follows from ext\d G Ln+i. So, (i) is implied by Proposition5.3(i).

Recall the result of [5] that £^J<5(0 is the unique Z+ -realization of td(Kn) for event and m > (£2/4) + (t/2) -f 3. Using this fact, we get that any Z+-realization of ext™dcontains t/2 cuts S(i) for some i if m is large enough. •

6. Quasi-h-points of 0-lifting of the main facet

Consider the main facet

F0(n) = Hypn{\\ - I " " 3 , n - 4) = Hypn(b%

where b°x = b°2 = 1,6? = - 1 , 3 < f < n-\, b°n = n-A. The cut vectors d{S) lying in the facetare defined by equations b(S) = J2tes hi = 0 ov 1. We take S not containing n. Then S G ^ ,

w h e r e

ST = { { 1 } , { 2 } , { 1 / } , {2 i}{12 /} (3 < i < n - 1), {I2ij} (3<i<j<n- 1 )} .

W e s e t

Every w-facet contains at least m cut vectors. Since the main w-facet contains exactly mcuts, it is simplicial.

The 0-lifting of the main facet is the facet

Besides the above cuts <5(S), S G y , it contains, according to Lemma 3.1, only the cutsd(Su{n+l}\S G ^ , and^(n+ l ) .

Note that A(Jf) = 0 for the main n-facet (as for any simplicial C(Jf)).Now we consider even points having no 2E+ -realization. The simplest such points are

points having a (0,l)-realization. We call these points even (0,1 )-points.Let d° G Fo(n) be an even h-point, and let J2SG^0 ^S$(S) be one of its Z+-realizations.

Consider a minimal set of comparisions mod 2 that ks's have to satisfy. The comparisionsare implied by the conditions d\j = 0 for all pairs (ij). Since d° G Lm we have d\j = d^+djk(mod 2) for all ordered triples (ijk). Hence independent comparisions are implied by thecomparisions d\n = 0 (mod 2), 1 < i < n — 1. The comparisions are as follows. (For thesake of simplicity, we set A|/; i = A,;... and omit the indication (mod 2)).

Mi + hi + ^12/ + ^ A12/7 = 0, 3 < / < fl — 1,

;^2»; = 0, (7)

^2+ X ( 2/ +^12/) +3</<«—1 3</<;<n-l

The system of comparisions (7) has n—\ equations with m = n(n — l ) / 2 — 1 unknowns.

Hence the number of (0,l)-solutions distinct from the trivial zero solution is equal to2m-(n-\) _ j = 2C21)-1 - 1.

This shows that all points of Fo(3) have 2Z+-realizations. The only even (0,l)-points ofFo(4) are 2 points 2d(K^) with dn = 0 or J23 = 0, and the point 2d(K4 — P(uO- There are31 even (0,l)-points in Fo(5).

Since there are exponentially many even (0,l)-points in Fo(w), we consider points of thefollowing type and call them special.

For these points the coefficients ks are

k\ = 0 1 , ki = «2» ^1/ = ^1? ^2/ — bi, A12/ = Ci, 3 < f < 7t — 1,

A1217 = C2, 3 < / < 7 < n — 1.

Here 0,-,b,-,c,-, / = 1,2, are equal to 0 or 1.

If we set

k = n-3, 1 =

then for the special points, (7) takes the form

— \)c2 = 0,

a 2 + k ( b 2 + c l ) + c 2 = 0 .

Since we have 3 equations for 6 variables, we can express 3 variables a\,a2,ci through theother 3 variables b\,b2,c2.

There are 4 families of the solutions of the system depending on the value of k (mod4). The solutions are as follows (undefined equivalences are taken by (mod 2)).

k = 0 (mod 4), a\ = a2 = 0, c\ = b\ + b2 + c2,

fc = 1 (mod 4), a\ = b2, a2 = b\,c\ = b\ + b2, c2 arbitrary,

k = 2 (mod 2), a\ = a2 = c2, c\ = b\ + b2 + c2,

k = 3 (mod 4), a\ =b2+ c2, a2 = b\+ c2, c\=b\+ b2.

In each case we obtain 7 nontrivial special even (0,l)-points.Turning our attention to the definition of 5^, for a = 0, +, we denote by tfk, Xa

k thefe-vectors with the components ^-, 3 < j < n— 1, i = 1,2, ^ 2 . , 3 < 7 < n— 1, respectively.Similarly, ^ is the /-vector with the components Xa

xli^ 3 < i < j < n — 1.In this notation a special point d° has a (0,l)-realization /° such that ^ = a^ /f?k =

bjk, i = 1,2, X°k = c\jk and tf = c2jh

Recall that special points are simplicial. Therefore their size is equal to Ylse^^s- Weshow below that the ^-extension of 2 special points with (a\,a2,b\,b2,c\,c2) = (1,1,0,0,0,1)and (0,1,0,1,1,1) are quasi-h-points for n = 2 (mod 4).

For n = 6 the points d° are d(Ke — Pi) and ant\^(ext^d(K/i}). Another example of d e A®is ant^(ext^d(Ks)) = d5'3 in terms of Corollary 6.6 below.

Proposition 6.1. Let d° be one of the 7 special points of the main facet Fo(n). Let t be apositive integer such that t > (1/2) ^ S G ^ ^S- Then the t-extension of d° is an h-point ifn^2 (mod 4), and if n = 2 (mod 4), then there is a point d° such that its t-extension is aquasi-h-point, namely the point with (ai,a2,b\,b2,c\,c2) = (1,1,0,0,0,1).

Proof. Recall that we can take Sf such that n $ S for all S e^.We apply equation (2) to the t-extension d. In this case the matrix A takes the form

B BA-(\D D jn

Here the first m columns correspond to sets S e Sf, the next m columns correspond to

sets S U {n + 1}, S G ^ , and the last (2m + l)th column corresponds to {n + 1}. The sizeof the matrix B is (!J) x m, and D, D are n x m matrices such that D + D = J, where J isthe matrix all of whose elements are equal to 1. Each column of the matrix J is the vectorjn consisting of n Vs. In this notation, we can write J as the direct product J = jn x j£.Hence for any m-vector a we have Ja = (jm,a)jn.

The rows of D and D are indexed by pairs (i,n + 1), 1 < i < n. The 5-column of thematrix D is the (0, l)-indicator vector of the set S. Since n £ S for all S e Sf, the last rowof D consists of 0's only.

We look for solutions of the system (2) for this matrix A such that X is a nonnegativeintegral (2m+l)-vector. We set

Us = hu{n+i}, S € Sf, y = A{n+i}.

Then the system (2) takes the form

Now, if we set A+ = k + fi, k~ = k — fi, y\ = y + (jm,n), and recall that dl = tjn, weobtain the equations

£> ~ + yiA = tjn. (8)

Recall that the last row of D is the 0-row. Hence the last equation of the system (8) givesy\ = t, and the equation (8) takes the form

Dk~ = 0.

A solution (k+,k~,y\) is feasible if the vector (k,fi,y) is nonnegative. Since

k = l-{k+ + k~\ ii = i(A+ - k~), and y = t - (/«,//),

a solution (/l+,A~,yi) is feasible if

A + > 0 , | r | < ^ + , andr>Ow,/ i ) . (9)

Since the main facet F0(n) is simplicial, the system Bk+ = d° has the full rank m suchthat k+ = k° is the unique solution.

We try to find an integral solution for k~. By (9), we have that \k~\ < k°. This impliesthat kj ^ 0 only for sets S where k°s =£ 0. Since k° is a (0,1)-vector, an integral k^ takesthe value 0 and ±1 only.

We write the matrix (DJn) = Dn explicitly:

Dn =

(

\

1000

0100

A0h0

0

Jkh0

Jiikh0

JljjGk

0

11

jk1 /

The first, the second and the last rows of the matrix Dn are indexed by the pairs(l,w + 1), (2,/t + 1) and (n, n + 1), respectively. The third row consists of matrices with krows corresponding to the pairs (/, H + 1) with 3 < i < n— 1. The columns of Dn are indexedby sets 5 € 5^0U{n+l} in the sequence {l},{2},{li},{2i},{12i}, 3 is an obvious submatrix of Dn, for n' < n.

In the above notation, the equation DX~ = 0 takes the form

= 22jJ. The

+ JiTn- -T i - = 0, i = 1,2,

Since ][Gk = 2jf, the last equality implies that

Hence the above system implies

Recall that we look for a (0, ±l)-solution. Note that if Xj" = 1 and A^ = 0, then^s = / s = 1/2 is nonintegral. Hence we shall look for a solution such that A^ = ±/Ps. So,such a solution is nonzero where 2PS is nonzero.

The main part of the above equations is contained in the term G^j". We can treatthe (±l)-variables (A~),; == X^iij a s labels of edges of the complete graph Kn. Now theproblem is reduced to finding such a labelling of edges of Kn that the sum of labelsof edges incident to a given vertex is equal to a prescribed value, usually equal to 0 or±1. The existence of such a solution depends on a possibility of factorization of Kn intocircuits and 1-factors.

Corresponding facts can be found in [16, Theorems 9.6 and 9.7].A tedious inspection shows that a feasible labelling exists for each of the 7 special

points if n ^ 2 (mod 4) (i.e. if k ^ 3 (mod 4)), and for 5 special points if n = 2 (mod 4).For the other point with (ai,ci2,^1,^2^1,^2) = (1,1,0,0,0,1) there is no feasible solution,i.e. there are S such that A5 = 0 ^= ±l°s.

Now the assertion of the proposition follows. •

In the table below, t-extensions of some special points are given explicitly. The lastcolumn of the table gives a point of A®m_{ for any m>2.

n(mod4) = 3 0 1 2

dn n - 3 0 n - 1 2

dn (3 < i < n - 1) (V) + 1 (V) (V) + 2 (V) + 1

d2,- (3 < i < n - 1) (""3) (V) (n23) + 1 (V) + l

dij(i ± j) (3 <ij<n- 1) 2(n - 4) 2(n - 5) 2(n - 4) 2(n - 5)

i (n~3\ (n-^ (n~2\ _L 1 (n~3\ _L 1

« 1 W { 2 ) { 2 ) V 2 j + 1 ( 2 j + !

j (n—2\ (n—3\ /«—2\ , 1 /«—3\ . i

2n ( 2 ) ( 2 ) ( 2 ) + 1 ( 2 ) + 14 , ( 3 6, (/.e., for n = 6) distance d is the3-extension of de = 2d(K6 — P( 1,6,2)), corresponding to the special point (1,1,0,0,0,1). Onthe other hand, the 3-extension of 2d(K5 — P( 1,2,5)) by the point 6 is an h-point.For n = 0 and n = 3 (mod 4) this d is an antipodal extension at the point 2, i.e.,din + d2i = d2n for all i.

(b) If we consider )§ such that X\2ij = 0 or 1, the problem is reduced to a factorization ofthe graph whose edges are pairs (ij) such that A°12ij =£ 0.

(c) In fact, we can take t slightly smaller. By (9), we must have t > (jm, n). Let r be thenumber of S e 6^0 such that Xs = 1. Then {jm,fi) < (l/2)(52sey ^s ~ r)-

Proposition 6.2. Let JT be the family of cuts lying on the 0-lifting F(n) of the main facetF0(n). Then A(X) = 0 if and only ifn<5.

Proof. By Lemma 6.1, F(6) has quasi-h-points, and (6) implies that quasi-h-points existin all F(n) for n > 6. We prove that there is no quasi-h-point on F(n) for n < 5.

We use the above notation and the equations B(A + //) = d°, Dn(k — //) + y\jn = d{.The first equation has the unique solution X + ji = /?. Hence 2DnA — Dn/P + y\jn = d\where y\ = y + (jmo^°) — Umo,X). The last row gives 71 = dn#+\- Hence the ith row of theequation with Dn takes the form

(Dnk)i = ^((DnA°)i + dI>+1 - dnjn+x).

It can be shown that the condition of evenness (3) implies that the right-hand side is aninteger for n < 5. Moreover, for n < 5, the matrix Dn is unimodular, i.e., |detZ)'| < 1 foreach n x n submatrix D' of Dn. Therefore any solution / is an integer. This implies that nand y = dn,n+\ - Urn*!*) a r e integers, too.

So, all points d e Ln+\ Pi F(n) have a Z+-realization (A,//,y) for n < 5. D

We now give some other examples of Z+-realizations of ^-extensions of even h-points.

Using the fact that Ylievn S(i) is the unique Z+-realization of 2d(Kn) for n ^ 4 , (see [5]),we obtain the following lemma.

Lemma 6.3. The only ^-realizations of extt(2d(Kn)), n > 5, t e Z+, are

(D y^d(i) + (t—l)d(n+l) for t > 1,

(1') ^ 5(i, n+l)+(t-n+l)<5(n+l) /or r > w-1.

Proof. Note that d° = 2rf(Kn) is an even (0,l)-point of Cn. The coefficients of its (CD-realization 2° are as follows: 2°s = 1 if S = {*}, 1 < i < n - 1, or S = Vn_i, and Ag = 0 forother 5. (Recall that we use S such that n ^ 5.) Since it is a unique Z+-realization of d°,the equation 2U+ = d° has the unique integral solution 1+ = A0.

The submatrix of D consisting of columns corresponding to S with 2 j ^ 0, and withoutthe last zero row, has the form D = (In_i,jn_i). Hence the unique (±l)-solutions of theequation Dk~ = 0 are as follows:

(1) A" = 1, 1 < i < n - 1, Xyn_x = - 1 , and(2) X~ = - 1 , 1 < i < n - 1, lyn_{ = 1.

Since (jm^) = 1 in the first case, and (ym,ju) = n — 1, in the second, we have y = t — 1,and y = f — n + 1, respectively. These solutions give the above Z+-realizations (1) and (1').

•

If we define dHyt = ant2textt(2d(Kn-\)), we obtain

dV = 2, 1 < i < j < n - 1, dUn = diin+i = r, 1 < i < n, 4,.+i = 2f.

If we apply (4) to (D and (lr) of Lemma 6.3 (where n is interchanged with n — 1), weobtain (2), and (2) with n and n + 1 interchanged, of Lemma 6.4 below. Summing thesetwo expressions, we obtain the symmetric expression (3) of that lemma.

Lemma 6.4. For dn^ the following holds

(2) dn^=

(3) 2dn" = ^2 (S(Un)+5(i9n+l))+(2t-n+l)(5(n)+S(n+l)).

Lemma 6.5. For n > 6, dn^ is h-embeddable if and only ift>n — 2. Moreover, for t > n — 2,the only ^-realizations are (2) and its image under the transposition (n,n+l).

Proof. In fact, if we use Lemma 6.4, the restrictions of an h-embedding of dn>t ontoVn+i - {n} and Vn has to be of the form (1) and (1') or (V) and (1). •

The realizations (2) and (3) of Lemma 6.4 imply

Corollary 6.6. dn'1 is a quasi-h-point of Cn and (antCn) Pi Cn'+l having the scale 2 iff(n — 1)/21 <t < n - 3 , n > 5.

In fact, for n = 1 we only have to prove that 2d(K^ — P(5,6)) is a quasi-h-point of scale2, and this will be done in Section 7. For n > 8 we use (2), (3) and Lemma 6.4.

Remark. dn~lt2 = 2d(Kn — Pi) and it is a quasi-h-point for n > 6. Its scale lies in thesegment [\n/A\, w/2). dn~12 G Z{antJTn-i njf1/) (see Remark (c) following Lemma 7.1below) for n > 6, but d""1'2 € R+(antJTn-i n Jf*i'2) only for n = 6.

The cone (anrCn_i)nC^2 has excess 1. It has 2n — 2 cuts d(i,n— l),8(i,n),S(n — l),S(n),for i e Fn_2,,its dimension is 2n — 3, and there is the following unique linear dependency:

Y^ S(U n - 1) + (n + 4)S(n) = ^ HU n) + {n- 4)S(n - 1).

The two sides of this equation differ only by the transposition (n— 1, n).The number of quasi-h-points in (antCn-\) n C,p is 0 for n = 5 (since it is so for the

larger cone Cs) and > n — 2 — \n/2] = [n/2\ — 2, which is implied by Corollary 6.6.Perhaps, it is exactly 1 for n = 6,7.

7. Cones on 6 points

Consider the following cones generated by cut vectors on 6 points:

C6, C6\ C62 = Ei*nC6, C6

3, C6U, C6

U = OrfrfC6, C62'3, anrC5.

Recall (see Section 3) that the facets of Ce are, up to permutations of V^ as follows:

(a) 3-fold 0-lifting of the main 3-facet, 3-gonal facet Hyp6(l2 , - l ,03) ,(b) 0-lifting of the main 5-facet, 5-gonal facet tfyp6(l

3,-l2,O),(c) the main 6-facet and its 'switching' (7-gonal simplicial facets) Hype(2,12,— I3) and

tfm(-2,-l,l4).

Let

d6--2d(K6-P{5,6]).

Recall that (up to permutations) d(, is the only known quasi-h-point of C&.The following lemma is useful for what follows. It can be checked by inspection. Recall

that Vn = { 1,2,...,«}.

Lemma 7.1.(1) All TL+-realizations of2de are

(la) 24 = J2(HhS)+S(i,6)) e Z+(JT26) = Z+(EvenJf6),

ieV4-{j}

(2) Some representations of de = 2d(K6 — P(5,6)) in L& are

(2a) d6 =ieV4

(2b) d6 = 23(5) + 23(6) + ^ 3(i) - 3(5,6) e L*'2,ieV4

(2c) d6 = ] T 3(V4 - {/}) - 5(5,6) - 5^(5(i, i + 1 , 6 ) - 5(i, / + 1)) € Lf.iev4 iev4

Here i + 1 is taken by mod 4.

Remarks.

(a) The projection of 2(a) onto V& — {1} gives the Z+-realization 2d(Ks — P(s,6)) =^(5) + Sj=2 3 4^0*>6); it and its permutation by the transposition (5,6) are the onlyZ+ -realizations of the above h-point.

(b) 'Small' pertubations of de do not produce other quasi-h-points. For example, one cancheck that

i 6 +3(1,2) = <J(1) + 5 ( 2 ) + 5(6)+ 5(1,2,5)+ 3(3,5)+ 5(4,5);

it and its permutation by the transposition (5,6) are the only Z+-realizations of thish-point.

(c) Actually, 2(a) is the case n = 5, a = 4 of

ant*(2d(Kn)) = 3(n) + ^ 5(i, n + 1) - (n - a)3(n + 1)

E \ ) + 3({n + 1}) - 5({n, n + 1})).

(d) One can check that Lf1 a Ln strictly, and 2Z15 c L^1 strictly. Note that L^'3 = LfKOn the other hand, Ltf = Ln if and only if (ij) = (1,2).

(t) By l(a) and l(b) of Lemma 7.1 we have

2d6 € /zC62 and 2d6 e fcC6

13,

but 2 4 ^ L\ U L^'3 = L(EvenJf6) U L(OddJf6).

We call a subcone of Cn a cur subcone if its extreme rays are cuts.

Lemma 7.2. Ler d e A(C/f) and let X(d) be the set of cuts of a minimal cut subcone of Cn

containing d. Then

(i) d e A(JiTr) for any Jff such that Jf(d) ^Jf'^JT,(ii) e(X') = 1 implies X1 = X(d\

Proof. In fact, d $ Z+(Jf (d)) implies d $ Z+(Jf'), and d e Z(JfT(d)) n C(Jf(d)) impliesd G Z(JT') n CpT) , and (i) follows. If g(jf') = 1, any proper cut subcone of C(Jf) issimplicial and has no quasi-h-points. •

Now we remark that the cone C6' Pi ant Cs has excess 1, since it has dimension 9 andcontains 10 cuts S(5)9S(6),8(i,5),5(U6), 1 < / < 4, with the unique linear dependency

5^(5(i, 5) — 5(i, 6)) = 2(5(5) — 5(6)).ieVn

Proposition 7.3. ds = 2d(Ks — P2) £ A(Jfs) and it is a quasi-h-point of the following propersubcones of C6: C6

U, C2'3, ant C5, the triangle facet Hyp(l2,-l,03) and Cl62 n ant C5

(which is a minimal cut subcone of Cs containing d).

Proof. The point ds, is the antipodal extension ant^ds) of the point ds := 2d(K5).The minimum size of Z+-realizations of ds is equal to z(ds) = z\ = 5, since its onlyZ + -realization is the following decomposition 2d(K5) = YM=I ^(0-

The minimum size of 1R+-realizations of ds is s(ds) = a\ = 10/3, which is given by theR+ -realization d5 = (1/3)/ X i</<7<5 ( 7)-

Since 10/3 < 4 < 5, we deduce that d6 = 2d(K6 - P{5fi]) £ Z+(C6).But d6 eC6n L6, from (1) and (2) of Lemma 7.1. So, d6 e A°6. Now, from l(a) and (2)

of the same lemma, we have ds G C(^T^2 Pi ant Jf$) n L(Jfl6'

2 n ant Jfs), and so, using (ii)of Lemma 7.2, we get that Jf ^2 n ant Jf 5 is a minimal subcone Jf(d).

Using (i) of Lemma 7.2, and the fact that ant Cs is the intersection of some triangularfacets, we get the assertion of Proposition 7.3 for C^'2, ant Cs and the triangle facet.Finaly, l(a) and 2(c) of Lemma 7.1 imply that d6 e A(Jf2

6'3). •

Remarks.

(a) On the other hand, the following subcones C(J^) of Cs have A(Jf) = 0 :5 simplicialcones Q , i = 1,2,3, both 7-gonal facets, and nonsimplicial cones: C5, C6' = OddCs,and 5-gonal facet.

(b) Nonsimplicial cones C6, Q1'2,^'3, C6

U, C5,anr C5, //y/76(l2,-l,03), / /yp6( l 3 , - l 2 ,0)have excess 16, 6, 10, 1, 5, 5, 9, 5, respectively. The cones CsX^X^Xl^Xs have,respectively, 210, 495, 780, 60, 40 facets and the facets are partitioned, respectively,into 4, 5, 8, 1, 2 classes of equivalent facets up to permutations.

8. Scales

In this section we consider the scale n°(anta2d(Kn)), which is, by Proposition 4.1(iii), equalto min{r G Z + : at > z£}, especially for two extreme cases a = 4 and a = n — 1. Thenumber t below is always a positive integer.

Denote by H(4t) a Hadamard matrix of order 4r, and by PG(2, t) a projective plane oforder t.

It is proved in [5] that t ^ J <5({0)is t h e unique Z+ -realization of 2td(Kn) if n > r2 + r+3,

and that for n = t2 + t + 2, 2td(Kn) has other Z+-realizations if and only if there exists aPG(2,t). Below, in (iv\) — (iv3) of Theorem 8.1, we reformulate this result in terms of A\,nl(2d(Kn)), z£, using the following trivial relations

nl(2d(Kn)) >t+lo 2td(Kn) ^A\ozln = nto

<=> t Y^<5({0) is the unique Z+ -realization of 2td(Kn).I

(iiii) of Theorem 8.1 follows from a result of Ryser (reformulated in terms of zln in [9,

Theorem 4.6(1)]) that zln>n—\ with equality if and only if n = At and there exists an

H(At).

Theorem 8.1.

(h) antp2td(Kn) e Cn+l if and only if p > j ^ f t ;

(i2) antp2td(Kn) e A0 if and only if ^ ^ z^ p eZ+;

(i4) anta2d(Kn) e Cn+l n Ln+l if and only if ^2^/21 ^ a ' a G Z+-

Moreover, if d = ant^2d{Kn) G Cn+i Pi Ln+i, f/ien(»i) eiffter n = 3 , r f e 4 is simplicial, d = ant32d(K4) (so nl(d) = 1 /or i > 0),

or d e Aln, d is not simplicial, a > n > 4 ("so f/°(d) = l),or d € A® (so rj°(d) > 2),

(n2) ^°(d) = min{t : z < at).(iih) n°(ant42d(Kn)) = n°(2d(Kn+l - P(U))) = f/°(2rf(Xnx2));(m2) \n/A] < n°{ant42d(Kn)) < min{t e Z+ : n < At and there exists a H(At)} < n/2;(iih) For n = At, At — 1, we have n°(ant42d(Kn)) = \n/A] = t if and only if there exists

an H(At);

(ivi) ri°(antn-i2d(Kn)) = nl(2d(Kn)) < min{n - 3,nl(2d(Kn+l))};

(iv2) |"(l/2)(V5w^7 - l)j ^ min{r G Z + : n < t2 + t + 2}

< ri°(antn-i2d(Kn))

< min{t e Z+ : n < t2 + t + 2 and there exists a PG(2, t)};

(iv3) For n = t2 + t + 2, we have n°(antn-i2d(Kn)) = ["(l /2)(V4n-7-1)1 = t if and only

if there exists a PG(2,t).

Remarks.

(a) For i > 0, we have rjm(2d(K4)) = i + 1, but n\ant3(2d(K4))) = 1, since ant3(2d(K4))is a simplicial point- For i > 0 and n > 5, we have ^/+1(2^(Xn)) < f/I'(awtn_i(2d(Kri)))with equality for i = 0 and for some pair (i,n) with i > 1. Propositions 5.9-5.11 of [9]imply that

ni+l(2d(K5)) = nl(ant4(2d(K5))) = 2 for i = 0,1;

^3(2d(X5)) = >/Vt4(2d(K5))) = */4(^(K5)) = 3;

n5(2d(K5)) = rj4(ant4(2d(K5))) = rj3(ant4(2d(K5))) = 4.

(b) Using the well-known fact that H(4t) exists for t < 106, we obtain

n°(ant4(2d(Kn))) = n°(2d(Kn+1 - P2)) = n°(2d(Knx2)) = \n/4] for n G [4,424];

(c) Using the well-known fact that PG(2, t), t < 11, exists if and only if t ± 6,10, we get foran = no(antn-{(2d(Kn))) = n\2d(Kn)\ that 6 < an < 1 for 33 < n < 43, 10 < an < 11for 93 < n < 111, and an = |"(l/2)(V4n-7 - 1)~| for all other n G [4,134].

(d) (iii), (iv) of Theorem 8.1 imply that

n°(d(K2tx2)) > 2t with equality if and only if there exists H(4t),

nl{d(Kt2+t+2)) > 2t with equality if and only if there exists PG(2, t).

Note also that an <n — 3 with equality if and only if n = 4,5.

Proof of (iv\). For n > 4 we have

r i ( ^ 4 n _ 7 - l) < n\2d{Kn)) = ^{antn-xthKKn))) < n - 3.

In fact, we have

{eZ+ : z\ < nt},

ri°(antN(2d(Kn))) = min{t G Z + : zln < Nt},

since 2td(Kn) has the following Z+ -realization tY^l^({i}) of maximal size nt, and sincet(antN(2d(Kn))) e /iCn+i if and only if 2td(Kn) admits a Z+-realization of size at most Nt.Denote

p = n\2td(Kn)), q = rjo(antn-{(2d(Kn))).

Then p < q, because z% < (n—l)q implies z^ < nq. Also, q < n — 3, because 2(n — 3)d(Kn)

has the ^-realization E T " 1 ^ - 4)^({0) + ^({^ «})) o f s i z e (n ~ 3)(" ~ !)• O n t h e o t h e r

hand, p > q, because z£ < np implies z% <np — (n — 3), which is proved in [9, Proposition

5.3]. So zvn < np — q < np — p. We have p > (l/2)(-N/4w — 7 — 1) , because otherwise

n > p2 + p + 3, and using [5], 2td(Kn) has exactly one Z+-realization, in contradictionwith the definition of p. •

Theorem 8.2. Let n[n = ni(2d(Kn)). Then

(i) tf < co for d e Ln n Cn,(ii) rj*-1 \nl

n for i > 1, and n^ | ^ for n > 5,

(iii) ^(ad) = [f/!'(d)/fl] /or deCnULn, i > 0, « e Z+.

Proof, (i) Define

Y =Lnncnn{J2^sSW : ° ^ ^ < I}-Clearly, 7 is finite, and one can find X e Z + such that irf is an h-point for every d G 7 .

Let d G LnDCn have an ]R+ -realization d = J^VsdiS)- Clearly the coefficients ii$ arerational numbers. We have d = d\ + d2, where d\ = ^ [^J 5(S), and d2 = YKl*s —llis\)8(S). By the construction, d\ is an h-point. Since d2 = d — d\ and d G LnD Cn, di G

Ln n Cn, we obtain d2 G Y. Hence there is X such that Xd2 G hCn, and we obtain thatXd = Xd\ + Xd2 is an h-point, too.(ii) Obvious.(iii) Take X = r\l(ad\ that is X(ad) has at least i + 1 Z+ -realizations. Hence Aa > rjl(d)implies X > \rjl(d)/a\9 that is, rjl(ad) > \rjl(d)/a\.

Now, take X = \rjl(d)/a~\. So, X — 1 < rjl(d)/a < X => (X — \)a < rjl(d) < Xa. Hence Xadhas at least i + 1 Z+-realizations, implying that X > rjl(ad), and so |V (d)/a] > rjl(ad). D

Remarks.

(a) r\\ = rjl(2d(K4)) = i for i > 1; rj°n = 1 if and only if n = 4,5.

(b) For d ^ Ln and 1 € Z + , we have Xd G Ln implies that X is even (because (>W;7 + Xdik +Xdjk)/2 = X(dij + dik+djk)/2). Hence, for d G Z(") — , we have either d $. Ln (so f7°(<i)is even), or rj°(d) = 1 (i.e. d G /iCn). Since d(G) $. A® for any connected graph G on nvertices (see [14]), we have either rj°(d(G)) = 1 or rj°(d(G)) is even. But, for example,tl°(2d(Kl0 - Pi)) = rj°(2d(K9x2)) = 3.

It will be interesting to see whether rft and max{rj°(d) : d G A®} are bounded fromabove by const x n.

The best-known lower bound for the last number is rj°(d(Kn — P2)), which belongs tothe interval [2 \(n - l ) / 4 ] , n - 2].

It is proved in [19] that for a graphic metric d = d(G), we have

(i) */°(d)<: " - 2 if d(G)eC n ,(ii) rj°(d) G {1,2}, that is, G is an isometric subgraph of a hypercube or a halved cube if

d(G) is simplicial.

9. h-points

Recall that any point of Z+(Jf\,) = hCn is called an h-point.A point d is called k-gonal, if it satisfies all hypermetric inequalities Hypn(b) with

The following cases are examples of when the conditions d G Ln and hypermetricity of

d imply that d is an h-point.

(a) [14], [17]: If d = d(G) and G is bipartite, then 5-gonality of d implies that d G hCn\

(b) [1]: If {dtj} G {1,2}, 1 9 and {<///} G {1,2,3}, 1 < i < 7 < n, then d G Ln and < 11-gonality of d

imply that d G hCn.

So, the cases (a), (b), (c) are among known cases when the problem of testing membershipof d in hCn can be solved by a polynomial time algorithm. The polynomial testing holdsfor any d = d(G) (see [19]) and for 'generalized bipartite' metrics (see [7] which generalizesthe cases (b) and (c) above).

Cases (a), (b) and (c) imply (i), (ii) and (iii), respectively, of

Corollary 9.1. IfdG A®, then none of the following hold

(i) d = d(G) for a bipartite graph G,

(ii) {dij}e {1,2}, l<i<j<n,(iii) {dij} G {1,2,3}, l<i<j<n,ifn>9.

A point d G Z+(JfM) = hCn is called rigid if d admits a unique ^-realization. In otherwords, d is rigid if and only if d G Al

n. Clearly, if d G hCn is simplicial, d is rigid. Rigidnonsimplicial points are more interesting. Hence we define the set

A\ := {d G Axn : d is not simplicial},

and call its points h-rigid.

Theorem 9.2.

(i) A°n = 0 for n < 5, 2d(K6 - P2) G A°6, \A°n\ = oo for n > 7,

(ii) J4j = 0 for n < 4, J4j = (2d(K5)}, |J4j| = oo for n > 6,(iii) for / > 2, A^ = 0 if n < 3, |4J = oo if n > 4.

Proof, (i) and (ii) The first equalities in (i) and (ii) are implied by results in [3]. Theinclusion in (i) is implied by [1]. The second equality in (ii) is proved in [13]. We have\A®\ = oo for n > 7, because A® ^ 0 and |j4j,+11 = oo whenever Ai

n ^ 0 from (6).We prove the third equality of (ii): \A[

n\ = oo for n > 6. The equality is implied by thefact that ant^diKn)) G A{

n+l for any n > 5, a G Z+, a > n. We prove the inclusion.Recall that 2td(Kn) has the unique Z+-realization of size tn if n > t2 + t + 3. (See [5]

or the beginning of Section 8). For t = 1 we obtain the equality z(2d(Kn)) = n for n > 5Using the fact that 2d(Kn) is not simplicial for n > 4, and (iv) of Proposition 4.1 we obtainthe required inclusion.(iii) Since C3 is simplicial, ^ 3 = 0 for / > 2. Consider now n = 4. We show thatA\ = {2(i — l)d(K4) + d : d is a simplicial h-point of C4}. This follows from the fact thatthe only linear dependency on cuts of C4 is, up to a multiple, <5(1) + <5(2) + 3(3) + 3(4) =

So, \A\\ = 00, because there are an infinity of simplicial points, e.g., /J(K2,i) for k G Z + .Finally we use (6). •

Some questions.

(a) Is it true that all 10 permutations of de = 2d(K^ — Pi) are only quasi-h-points of C^l

If yes, these 10 points and 31 nonzero cuts from Jf6 form a Hilbert basis of Ce-(b) Does there exist a ray {M : X G IR+} c Cn containing an infinite set of quasi-h-points?

Recall that we got in Section 6 examples of rays {d° + tdl : t > 0} containing infinitelymany quasi-h-points.

Lemma 9.3. Let d G A®, and let d = ant^d' where d' <£ A®_{. Then d1 is an h-point andz(d') > \s(d')\ + 1.

Proof. In fact, d e CnnLm so d' G Cn-i nLn_i. But d' $. A°n_{, so d' is an h-point of Cw_i.Hence by Proposition 4.1(ii), a G Z + , s(^) < a < z(d').

Note that for n > 5 we have 2d(Knx2) G A\n, 2d(Knx2) = ant4d\ where d' G A\n_{ andd' = ant4d" for d"G A\n_2, and so on.

So, d' is neither a simplicial point nor an antipodal extension (i.e. d' ^ JR+(antJf n_2)),nor d' G Z+(Jf^_{), m = [(n— 1)/2J, because in each of these 3 cases we have for an h-point d!', z(d') = s(d'); this also implies that by Proposition 4.1(iv), d itself is not simplicial.

•

The following proposition makes plausible the fact that the metric d^ = 2d(Ke — P2) isthe unique (up to permutations) quasi-h-point of C&.

Proposition 9.4. Let d G A% d = antadf and d ^ d^. Then

(a) both d and d' are not simplicial;(b) d! $ R+(flntJf4), d! <fr Z+(Jf2

5);(c) d' =/= Xd(G) for any X G Z+ and any graph G on 5 vertices;(d) df has at least Wo ^^-realizations.

Proof. Since ,4° = 0 by [3], we can apply Proposition 9.3, and (a) and (b) follow. Onecan see by inspection, that among all 21 connected graphs on 5 vertices, the only graphsG with nonsimplicial d(G) e Ce are the following 3 graphs: X5, K5 — P2, and K4.K2 = K4with an additional vertex adjacent to a vertex of K4. For these graphs, M(G) is an h-pointif and only if X G 2Z+.

Since 2d(K5 - P2) = ant4(2d(K4)l then, according to (b), d ^ M{K5 - P2).Since for any k e Z + we have z(2M(K4.K2)) = 5A = s{2Ad{K4.K2)\ and (by Proposition

9.3) s(d!) < z(d'\ then dr j= M(K4.K2).There remains the case d' = M(K5). We have s(df) = A5/3 , z(d') = 5 for A = 2 and

z{d') = s{dr) for A G 2Z+, A > 2. (See [9, Proposition 5.11]). So s(d') < a < z(d') impliesX = 2, a = 4, i.e., exactly the case d = ant4(2d(K5)). This proves (c).

Finally, (d) follows from the fact (see [13]) that 2d(K5) is the unique nonsimplicialh-point of C5 with unique Z+ -realization. •

References

[1] Assouad, P. and Deza, M. (1980) Espaces metriques plongebles dans un hypercube: aspectscombinatoires Annals of Discrete Math. 8 197-210.

[2] Avis, D. (1990) On the complexity of isometric embedding in the hypercube: In Algorithms.Springer-Verlag Lecture Notes in Computer science, 450 348-357.

[3] Deza, M. (1960) On the Hamming geometry of unitary cubes. Doklady Academii Nauk SSSR134, 1037-1040 (in Russian) Soviet Physics Doklady (English translation) 5 940-943.

[4] Deza, M. (1973) Matrices de formes quadratiques non negatives pour des arguments binaires.C. R. Acad. Sci. Paris 111 873-875.

[5] Deza, M. (1973) Une propriete extremal des plans projectifs finis dans une classe de codesequidistants. Discrete Mathematics 6 343-352.

[6] Deza, M. and Laurent, M. (1992) Facets for the cut cone I, II. Mathematical Programming 52121-161, 162-188.

[7] Deza, M. and Laurent, M. (1991) Isometric hypercube embedding of generalized bipartitemetrics, Research report 91706-OR, Institut fiir Discrete Mathematik, Universitat Bonn.


[8] Deza, M. and Laurent, M. (1992) Extension operations for cuts. Discrete Mathematics 106-107163-179.

[9] Deza, M. and Laurent, M (1992) Variety of hypercube embeddings of the equidistant metricand designs. Journal of Combinatorics, Information and System sciences (to appear).

[10] Deza, M. and Laurent, M. (1993) The cut cone: simplicial faces and linear dependencies.Bulletin of the Institute of Math. Academia Sinica 21 143-182.

[11] Deza, M , Laurent, M. and Poljak, S. (1992) The cut cone III: on the role of triangle facets.Graphs and Combinatorics 8 125-142.

[12] Deza, M. and Laurent, M. (1992) Applications of cut polyhedra, Research report LIENS 92-18,ENS. J. of Computational and Applied Math (to appear).

[13] Deza, M. and Singhi, N. M. (1988) Rigid pentagons in hypercubes. Graphs and Combinatorics4 31-42.,

[14] Djokovic, D.Z. (1973) Distance preserving subgraphs of hypercubes. Journal of CombinatorialTheory B14 263-267.

[15] Koolen, J. (1990) On metric properties of regular graphs, Master's thesis, Eindhoven Universityof Technology.

[16] Harary, F. (1969) Graph Theory, Addison-Wesley.[17] Roth, R. L. and Winkler, P. M. (1986) Collapse of the metric hierarchy for bipartite graphs.

European Journal of Combinatorics 1 371-375.[18] Schrijver, A. (1986) Theory of linear and integer programming, Wiley.[19] Shpectorov, S. V. (1993) On scale embeddings of graphs into hypercubes. European Journal of

Combinatorics 14.

The Growth of Infinite Graphs: Boundedness andFinite Spreading

REINHARD DIESTELt and IMRE LEADER*

faculty of Mathematics (SFB 343), Bielefeld University, D-4800 Bielefeld, Germany* Department of Pure Mathematics, University of Cambridge,

16 Mill Lane, Cambridge, CB2 1SB England

An infinite graph is called bounded if for every labelling of its vertices with natural numbersthere exists a sequence of natural numbers which eventually exceeds the labelling alongany ray in the graph. Thomassen has conjectured that a countable graph is bounded ifand only if its edges can be oriented, possibly both ways, so that every vertex has finiteout-degree and every ray has a forward oriented tail. We present a counterexample to thisconjecture.

1. The conjecture

For two N —• N functions / and g, let us say that / dominates g if f(n) > g(n) for everyn greater than some no € N.

An infinite graph G is called bounded if for every labelling of its vertices with naturalnumbers, there is an N —• N function that dominates every labelling along a ray (one-wayinfinite path) in G. More precisely, G is bounded if for every labelling / : V(G) —• N thereis a function / : N —• N such that for every ray XQX\ ... in G the function n i—• £(xn) isdominated by / . Otherwise G is unbounded.

Let us see some examples of bounded and unbounded graphs.Every locally finite connected graph is bounded. Indeed, given a labelling / , and given

any fixed vertex v of G, it is easy to define a function fv that dominates all the rays startingat v: just take as fv(n) the largest label of the vertices at distance at most n from v. NowG has only countably many vertices, so there are only countably many functions fv, say/ ° , / \ Setting f(n) = max,<n/'(n), we obtain a function / : N —> N that dominatesevery /„, and hence dominates every ray in G.

The complete graph on a countably infinite set of vertices, Kw, is clearly unbounded:just choose any labelling that uses infinitely many distinct labels, and there will be

218 R. Diestel and I. Leader

B

Figure 1 The unbounded graphs B and F

rays whose labellings grow faster than any fixed N - • N function. The regular tree ofcountably infinite degree, Tw, is another simple example of an unbounded graph: justlabel its vertices injectively, that is, so that any two labels are different.

Two further examples of unboundedness are found in the graphs B and F shown inFigure 1; again, any injective labelling will show that these graphs are unbounded.

Bounded graphs were first introduced by Halin around 1964, in connection with Rado'swell-known paper on Universal graphs and universal functions [4]. Halin conjectured that acountable graph is bounded if and only if it has no subgraph isomorphic to a subdivisionof any of the three graphs 7^, B and F. Halin himself proved this for some specialcases [2, 3]; the conjecture was recently proved by the authors [1]. (We remark that [1]also contains an uncountable version of this result. In the present paper, however, weare only interested in countable graphs.) An interesting aspect of this 'bounded graphtheorem', typical for a characterization by forbidden configurations, is that it provides uswith simple 'certificates' for unboundedness: all we need do to convince someone of theunboundedness of a particular countable graph is to exhibit in it one of the three typesof forbidden subgraph. For boundedness, by contrast, no such 'certificates' are known.

C. Thomassen has recently proposed the following attractive conjecture, which wouldhave provided not only another elegant characterization of the bounded graphs but alsosomething like a certificate for boundedness:

Conjecture. (Thomassen) A countable graph is bounded if and only if its edges can beoriented, each in one or both or neither of its two directions, so that every vertex has finiteout-degree and every ray has a forward oriented tail

(A tail of a ray xo*i... is a subray xnxn+i..., and it is forward oriented if every edgeXmXm+i {m > n) is oriented from xm towards xm+i (and possibly, but not necessarily, alsofrom xw+i towards xm).)

An orientation as above will be called admissible. We remark that any admissibleorientation can be extended to one in which every edge has at least one direction: sincethe graph has only countably many vertices, vo,v\9... say, local finiteness will be preservedif every unoriented edge vtVj with i < j is oriented from Vj to vt.

Intuitively, an admissible orientation identifies in the graph a locally finite substructure

The Growth of Infinite Graphs: Boundedness and Finite Spreading 219

mapping out the preferred directions of rays: eventually, every ray in the graph will followa ray indicated by the orientation. Much of the attractiveness of Thomassen's conjecturelies in its promise that the boundedness of any bounded graph can be tied to such adefinite and simple substructure - one that is obviously itself bounded (by local finiteness),and at the same time accounts for the boundedness of the entire graph.

The 'if direction of Thomassen's conjecture is clearly true: to prove it, we just imitatethe proof that locally finite connected graphs are bounded. More precisely, given anadmissible orientation of the graph and any labelling of its vertices, we first find afunction / that dominates every forward oriented ray (as in our local finiteness proof);the function g defined by

then dominates every ray in the graph.Note also that the conjecture is trivially true for locally finite graphs themselves, as

we may simply orient every edge both ways. The provision for 2-way orientations inthe definition of admissible is, however, essential: the infinite ladder is an example of abounded graph whose edges cannot be 1-way oriented in such a way that every ray hasa forward oriented tail.

Finally, it is not difficult to prove the conjecture for trees; this was first observed byThomassen[5].

Unfortunately, Thomassen's conjecture is not true in general: in the next section weshall exhibit a graph that is bounded but allows no admissible orientation of its edges.

2. The counterexample

Let S be the graph constructed as follows (see Fig. 2). For every n e N, let Rn = VQV^V^ ...be a ray. Let these rays be pairwise disjoint, except that vfi = v® for every n. For everyodd n, make the pair (Rn,Rn+l) into a ladder by adding the edges v"v"+l for all / > 0, asrungs. Finally, for every even n > 0, add a new vertex xn and join it to every vertex of Rn

except VQ.

Figure 2 The graph S

220 R. Diestel and I. Leader

Theorem. The graph S is bounded but allows no admissible orientation of its edges.

Proof. It is not difficult to see that the edges of S cannot be admissibly oriented. Indeed,as the vertices xn all have infinite degree, any admissible orientation would leave each xn

incident with an edge en = xnv" (for some i) that is not oriented from xn towards r". It isthen easy to find a ray in S that traverses every such edge en from xn towards t;", that isagainst its (possible) orientation.

It remains to show that S is bounded. Using the above-mentioned bounded graphtheorem, all we need to show is that S contains no subdivision of Tw, B or F. Thisis easily done. The cases of T^ and B are trivial. Now suppose we have embedded asubdivision of F into S. The bottom ray of F will then be mapped to a ray R c S thatcontains infinitely many of the vertices x", since these are the only vertices of S that haveinfinite degree. For each of those n (except possibly the first), the initial segment Rxn of Rseparates its tail xnR from all but finitely many neighbours of xn in S. As this is not thecase for the bottom ray and the vertices of infinite degree in F, we have a contradiction.

•Actually, it is not much more difficult to verify the boundedness of S directly. Let

i\ V(G) - ^ N b e a labelling of S; we shall define a function / : N —> N that dominatesevery ray in S with respect to / . Let g and / be defined by

g: and g(2n).

Note that g is increasing and dominates every Rn. Therefore / dominates every ray thathas a tail in

S = S -{x 2 ,x 4 , . . . } .

Now let R be an arbitrary ray in S. If R has a tail in 5, then / dominates R. Otherwise,R contains infinitely many xn. It is easily seen that g dominates any ray that starts at v®and contains infinitely many xn. Since R contains a tail of such a ray, it follows that /dominates R.

Of course, the question now arises as to which graphs can be admissibly oriented. Togive this property a proper name (at last), let us say that a countable graph is finitely

Figure 3 The graph S'

The Growth of Infinite Graphs: Boundedness and Finite Spreading 221

spreading if its edges can be admissibly oriented. Thus finitely spreading graphs arebounded, but not vice versa.

It is natural to ask whether S is essentially the only counterexample to Thomassen'sconjecture. More precisely, is it true that every bounded graph that is not finitely spreadingcontains a subdivision of the graph S' of Fig. 3? (Note that S contains a subdivision of S",but not conversely.) It turns out that this is indeed the case, and a proof will be givenelsewhere by the first author.

References

[1] Diestel, R. and Leader, I. (1992) A proof of the bounded graph conjecture. Invent. Math. 108131-162.

[2] Halin, R. (1989) Some problems and results in infinite graphs. In: Andersen, L. D. et al, (eds.)Graph Theory in Memory of G. A. Dirac. Annals of Discrete Mathematics 41.

[3] Halin, R. (1992) Bounded graphs. In: Diestel, R. (ed.) Directions in infinite graph theory andcombinatorics. Topics in Discrete Mathematics 3.

[4] Rado, R. (1964) Universal graphs and universal functions. Acta Arith. 9 331-340.[5] Thomassen, C. (private communication).

Amalgamated Factorizations of Complete Graphs

J. K. DUGDALE1 and A. J. W. HILTON'+ Department of Mathematics, West Virginia University, PO Box 6310, Morgantown WV 26506-6310, U.S.A.

• Department of Mathematics, University of Reading, Whiteknights, PO Box 220, Reading RG6 2AX. U.K.

We give some sufficient conditions for an (5, t/)-outline T-factorization of Kn to be an(S, £/)-amalgamated T-factorization of Kn. We then apply these to give various necessaryand sufficient conditions for edge coloured graphs G to have recoverable embeddings inT-factorized AT's.

1. Introduction

In [9] (see also [12]) the second author developed the idea of an outline latin square, andshowed that every outline latin square is an amalgamated latin square. In [4] the secondauthor and A. G. Chetwynd described various analogues of this result for symmetric latinsquares. Since a latin square can be viewed as a proper edge colouring of Knn with ncolours, it is also very natural to consider similar analogues for edge-coloured Kn's. Thishas already been done to a limited extent by Andersen and Hilton [1,2,3] and later byRodger and Wantland [21] (who were concentrating on other aspects) but a more roundedand complete account is given here.

The ideas here began with the joint work of L. D. Andersen and the second author onGeneralized Latin Squares [1,2,3]. The amalgamation idea has been taken further indifferent directions by various authors. As well as the authors already mentioned, furtherdevelopments have been due, at least in part, to R. Haggkvist, A. Johanson, C. St J. A.Nash-Williams and J. Wojciechowski (see the references).

Graphs will in general contain loops and multiple edges.Given a graph G, an amalgamation of G is a graph G* and a surjective map <j>: V(G) ->

V(G*) together with a bijective map r/ : E(G) -> E(G*) such that ifeeE(G) and e = vw then

224 / . K. Dugdale and A. J. W. Hilton

rj{e) = 0(v)(f)(w). Intuitively in an amalgamation of G, various of the vertices areamalgamated, or stuck together, whilst the original set of edges all remain distinct.

An edge colouring of a graph G is here simply a function i/r: E{G) -> <&, where ^ is a setof colours.

If G has an edge colouring i/r, then an amalgamated edge coloured G is an amalgamationG* of G, as above, together with an edge colouring ^* of G*, ^ * : £ ( G * ) ^ ^ * , and asurjection £ : # - • # * such that £(i/r(e)) = ^*0y(e)) (VeeE(G)). Intuitively, if G is edgecoloured, then in an amalgamated edge coloured G, the vertices are amalgamated, asabove, and various of the colours on the edges are combined [one could imagine, forexample, that the distinction between light blue, medium blue and dark blue edges isforgotten, and that these edges are simply taken together as being the blue edges].

G: e9

Y N N

e4 G*:

e 3 V4

Figure 1

This is illustrated in Figure 1. The amalgamation of G is given by <f>(vx) = w19 <f>(v2) ==ft (1 ^ / ^ 6). The colour set for G is # = {a, ^,7} and theA,^}. The surjection £:<^->^* is given by £(a) = A,

0(t?3) = <f>(vj = w2 andcolour set for G* is ^* =

= I1 an<i ^ has the required property that

- «a) = A = t/r*(/2) -

If each vertex of a graph G is incident with the same number, say t, of edges from a set Fof edges, then F is called a t-factor of G. If G is regular and if each colour class of an edgecolouring of G is a ^-factor, then the edge colouring is called a t-factorization of G. Similarlyif G is edge coloured with the q colours k1,...,kq, and, for 1 q, the edges of ki form

Amalgamated Factorizations of Complete Graphs 225

a ^-factor, then the edge colouring of G is called a T-factorization of G, where T = (r\,...,tq) is a composition of d(G), the degree of G.

A composition of a positive integer « is a vector whose components are positive integersthat sum to n. Let / denote the composition (1 ,1 , . . . , 1) (the appropriate value of n willalways be clear from the context).

If T = (*!,..., tq) and U = (r19..., ru) are.two compositions of the same number n, and,letting x0 = 0, if there is a composition X = (x1,...,xn) of # such that

then we call (7 an amalgamation of 7".We are concerned in this paper with T-factorizations of Kn (so that T = (tx,..., f9) is a

composition of « — 1).. Given a 7"-factorization of G = Kn, an edge coloured amalgamationG* of G may conveniently be described in the following way. We may suppose that thevertices of Kn are vv...,vn and that the colours used on E(G) are K1,...,KQ (the colourclass Kt is a /rfactor). Similarly we may suppose that the vertices of G* are w19..., vvs andthat the colours used on E(G*) are cl9..., cM. Let l^" 1 ^) ! = pt (1 ^ / ^ 5) and | ~1(cA.)| = xk

(1 < k ^ M). Let ^. + 1+ ... + ^. = r; (1 ^ y ^ u) so that U = ( r 1 ? . . . , r j is an amalgamationof 7. Without loss of generality, we may suppose that <j>~\w?) = {vp._+1,...,y } for1 ^ z'^ s, taking ^0 = 0. Let S = (/?l9...,/?s). Clearly much of the essence of the edgecoloured G* is described by S and £/, given that the T-factorization of Kn is known. We saythat the edge coloured G* is the (S, £/)-amalgamation of the T-factorized Kn.

Some obvious properties of an (S, ^/)-amalgamation of a T-factorized Kn are given in thefollowing proposition.

Proposition 1. Let Kn be given a T-factor ization. Let S = (/?l9 ...,ps) be a composition of nand let U = (r1?..., ru) be an amalgamation ofT— (tly..., tQ). Then the (S, U)-amalgamationof the T-factorized Kn has the properties:

(Pi) colour ck occurs on rkpi edges incident with wt (counting loops as two edges)(1 < A: < M, 1 < / ^ 5 ) ;

(Pii) there are ptpj edges joining wt to w} (1 ^ / <j ^ s);(P Hi) there are (.£') loops incident with wf (1 ^ / ^ s).

Proof. After the T-factorization of Kn is amalgamated to form a (/-factorization, colour ck

occurs on rk edges incident with each vertex v. When the vertices are amalgamated, a vertexwt is formed by amalgamating/^ of the vertices of the Kn, so the number of edges of colourck incident with wi is/?,.rk (counting a loop as two edges). This proves (Pi). Between the twodisjoint sets of/?,, vertices and pi vertices in Kn which are amalgamated to form w, and \vp

there are pipj edges, so there are/??/?; edges joining wt and wy This proves (Pii). Betweenany two of the set of/?, vertices in Kn which are amalgamated to form wi there are (.?') edges,and these become loops on wt. This proves (Piii). •

Suppose now that G* is an edge coloured graph with s vertices, w1?..., ws, whose edgesare coloured with u colours, q, . . . , cu, and suppose that there are compositions S = (p^...</?s) of ft and U = (r1? ...,ru) of «—l, where U is an amalgamation of T = (t19...,t), such that

226 J. K. Dugdale and A. J. W. Hilton

G* satisfies (Pi), (Pii) and (Piii) of Proposition 1. Then the edge coloured G* is called an(S, £/)-outline (f1?..., ^)-factorized Kn. Ift1 = ... = tq = t then this would be abbreviated toan (5, £/)-outline /-factorized Kn.

The idea here is that if G* is an (S, £/)-outline (^ . . .^-factor izat ion of Kn, then G*satisfies all the numerical conditions that we know that an (S, £/)-amalgamated (715..., tq)-factorization of Kn would have, but we are not informed as to whether G* is actually an(S, t/)-amalgamated (/19...,^-factorization or not. It is not hard to construct exampleswhere an outline factorization is not an amalgamated factorization. However for somevalues of the various parameters, an (S, £/)-outline (/19..., ^-factorization of Kn is an(S, t/)-amalgamated (/1?..., ^-factorization of Kn.

We give various such values of the parameters in the following theorem which is provedin Section 3.

Theorem 2. Let S = (pt,...,ps) be a composition ofn, let T = (71?..., tq) be a composition ofn - 1, and let U = (r15..., ru) be an amalgamation of T. Suppose also that X=(x1,..., xu) isa composition of q such that, for ke{l,...,u}, rk = tx +1 + ... + tx . If either

(i) /7l9... ,ps are even (so that n is even), or(ii) u = q (so that U = T), or

(iii) for fce{l,...,M} either rk - ^_ l + 1 (= tx) or tXk_x+1,...JH are even,

then any (S, U)-outline T-factorization ofKn is the (S, U)-amalgamation of a T-factorized Kn.

2. Preliminary definitions and results about edge-colourings

We first need to give a number of definitions and results concerning edge colourings ofgraphs.

Suppose that a multigraph G is edge coloured with colours ^ = {cl9..., cu}. For ke{\,...,u} and ve V(G), let Ek(v) be the set of edges incident with v of colour ck, and for v, we V(G),v =j= w, let Ek(v, w) be the set of edges joining v and w of colour ck\ if v = w then Ek(v, \v)denotes the set of loops of colour k incident with v. We let \Ek(v)\ denote the number ofedges of colour ck incident with v, counting each loop as two edges. The edge colouring ofG is called equitable if

\\Ej(v)\-\Ek(v)\\^ 1 (VreK(G) and Vy, *e{l,...,w}).

The edge colouring of G is called balanced if in addition to being equitable, the edgecolouring has the property that

\\Ej(v,w)\-\Ek(v,w)\\ ^ 1 (Vi\ WE F(G), and Vy, A:G{1,...,M}).

If G is edge coloured with ^ = {q,...,cM}, then, for ke{l,...,w}, let Ek denote the set ofedges of colour ck. An edge colouring is called equalized if

An edge colouring with the property that no vertex has more than one edge of any colourincident with it is called a proper edge colouring.


For a graph G, let A(G) and 8(G) denote the maximum and minimum degree of Grespectively.

The first lemma we need is due to de Werra (see [24, 25]; another proof may be foundin [3]).

Lemma 3. Let k > I be an integer and let G be a bipartite multigraph (so G has no loops). ThenG has a balanced edge colouring with k colours.

The next lemma is essentially due to Petersen [20]; see also [4].

Lemma 4. Let G be a multigraph in which loops are permitted, and let k be a positive integersuch that, for each ve V(G), either (\/k)dG{v) is an even integer or (\/k)(dG(v) + 1) is an eveninteger. Then G has an equitable edge colouring with k colours.

Here as elsewhere in this paper, a loop counts two towards the degree of a vertex.

A special case of Lemma 4 is the following well known theorem of Petersen [2].

Lemma 5. Let G be a regular multigraph in which loops are permitted. Let d(G) = 2k. ThenG can be 2-factorized.

Lemma 6. Let xl9...,xl be positive even integers. Let H be a graph satisfying

and

where a is a positive integer. Then H has an edge colouring with I colours K1,...,K1 such that,with Hi denoting the spanning subgraph of H whose edges are the edges of H coloured Kf,

and

Proof. The number of vertices of H with odd degree is even, say 2y. From H form a graphH+ by adding in y further edges in such a way that the degree of each vertex of H+ is even.Now form a graph H++ by adding sufficiently many loops at each vertex so that H++ isregular of degree x1 + ... + xl [each loop counts two]. By Petersen's theorem (Lemma 5) H++

can be 2-factorized; thus we can edge colour H++ with colours 7i + . . .+y p , where p =l(xl + . . . +xt), in such a way that the spanning subgraph, whose edges are the edges of H++

coloured yt, is regular of degree two (\ ^ i ^p). Let Jt (1 ^ / ^p) denote the spanningsubgraph of H whose edges are coloured yv Then A(^) ^ 2 (1 ^ / ^ p).

We can now change the edge colouring of H with y19..., yp so that it is equalized and stillhas the property that the maximum degree in each colour is at most two. For suppose thereare two colours, say yx, and y2, such that \EiJJl ^ \E(J2)\ + 2. Consider the graph Jx U J2. Ifthis has 2y vertices of odd degree, form (Jx U J2)

+ by inserting y edges so that, in (Jx U J2)+,


each vertex has even degree. Then each component of {Jx U J2)+ is Eulerian. Going round

an Eulerian cycle in each component and leaving out the y extra edges, produces trails inJx U J2 which begin and end with distinct vertices of odd degree, and cycles. Note that if acycle is regular of degree four it has an even number of edges. We colour each cycle andtrail alternately y1 and y2. If a cycle has an odd number of edges, we ensure that some vertexof degree 2 is the starting and finishing vertex (so the two edges incident with it receive thesame colour). Let J[ and / 2 denote the spanning subgraphs of Jx U J2 coloured yx and y2

after this recolouring. Then A(J[) ^ 2 and A(/2) ^ 2. We may arrange that the cycles andtrails with an odd number of edges were paired off (with possibly one over) so that if onehas one more yx edge than y2 edges, the other has one more y2 edge than yx edges. If thisis done then \E{J^\-\E(J'2)\\ ^ 1.

Repeating this as often as necessary, produces an edge colouring in which

where / " denotes the spanning subgraph of H coloured yt eventually obtained (1 ^ zSince \E(G)\ ^ a(xx +... +xz) = lap it follows that

Now for 1 ^ 7 ^ / we form disjoint unions of \xj of these colour classes. Calling theseunions H1,...,Hl we have that Hx U ... U Hl = // ,

as required. D


In this section we prove several lemmas, and these lead to a proof of Theorem 2.

Lemma 7. Let S = (px,... ,ps) be a composition of n, let T = (/\,..., tq) be a composition ofn—l, and let U = (rl5..., ru) be an amalgamation of T. Let X = (x1?..., xu) be a compositionof q such that

rk = tZk_i+1 + ... + tZk.

If, for each k e {1, . . . , M}5 either

or tXk_i+1,...,tXk are all even, then any (S, £/)-outline T-factorization of Kn is the (S, U)-amalgamation of an (S, T)-outline T-factorization of Kn.

Proof. Consider an (5, t/)-outline T-factorization G* of Kn. Let the vertices of G* be w1?...,ws and the colours used be c15...,cu. If rk = t +1 for all ke{1,...,u) then T= Uand thereis nothing to prove. If rfc #= tx _ +1 for some k then tx _ +1,..., tx are, by hypothesis, all even.Let Gk denote the subgraph of G* induced by the edges coloured ck. Then, by property


(Pi), in Gk each vertex wi has degreepjk. We give G* an equitable edge colouring with \rk

colours, y15..., yir ; such an edge colouring exists by Lemma 4. Then, for each/ e {xk_1 + 1,...,xk), we combine a disjoint set of \tj of yl 5 . . . , ys together to produce colours fix _ +1,..., /?x..Then there are tjpi edges of colour fa incident with each vertex wt. Doing this for eachcolour ck for which rk 4= tx +1 produces an (S, r)-outline ^-factorization of Kn of whichG* is an (S, £/)-amalgamation. D

Lemma 8. Let S = (px,... ,pn) be a composition of In with p19... ,pn all even, let T = (tx,...,

tQ) be a composition ofln—\, and let U = (r19... , r j be an amalgamation of T. Then any

(S, U)-outline T-factorization of K2n is the (S, U)-amalgamation of an (S, T)-outline

T-factorization of K2n.

Proof. Consider an (S, £/)-outline T-factorization Y of K2n. For each colour ck whererk > 1, let Gk denote the subgraph of Y induced by the edges coloured ck. Then, byproperty (Pi), vertex wt has degree rkpt. We give Gk an equitable edge colouring with rk

colours; such an edge colouring exists by Lemma 4, sincepx,... ,ps are all even. Doing thisfor each colour ck produces an (S, /)-outline 1-factorization of K2n. By amalgamatingcolours appropriately we obtain an (S, r)-outline 7"-factorization of Kn of which theoriginal (S, £/)-outline T-factorization of K2n is the (5, £/)-amalgamation. D

Lemma 9. Let S = (pr, ...,ps) be a composition ofn and let U = (r15 ...,ru) be a composition

of n — 1. Any (S, U)-outline U-factorization of Kn is the (S, U)-amalgamation of a U-

factorization of Kn.

Proof. Suppose we have an (S, £/)-outline U-factorization G* of Kn. Let the vertices of G*be w1,...,ws and the colours be cl,...,cu and suppose that G* satisfies properties (Pi),(Pii) and (Piii) of Proposition 1. If S = I there is nothing to prove, so we may supposethat S ^ I. We may suppose without loss of generality that ps ^ 2.

Our object will be to change the edge coloured graph G* on the vertices wx,..., ws to anedge coloured graph G** on vertices w15..., ws_1? wsl, ws2 by 'splitting' the vertex ws intotwo vertices wsl and ws2. For ie{\,...,s—l} the ptps edges joining wt to vvs will beredistributed so that p{ of them join wt to wsl and the remaining (ps— \)pt join wt to ws2;(ps — \) of the (Is) loops on ws in G* will become edges joining wsl to ws2, and the remaining(i°s) — (ps—l) — d5"1) loops on ws in G* become loops on ws2 in G**. The colours on thesubgraphs induced by w1,...,ws_1 in G* and in G** are the same. For ie{l,...,s—l} andk e {1,..., u), the number of edges of colour ck joining wt to ws in G* equals the number ofedges of colour q. joining wt to wsl or ws2 in G**. Also the number, say l(k), of loops of eachcolour ck on ws in G* equals the number of edges of colour enjoining wsl to ws2 in G** plusthe number of loops of colour ck on ws2 in G**. Finally we arrange that the number of edgesof colour ck incident with wsl is rk and that the number of edges of colour ck incident withws2 is (ps—l)rk (counting loops as two edges). [Recall that, by (Pi), the number of edgesof colour ck incident in G* with ws was psrk.] This process keeps the number of edges ofcolour ck incident with each of wv..., ws_1 the same as it was in G**.

It is easy to check that if this process is carried out successfully, then the resulting edge

coloured graph G** is an (S\ £/)-outline ^/-factorization, where S' = (p1,...,ps_1, \,ps— 1),and it is easy to see that G* is an amalgamation of G**. Repetition of this process willeventually produce an (/, U) outline ^/-factorization of which G* is the (S, U)-amalgamation.

Of course the construction of G** from G* is no problem; the aspect we need toconcentrate on is the colouring of the edges incident with wsl and ws2. As an aid in seeinghow to colour these edges, we construct the following bipartite graph //. We let the vertexsets of H be {p'19... ,/Vi) a n d {r[,..., r'u), and we join vertex p\ to vertex rk with x edges ifthere are exactly x edges of colour ck joining w{ and ws in G* (1 ^ k ^ u, 1 i < s— 1).Using Lemma 3 we give H an equitable edge colouring with the ps colours y15..., yp . [Notethat in various analogues we must require at this point that H has further properties. Forexample in [4] it is required that the analogous graph be balanced; however this is notneeded here.]

If there are z edges in //joining p{ to rk coloured yv then we colour z edges of G**joining wsl to w1 with colour ck, and corresponding to each of the remaining rkps — 2l(k) — zedges of //incident with r'k we colour an edge of G** joining ws2 to w{ with colour ck

(1 < / ^ s— 1, 1 ^ k ^u). Since dH(p{) = ptps, there are pt edges coloured yx incident withPi, and so the pi edges in G** joining wi to wsl each receive a colour, and similarly thePi(ps—l) edges joining wt to ws2 each receive a colour. The number yk of edges of Hincident with rk coloured yx satisfies

yk < |

Therefore the number (yk) of edges of G** incident with wsl coloured ck is at most rk. Inorder to make the number of edges of G** incident with wsl coloured ck be exactly rk, rk—yk

edges joining wsl to ws2 are coloured ck. The number of edges (excluding loops) incidentwith ws2 coloured rk is therefore

(r,ps - 2l(k) - v , ) + (r, - v,) = rk{ps - 1) - 2( vA. + /(/:) - rk).

But1

^' " A ^+ l(k)-rt

-(A/,-2/W)

2I(k)

l(k)-rt

2l{k)

Ps

Therefore the number of edges of G** incident with ws2 coloured ck (excluding loops) is atmost (ps-l)rk.

We now colour the loops on ws2 in such a way that the c^-degree of ws2 (i.e. the numberof edges coloured ck incident with ws2, counting loops as two edges) is (ps—\)rk. Since thenumber of edges of colour ck in G* joining ws to {w19..., vv -J equals the number of edgescoloured ck joining {wsl,ws2} to {w15...,ws_x}, and since the c^-degree of ws in G* (psrk)equals the cfc-degree of wsl (rk) in G** plus the required cfc-degree of ws2 ((ps—\)rk), thenumber of loops on ws2 we need to colour ck equals l(k) — (the number of edges colouredck joining wsl to ws2), i.e. l(k) — {rk—yk) = l(k)+yk — rk ^ 0 (as above). This colouring isclearly possible and exactly uses up all the (Ps

2~1) loops on wsl.

Proof of Theorem 2. In case (ii), namely when u = q, so that T = U, Theorem 2 is simplyLemma 9.

In case (i), when pl,...,ps are all even, then, by Lemma 8, any (5, £/)-outline T-factorization of K2n is the (S, £/)-amalgamation of an (S, 7>outline 7-factorization ofK2n. By Lemma 9, any (S, 7>outline T-factorization of K2n is the (S, T)-amalgamation ofa T-factorization of K2n, Therefore any (5, (7)-outline T-factorization is the (5, t/)-amalgamation of a T-factorized ^C2w.

In case (iii), by Lemma 9, any (S, £/)-outline T-factorization of ^ is the (S, U)-amalgamation of an (/, £/)-outline T-factorization of Kn. By Lemma 7, when eithert M, = r, ov t +,,..., fr are all even, each (/, £/)-outline T-factorization of Kn is the(/, £/)-amalgamation of a T-factorization of Kn. Therefore, in case (iii), any (5, (7)-outlineT-factorization of Kn is the (S, (7)-amalgamation of a T-factorization of Kn. [In case (iii),we could equally well apply Lemma 7 first and Lemma 9 afterwards.]

4. Embedding an edge coloured Kr

Theorem 2 has a number of interesting applications to do with embedding edge colouredgraphs G inside T-factorizations of Kn, where T = (tx, ...,tq) and the colours used are q, . . . ,cq. The embeddings are such that the /-th colour class, consisting of the edges coloured c,,becomes part of the corresponding rrfactor of Kn.

The simplest such application is the following result, which generalizes a theorem ofAndersen and Hilton [2, Corollary 4.3.4]. The Andersen-Hilton result was rediscoveredrecently by Rodger and Wantland [21].

Theorem 10. Let T= (t1,...,tq) be a composition of n—\. Let Kr be edge coloured with qcolours, cv ..., cq, and let Gt be the spanning subgraph of Kr whose edges are the edges of Kr

coloured ct (1 ^ / ^ q). Then the edge coloured Kr can be embedded in a T-factorized Kn,with G( forming part of the corresponding t-factor, if and only if

(i) \E(Gt)\^tfr-±tfn (1 < / q\

(ii) ttn is even (1 ^ / q),

and

(iii)

Proof. It follows from Proposition 1 and Theorem 2(ii) that the edge coloured Kr can beembedded in Kn with each Gf in the corresponding rrfactor if and only if we can constructan (S, r)-butlineT-factorization G* of Kn, with S = (\,l,...,\, n — r) being a compositionof n, in the following way. Let w1?..., wr be the vertices of Kn and let wr+l be a further vertex.Join wr+1 to wf (1 / r) by n-r edges, and place (\r) loops on wr+1. For each /e{l,...,r} and ye{l, . . . ,g), colour tj — dG(w^) edges joining vv?. to wr+1 with colour cr If thisis done, then since £ j = 1 (tj-dG_(wi)) = (n- \)-(r- 1) = n-r, all edges between w?. and wr+1

are coloured. This colouring is possible if and only if t. A(Gj) ^ dG(wt), which iscondition (iii). After this, colour sufficient loops incident with wr+l with colour ci so that thenumber of edges of colour ci incident with wr+1 becomes equal to tf(n — r). [Here a loopcounts as two edges.] If this is done then, since Yf)=\ tj(n~r) = (n~ ^)(n~ rX all loops andedges incident with wr+1 are coloured. This is possible if and only if

and

'M-^-ib-dciwM^O (mod2)

The first condition here can be rearranged to give

which is equivalent to (i), and the second condition yields

(2r - n) tj - 2 \E{G$ = 0 (mod 2)

which is equivalent to (iii). This proves Theorem 10. •

We can use Theorem 10 to show that an edge coloured graph of order r can be embeddedin a 7-factorized Kn if n ^ 2r. The argument to show this is essentially the same as that usedby Evans [7] to deduce from Ryser's theorem [22] that an incomplete latin square of sider can be placed in a complete latin square of side n if n ^ 2r.

Theorem 11. Let T = (f19... ,tQ) be a composition ofn — 1, and let ttn be even (1 ^ / ^ q). Let

G be a simple graph with r vertices, where r ^^n. Let G be edge coloured with colours cx,...,cq

in such a way that, with Gt denoting the spanning subgraph of G whose edges are the edges

ofG coloured ct, A(GJ ^ ti (1 ^ / < q). Then the edge coloured graph G can be embedded in

a T-factorized Kn, with each Gt becoming part of the corresponding tffactor.

Proof. We may first extend the edge colouring of G to an edge colouring of Kr with c15...,cQ with the property that, with Hi denoting the spanning subgraph of Kr whose edges arethe edges of Kr coloured ct, A(Ht) ^ tt (1 ^ i q). We do this by colouring the edges of Gone by one. If an edge e = {u, v} of G has yet to be coloured then there are at most 2(r — 2)

coloured edges incident with one or other of u and v, and so there is a colour ct which isneither used on ti edges incident with w, nor on tt edges incident with v. We may thereforecolour e with this colour cv Proceeding in this way, all edges of Kr are coloured. Since, for1 ^ / < q, ttr — \ttn ^ 0 as r \n, it follows that all of the conditions (i)-(iii) of Theorem10 are satisfied, and so the edge coloured Kr can be embedded in a J-factorized Kn.Therefore the edge coloured graph G is embedded in -a T-factorized Kn, as required. •

5. Recoverable embeddings of edge coloured graphs

Now consider a simple graph G with vertex set {i?15..., vr} which is edge coloured with scolours. Suppose that G can be embedded inside a T-factorized Kn, where T = (t19..., tq)and q s, which has vertex set {v19...,vn}in such a way that the edges of G coloured ci areall within the corresponding /rfactor, for each ie{\,... ,s}. We say that the edge-colouredgraph G is recoverable from the T-factorized Kn if the colours used on the edges of G (thecomplement of G with respect to the vertex set V(G)) are all in {cs+1,...,cq}; we say that Gis recover ably embeddedr in Kn. The word recoverable is used because if all the new vertices(i.e. the vertices vr+1,...,vn) are removed from Kn and the edges with the new colours (i.e.E{G)) are also removed, the original edge coloured graph G is what remains.

Theorem 12. Let T = (t19..., tq) be a composition ofn—l, with ts+l,..., tq all being even. LetG be a simple graph with r vertices, and let G be edge coloured with s colours cv..., cs. LetG{ be the spanning subgraph ofG whose edges are the edges ofG coloured ct(\ ^ / s). Thenthe edge coloured graph G can be recover ably embedded in a T-factorized Kn, with Gt formingpart of the corresponding tt-factor (1 ^ / s), if and only if

(i) (r)-\E(G)\> t (tj-^n),\^J i=s+l

(ii) \E(G()\^t(r-$t(n (1 «S i s? $),

(iii) ttn is even (1 ^ i s),

(iv)

We give two proofs of Theorem 12. The first is more elegant, but the second is useful asa model for some later results. For the second proof we need Lemma 6, which is not neededfor the first.

First proof of Theorem 12By Theorem 2 (iii), since ts+1,...,tq are even, G can be recoverably embedded in a T-

factorized Kn in the way described if and only if G can be embedded in an (S, £/)-outliner-factorization of Kn, where S = ( 1 , . . . , l,n — r) is a composition of n and U = (/1?...,fs,

'.+i + ..- + '„).Let the vertex set of G be {w15..., wr}. Let t* = ts+1 + ... + tq. Let G be the complement of

G with respect to {wv..., wr}. Colour each edge of G with colour c*. For 1 / r, join \v{

to a further vertex w* with n — r edges, and introduce (V) loops onto vv*.

For 1 ^ / ^ r, colour sufficient edges from w* to wt with colours ci (1 ^y ^s) and c* sothat the number of edges of colour cj incident with wt is tp and the number of edges ofcolour c* is t*. Since (r—l) + (n — r) = n—l = t1 + ... + tq = t1 + ... + t8 + t*, there areexactly the right number of edges joining wi to w* for this to be possible, with every edgereceiving a colour. So this process is possible if and only if

and

da(wf) ^ t* (1 < i < r),

or in other words, if and only if condition (iv) is satisfied and

which is condition (v), is satisfied.When this is done, colour the loops on w* in such a way that the number of edges of

colours Cj (1 ^ 7 ^ s) and c* incident with w* is t^n — r) and t*{n — r) respectively, wherea loop counts as two edges. The number of non-loop edges of colours Cj and c* incidentwith w* is rtj — 2|2S(CJ;)| and rt* — 2\E(G)\ respectively, so the number of loops we needto colour is

n — r2

so there are exactly the right number of loops on w* for this to be possible, with each loopreceiving a colour. It follows that we may colour the loops in the way described if and onlyif

(a) there are not already too many edges of some colour incident with u*, and(b) the number of edges of each colour joining w* to {\v\,..., ws} has the right parity.

Condition (a) is, more precisely, that

and

rt*-2\E(G)\ < t*(n-r),

and these are equivalent to conditions (ii) and (i) respectively. Condition (b) is, more


precisely, that

,)| = f,(*-r) (mod2) (l^j^s)

and

rt* - 2 \E(G)\ = t*{n - r) (mod 2);

the second of these is always true, and the first is true if and only if condition (iii) is true.

•Second proof of Theorem 12Necessity. Suppose G can be recoverably embedded in Kn. Then G is edge coloured withcs+1,...,c9, so G and G together constitute a Kr edge coloured with c\,...,cQ, and theembedding can be viewed as being of Kr embedded in a T-factorization of Kn with each G,forming part of the corresponding rrfactor of Kn (1 ^ / q). [Here, for 1 / ^ q, G, is thespanning subgraph of G whose edges are the edges of Kr coloured ct] Conditions (ii), (iii)and (iv) now follow from Theorem 10. By Theorem 10 (i)

\E(Gt)\ ^tj-\ttn ( 5 + 1 ^i^ql

so

\E(G)\> t O.r-frn),i=s+l

from which condition (i) follows. By Theorem 10 (iii),

so that

which yields condition (v).

Sufficiency. Suppose (i)-(v) hold. Since, by (v), A(G) ^ ts+1 + ... + tQ, since ts+l,..., tQ are alleven, and since, by (i),

\E(G)\> t t,{r-&),i=s+l

it follows from Lemma 6 that we can edge colour G with q — s colours cs+1,...,cQ in sucha way that, with G( denoting the spanning subgraph G whose edges are the edges of Gcoloured c\,

\E(Gf)\ ^ tt(r-±n)

and

This edge colouring of G, together with the given edge colouring of G, constitute an edge

colouring of Kr which satisfies Theorem 10. Therefore this edge coloured Kr can beembedded in a T-factorized Kn, where each Gi forms part of the corresponding ^-factor.Clearly the embedding of G that is included by the embedding of Kr is recoverable. •

Of course, Theorem 10 can be considered to be a corollary of Theorem 12 (just puts = q). A further pair of corollaries concerns the recoverable embedding of an edgecoloured graph G into a 2-factorized K2n+1 or a 2-factorized K*n, where K*n denotes thegraph obtained from K2n by removing a 1-factor (K*n is sometimes known as the w cocktailparty graph').

The first is straightforward from Theorem 12.

Corollary 13. Let G be a simple graph of order r which is edge coloured with s colours cx,...,cs in such a way that, if G is the spanning subgraph of G whose edges are the edges of Gcoloured ci then A(Gt) ^ 2 (1 < / s). Then the edge coloured G can be recover ably embeddedin a 2-factorized K2q+1 with each G{ forming part of a distinct 2-factor if and only if

(i) (^

(ii) \E{Gl)\^2r-2q-\ (\ ^ i s),

(hi) r-S(G)-\ ^2(q-s).

The second corollary is also fairly straightforward if you think of a 2-factorized K£n asbeing a T-factorized K2n, where T= (1,2,...,2) is a composition of 2n— 1.

Corollary 14. Let G be a simple graph of order r which is edge coloured with s colours cv ...,cs in such a way that, if Gt is the spanning subgraph of G whose edges are the edges of Gcoloured ci9 then A(Gf) ^ 2 (1 ^ / s). Then the edge coloured G can be recover ablyembedded in a 2-factorized K2{q+1), with each Gt forming part of a distinct 2-f actor, if and onlyif G has a partial matching T7* such that

(i) {£j-\E{G)\-\F*\ > (q-s) (2r-2q-2),

(ii) \E(Gi)\^2r-2q-2 (1 < / < s),

(hi) \F*\^r-q-l,

(iv) r - ^GuPj -U 2(q-s).

Proof. If G can be recoverably embedded in K*{Q+1) and if F represents the missing 1-factor,then for some F* cz F, with V(F*) a V(G), G U Z7* is recoverably embedded in a T-factorization of K*{q+1), where T = (1,2,...,2) is a composition of 2q+ 1. Then conditions(i)-(iv) follow from Theorem 12.

Conversely if conditions (i)-(iv) are satisfied, then G U F* can be recoverably embeddedin a T-factorization of K2iq+1), and so G can be recoverably embedded in a 2-factorization

If ts+1,..., tq in Theorem 12 are not all even, then the analogous result may not be true;it depends on the structure of G. We can replace (i) and (v) by requirements about thegraph G.

Theorem 15. Let T = (tv..., tq) be a composition of n— 1. Let G be a simple graph with rvertices and let G be its complement. Let G be edge coloured with s colours cx,..., cs and, for1 ^ / ^ s, let Gi be the spanning subgraph of G whose edges are the edges of G coloured c{.Then the edge coloured graph G can be recoverably embedded in a T-factorized Kn, with Gi

forming part of the corresponding tt-factor (1 ^ / ^ s), if and only if

(i) \E(Gt)\>tt(r-±n) (1 < i O ) ,(ii) ttn is even (1 ^ / ^ q),(hi) AiG^t, (l^i^s),(iv) G can be edge coloured with colours cs+1,...,cq so that, with Gf denoting the

spanning subgraph of G whose edges are the edges of G coloured ct,(a) ^ ( G J I ^(b) A(Gt)^tt

Proof. This is easy to see from the second proof of Theorem 12. •

Whether or not ts+1,..., tu are all even, conditions (i)—(iv) of Theorem 12, together withstrict inequality in condition (v), are sufficient for the edge coloured graph G to berecoverably embeddable in a T-factorized Kn.

Theorem 16. Let T= (t1,...,tQ) be a composition ofn — l. Let G be a simple graph with rvertices and let G be edge coloured with s colours cx,..., cs. Let Gt be the spanning subgraphof G whose edges are the edges of G coloured cf (\ ^ i ^ s). Then:

I. Ifr-S(G) ^ ts+l + ... + tQ, the edge coloured graph G can be recoverably embedded in aT-factorized Kn, with Gi forming part of the corresponding tffactor, if and only if

(i) (L)-\E(G)\> t a^-X),

(ii)

(iii) ttn is even (1 ^ / ^ q), and

(iv) A(G ? )^ / ? . ( 1 ^ / ^ j ) .

II. If r — S(G) — 2 ^ ts+1 + ... + tq, there is no such recoverable embedding.

Proof. First suppose that r-S(G) ^ ts+1 + ... + tq and that conditions (i)—(iv) all hold.If n is odd then condition (iii) implies that ts+1,..., tq are all even, and then Theorem 16

is implied by Theorem 12. So we may suppose that n is even.By Vizing's theorem [23] and the fact that A(G) + 1 = r-S(G) ^ ts+1 + ... + tq, we can

properly edge colour G with p = ts+1 -f ... + tq colours y1 ; . . . , yp. If there is a colour, say yx,which appears on at least two more edges than some other colour, say y2, then there is an


odd length path coloured alternately yx and y2 with one more y1 edge than y2 edges. Now

interchange colours on such a path. Repeating this as necessary with all the various colours,

we eventually obtain a proper equalized edge colouring with p colours. In view of

condition (i) and the fact that n is even, each colour class has at least r — \n edges. Now, for

y = 5H-l,...,w, form disjoint unions of t} of these colour classes. Recolouring the edges of

the y-th union with colour c} and letting G; denote the edges of G coloured c} (5 + 1 j ^ w),

we have that

and

\E{G^\^t}{r-\n)

It now follows from Theorem 10 that G can be recoverably embedded in the required way.

If r — 8(G) ^ ts+1 + ... + tq and G can be recoverably embedded in the way described, thenthe argument given in the necessity part of the second proof of Theorem 12 shows that(i)-(iv) all hold.

If r - S(G) - 2 ^ t8+1 + ... + tq then A(G) > tj+1 + ... + tq, so G cannot be edge coloured in

the required way with A(G,) ^ tf (5+ 1 / q), so there is no recoverable embedding.

We remark that the equalizing argument used in the proof of Theorem 16 goes back to

McDiarmid [17] and de Werra [24, 25].

References

[I] Andersen, L. D. and Hilton, A. J. W. (1980) Generalized latin rectangles I: construction anddecomposition. Discrete Math. 31 125-152.

[2] Andersen, L. D. and Hilton, A. J. W. (1980) Generalized latin rectangles II: embedding.Discrete Math. 31 235-260.

[3] Andersen, L. D. and Hilton, A. J. W. (1979) Generalized latin rectangles. In: Graph Theory andCombinatorics, Research Notes in Mathematics. Pitman, London, 1-17.

[4] Chetwynd, A. G. and Hilton, A. J. W. (1991) Outline symmetric latin squares. Discrete Math.97 101-117.

[5] Cruse, A. B. (1974) On embedding incomplete symmetric latin squares. / . Combinatorial Theory,Ser. A 16 18-22.

[6] Cruse, A. B. (1974) On extending incomplete latin rectangles. Proc. 5th Southeastern Conf. onCombinatorics, Graph Theory and Computing. Florida Atlantic University, Boca Raton, Florida,333-348.

[7] Evans, T. (1960) Embedding incomplete latin squares. Amer. Math. Monthly 67 958-961.[8] Haggkvist, R. and Johanson, A. (to appear 1994) (1,2)-factorizations of general Eulerian nearly

regular graphs. Combinatorics, Probability and Computing.[9] Hilton, A. J. W. (1980) The reconstruction of latin squares, with applications to school

timetabling and to experimental design. Math. Programming Study 13 68-77.[10] Hilton, A. J. W. (1981) School timetables. In: Hansen, P. (ed.) Studies on graphs and Discrete

Programming. North-Holland, Amsterdam, 177-188.II1] Hilton, A. J. W. (1982) Embedding incomplete latin rectangles. Ann. Discrete Math. 13 121-138.[12] Hilton, A. J. W. (1987) Outlines of Latin squares. Ann. Discrete Math. 34 225-242.[13] Hilton, A. J. W. (1984) Hamiltonian decompositions of complete graphs. / . Combinatorial

Theory, Ser. B 36 125-134.


[14] Hilton, A. J. W. and Rodger, C. A. (1986) Hamiltonian decompositions of complete regular5-partite graphs. Discrete Math. 58 63-78.

[15] Hilton, A. J. W. and Wojciechowski, J. (1993) Weighted quasigroups. Surveys in Combinatorics.London Mathematical Society Lecture Note Series 187 137-171.

[16] Hilton, A. J. W. and Wojciechowski, J. (submitted) Simplex Algebras.[17] McDiarmid, C. J. H. (1972) The solution of a time-tabling problem. / . Inst. Math. Applies. 9

23-34.[18] Nash-Williams, C. St J. A. (1986) Detachments of graphs and generalized Euler trials. In: Proc.

10th British Combinatorics Conf., Surveys in Combinatorics 137-151.[19] Nash-Williams, C. St J. A. (1987) Amalgamations of almost regular edge-colourings of simple

graphs. / . Combinatorial Theory, Ser. B 43 322-342.[20] Petersen, J. (1891) Die Theorie der regularen Graphen. Acta Math. 15 193-220.[21] Rodger, C. A. and Wantland, E. (to appear) Embedding edge-colourings into m-edge-connected

^-factorizations. Discrete Math.[22] Ryser, H. J. (1951) A combinatorial theorem with an application to latin squares. Proc. Amer.

Math. Soc. 2 550-552.[23] Vizing, V. G. (1960) On an estimate of the chromatic class of a /?-graph (in Russian). Diskret.

Analiz. 3 25-30.[24] de Werra, D. (1971) Balanced schedules. INFOR 9 230-237.[25] de Werra, D. (1975) A few remarks on chromatic scheduling. In: Roy, B. (ed.) Combinatorial

Programming: Methods and Applications. Reidel, Dordrecht. 337-342.

Ramsey Size Linear Graphs

PAUL ERDOS1, R. J. FAUDREE^, C C. ROUSSEAU*and R. H. SCHELP*

+ Mathematical Institute, Hungarian Academy of Sciences, Budapest V, Hungary

^Department of Mathematical Science, Memphis State University, Tenn. 38152 USA

A graph G is Ramsey size linear if there is a constant C such that for any graph H with nedges and no isolated vertices, the Ramsey number r(G,H) < Cn. It will be shown that anygraph G with p vertices and q > 2p — 2 edges is not Ramsey size linear, and this bound issharp. Also, if G is connected and q < p + 1, then G is Ramsey size linear, and this boundis sharp also. Special classes of graphs will be shown to be Ramsey size linear, and boundson the Ramsey numbers will be determined.

1. Introduction

Only finite graphs without loops or multiple edges will be considered. The general notationwill be standard, with specialized notation introduced as needed. For a graph G, the vertexset and edge set will be denoted by V(G) and E(G) respectively, and the order of G (thenumber of vertices in V(G)) and the size of G (the number of edges in E(G)) will bedenoted by p(G) and q(G) respectively. For graphs G and H, the Ramsey number r{G,H)is the smallest positive integer n such that if the edges of a Kn are colored either red orblue, there will always be a red copy of G or a blue copy of H.

The following Ramsey bound theorem was conjectured by Harary, and proved bySidorenko in [12].

Theorem 1. For any graph Hn of size n and without isolated vertices,

8 Research partially supported by O.N.R. Grant No. N00014-91-J-1085 and N.S.A. Grant No. MDA 904-90-H-4034

242 P. Erdos, R. J. Faudree, C. C. Rousseau and R. H. Schelp

Since r(K3, Tn+{) = In + 1 (see [2]), and r{K3,nK2) = 2n + 1 (see [11]), the bound inTheorem 1 cannot be lowered. Thus, for G = KT, the Ramsey number r(G,H) has anupper bound that is linear in the number of edges in H. It is natural to ask for whichgraphs G is this true. This motivated the following definition.

Definition 1. A graph G is Ramsey size linear if there is a constant C such that for anygraph Hn of size n without isolated vertices,

r(G,Hn) <C-n.

In Section 2, the maximum number of edges in a graph G that is Ramsey size linearwill be determined, as well as the minimum number of edges in a connected graph G thatis not Ramsey size linear. In particular, it will be shown that if q(G) > 2p(G) — 2, then Gis not Ramsey size linear. We will also prove that if G is connected and q(G) < p(G) + 1,then G is Ramsey size linear. Both Ramsey size linear graphs and graphs that are notRamsey size linear exist in the interval p(G) + 1 < q(G) < 2p(G) — 2. In fact, exampleswill be described to show that for each p + 2<q<2p — 3, there are connected graphs G,with p(Gi) = p and q(G{) = q for / — 1,2 such that G\ is Ramsey size linear and G: is notRamsey size linear.

In Section 3, special classes of graphs will be shown to be Ramsey size linear; morespecifically graphs G with extremal number ext(G,ri) = O(n3/2) will be shown to beRamsey size linear. In Section 4, some upper bounds for the Ramsey numbers r{G, Hn)will be verified for some special graphs G, such as even cycles, where Hn denotes a graphof size n without isolated vertices. Some open questions related to the results of the paperwill be discussed in Section 5.

2. Extremal problems

We start this section with a proof of Theorem 2, which will be the basis for showing thata graph of order p and size at least 2p — 2 is not Ramsey size linear.

Theorem 2. Let G be a fixed graph with p(G) = p > 3 and q(G) = q. There exists apositive constant C such that for n sufficiently large,

(q-\)/(p-2)

r(G, Kn) > C '

An immediate consequence of Theorem 1 is the following.

Corollary 1. If p(G) > 3 and q(G) > 2 • p(G) — 2, then G is not Ramsey size linear.

Theorem 2 can be found in [6], but we include it here since it is central to the results ofthis paper. In this proof, [N]k will denote the set of all /c-element subsets of (1,2, • • •, N}.Any 2-coloring of the edges [N]2 of the complete graph with vertices [N] will be denotedby (R,B), with R as the red graph and B as the blue graph. If S c [N], then the red

Ramsey Size Linear Graphs 243

subgraph (blue subgraph) induced by R{B) will be denoted by (S)R ((S)B). Central to thisproof is the following result of Erdos and Lovasz (see [8]). The form used in this papercan be found in [14].

L e m m a 1. (Erdos-Lovasz) Let C\, Ci, •--, Cn be events with probabilities P ( C ) , / = 1,2, . . . , n— 1, n. Suppose there exist corresponding positive numbers X\, xi, . . . , x n such thatXj P(Ci) < 1 and

\ogxl>YJXjP(Cj)> i= 12,...,n,

where the sum is taken over all j ^ i such that Cx and Cj are dependent. Then

P{f]Ci)>0.

With Lemma 1, we can give the proof of Theorem 2.

Proof of Theorem 2. The proof uses the Lovasz-Spencer method (see [14]). For anappropriately large N, we will verify the existence of a two-coloring (R,B) of [N]2 suchthat R -ft G and B > Kn. Randomly two-color [N]2, each edge being red with independentprobability r. For each S cz [N]p let As denote the event (S)R => G. Similarly, for eachT c [N]n, let BT denote the event (T)B => ^7.

The fundamental result to be used here is the Erdos-Lovasz local lemma (Lemma1). To implement Lemma 1 in the setting previously described, we make the followingsimplification.

For each Cx = As, let x,- = a, and for each C,- = Bj, let x,- = b. For a fixed As, let NAA

denote the number of Sf ^ S such that As and A$> are dependent. Similarly, define NABto be the number of T such that As and Bj are dependent. In exactly the same way,define NBA and NBB- Letting A and B denote typical As and BT respectively, note that thedesired conclusion follows if there exist positive numbers a and b such that a - P(A) < 1,b-P(B) < 1,

log a >NAA-a- P(A) + NAB - b • P(B), (1)

and

l o g f r > N B A - a - P(A) + N B B - b - P{B). (2)

Note that As and BT are dependent only if \S n T\ > 2. A similar observation holds forthe pairs (AS,AS>) and (BT,BT>).

For the purpose of this calculation, it suffices to use the following bounds:

NAB,NBB<

P(A) >p\rq, and

Let s = (p- 2)/(q - 1) and set

p = Ci • N~\ n = C2'Ns - log N,

fl = C 3 > l and ft = ^4^-0ogN)^

where C\ through C4 are positive constants. Then log a > 0,

A ^ • a • P(A) = 0(Np-2N~sq) = o(l)

and

NAB'b- P(B) < ei^+Ct-dcywiiogN?) = o ( 1 )

if C\C\I2 > C2 + C4. Similarly, both sides of equation (2) are of order

c • Ns • (log N)2

for an appropriate constant c. The constants C\ through C4 may be chosen so thatequation (2) holds. Thus, there is a two-coloring of [N]2 with no red G and no blue Km,where n = Ci ' Ns • log Af. Solving for N in terms of n, we get the stated result. Thiscompletes the proof of Theorem 2. •

There are graphs of order p and size q = 2p — 3 that are Ramsey size linear, as thefollowing result confirms.

Theorem 3. Let Tp_\ be any tree on p — 1 vertices (p > 2), Gp = Kx + Tp-\, and Hn beany graph of size n. Then,

r(Gp,Hn)<2n(p-2)+p{Hn).

If Hn has no isolated vertices, then p(Hn) < 2n. Thus, one immediate consequence ofTheorem 3 is the following corollary.

Corollary 2. If Hn is a graph of size n without isolated vertices, then

r(Gp,Hn)<2(p-l)n.

Proof of Theorem 3. The proof will be by induction on n. The result is trivial for n = 1,since r(Gp,Hn) = max{p,p(H\)}. Proceed by induction on n.

Let i? be a vertex of Hn of smallest degree, and let H'n = Hn — v. Two color the edges of aK2n{p-2)+P{Hn), and assume that there is no red Gp or blue Hn. By the induction assumption,there is a blue copy of H'n.

Let N be the neighborhood of v in the graph H'n.By assumption, each vertex of K2n{P-2)+P{Gn) — H'n is adjacent in red to at least one

vertex of N. On the other hand, no vertex of N can have red degree r(Tp_{,Kp[Hn)) =(p — 2)(p(Hn) — 1) + 1 in K2n(P-2)+p(Hn), since this would ensure either a red Gp or a blueHn. Therefore, using these counts on the number of red edges emanating from N gives

the following inequalities.

2n(p - 2) + 1 < \N$p - 2)(p(Hn) - 1) < - ^ - ( p - 2)(p(Hn) - 1).

This gives a contradiction that completes the proof of Theorem 3. •

The next result gives the minimum number of edges in a connected graph that is notRamsey size linear.

Theorem 4. If G is a connected graph with q(G) < p(G) + 1, then G is Ramsey size linear.In addition, there is a graph G with q(G) = p(G) + 2 that is not Ramsey size linear.

Some preliminary results and examples will be needed in the proof of Theorem 4,which we now give.

Lemma 2. If G\ and G^ are Ramsey size linear graphs, then the graph G\ • Gi obtainedby identifying precisely one vertex from each graph is also Ramsey size linear.

Proof. Let Hn be a graph of size n. We can assume for i = 1, 2 that r(Gj,Hn) < c-xn forpositive integers with c\ < c2. We will show that

r(Gx 'G2,Hn)<(c2p(Gi)+ci)n,

and so G\ • Gi is Ramsey linear. Let m = (c2p(G$ + c\)n, and 2-color the edges of a Km

with red and blue. Assume there is no blue copy of Hn. Therefore, using r{G\,Hn) < c\n,there must be c^n vertex disjoint red copies of G\. If v\ is the vertex of G\ that is to beidentified with the vertex v2 of G2i let S be the set of cin vertices that represent v\ ineach of the cjn copies of G\. Since r(G2,Hn) < c2n, there is a red copy of Gi using onlyvertices in S. In this red copy of G2, the vertex identified with v2 is the same as the vertexidentified with v\ in some copy of G\, and this gives a red copy of G\ • G2. This completesthe proof of Lemma 2. •

An immediate consequence of Lemma 2 is the following.

Corollary 3. / / G is a graph such that each of its blocks is Ramsey size linear, then G isRamsey size linear.

Next we describe a family of examples that will be needed to verify the sharpness ofthe result in Theorem 4.

Example 1. Let G be any graph that contains K4 as a subgraph. Then, since r(K^,Hn) >C ( j ^ ) 5 / 2 , any graph G that contains a K4 is not Ramsey size linear. Thus for any treeTp-3 on p — 3 vertices, the graph K4 • Tp-i is a connected graph of order p and size p + 2that is not Ramsey size linear. Clearly, any graph G that is a supergraph of X4 • Tp-i is notRamsey size linear.

Proof of Theorem 4. If there is a vertex v in G such that G — v has no cycles, thenG is Ramsey size linear by Corollary 2. This will always be true if q(G) < p(G). Ifq(G) = p(G) + 1, and no such vertex v exists, then G will contain two vertex disjoint cycles.Thus, each block of G will be either an edge or a cycle, both of which are Ramsey sizelinear. Hence, by Corollary 3, G is Ramsey size linear. Example 1 gives the sharpness ofthe result. This completes the proof of Theorem 4. •

Let G be a connected graph of order p and size q. To summarize, we know that: ifq 2p — 2, then it is not Ramsey size linear.

3. Special classes of graphs

In this section special classes of graphs will be shown to be Ramsey size linear. We startwith a class of graphs defined by their Turan extremal numbers. Recall that ext(G,n), theTuran extremal number, is the maximum number of edges in a graph of order n that doesnot contain a copy of G. An excellent survey of results in Turan extremal theory can befound in [13], and a more general survey of extremal theory in [1].

Theorem 5. If ext(G,n) < cn3^2, then for every graph Hn of size n without isolates,

r(G,H) < (32c2 + 8)n.

Proof. Let (R,B) be a two-coloring of the edges of K\, where N > (32c2 + 8)/7. Suppose(R) ^ G. Then (R) has at most cN3^2 edges. Sequentially delete vertices of degree at least2c\/~N in the current red graph until none remain. After M vertices have been deleted, atleast 2cy/NM red edges have been removed from the original two-colord KA, so at mostN/2 vertices are deleted before the process terminates.

Now we have a two-colored complete graph with at least N/2 vertices in which thered graph has no vertices of degree 2c\f~N or more. We wish to show that there is anembedding of Hn into the blue graph. Embed Hn into the blue graph one vertex ata time, starting with the largest degree vertex of Hn and continuing so the sequenceis non-increasing by degree. Suppose that this process terminates. Then some inducedsubgraph of Hn has been embedded and the process cannot be continued because thereis no external vertex that can play the role of the next vertex of Hn in the sequence. Wemay suppose that Hn has p vertices altogether; since Hn has no isolates, p < 2/?. Supposethat the vertex needed to continue the embedding has degree k in Hn. Thus Hn has k + 1vertices of degree k or more, so k(k + 1) < 2n and k < \/2n. In the two-colored completegraph, there are more than N/2 — 2n vertices external to the subgraph of Hn that isembedded. By assumption, there are k vertices in the embedded subgraph of Hn that inthe blue graph have no common neighbor among these external vertices. Thus, at leastone of the k vertices has degree \(N/2 — 2n)/k] or more in the red graph, and we have

In

soN I2N

4c\ — < 4.n V n

But the left-hand side is an increasing function of N/n for N/n > 8c2, so, since N/n >32c2 + 8,

> 32c2 + 8 - 32c2 ( l + ^ ) = 4,

and a contradiction has been obtained. This completes the proof of Theorem 5. •

N • ' " " > 3 2 r + 8 - .n

The Turan extremal numbers for bipartite graphs have been studied extensively, andthere are several graphs of interest that have extremal numbers O(rc3/2), and thus areRamsey size linear. In [9] and [13] many families of such examples can be found, andsome particular families can be found in [3] and [4]. For example it is known that K33 — c,and Q3 — e (where Q3 is the 3-dimensional cube) have Turan extremal numbers equal to0{n3'2).

Using Corollary 1, Theorem 4, Corollary 2, and Theorem 5, it can be determined withjust one exception if a graph of order at most 5 is Ramsey size linear. All graphs of orderat most 4 are Ramsey size linear with the exception of K4, which is not Ramsey sizelinear. All graphs of order 5 that do not contain a K4 or have at least 8 edges can beshown to be Ramsey size linear with the exception of K5 — (K2 U £1,2). It is not known ifthis graph is Ramsey size linear. Also, it is not known if K33, a graph with 6 vertices and9 edges, is Ramsey size linear.

More generally, it would be of interest to know if a graph G is Ramsey size linear if itsatisfies the density condition that each subgraph H of order m has size at most 2m — 3.

The graph K4 is not Ramsey size linear, but the deletion of any edge leaves the graphB2, which is Ramsey size linear. Graphs with this property are of interest, and thus wegive the following definition.

Definition 2. A graph G is minimal Ramsey size linear if G is not Ramsey size linear, butif any edge is deleted, then the resulting graph is Ramsey size linear.

If any of the graphs X5 — (Ki U ^1,2), K3.3, and the 3-dimensional cube Qi are notRamsey size linear, then they would be minimal, since all of their proper subgraphs areRamsey size linear.

4. Upper bounds for special graphs

In this section we will consider some special graphs G that we know are Ramsey sizelinear, and determine an upper bound on the Ramsey number r(G,//„), where Hn is agraph of size n with no isolated vertices. Of course, Corollary 2 gives upper bounds forthe books and fans (where the book Bk = K\ + K\± and the fan Fk = K\ + i \ ) . We have

r(Bk,Hn) < 2{k + \)n and r(Fk,Hn) < 2kn.

We next look at even cycles C2k, and in particular, C4.

Theorem 6. If k > 2 and Hn is a connected graph of size n, then for n sufficiently large

r(C2k,Hn) <H

An immediate consequence of Theorem 6 is the following corollary.

Corollary 4. If k > 2 and Hn is a graph of size n without isolated vertices, then for nsufficiently large

r(C2k,Hn) <2n + k- 1.

Note that if Hn = nK2, then r(C2k,Hn) > In + k — 1. If a K2n+k-2 is colored such thatthere is a blue K2n-\ and the remaining edges are red, then there is no red C2k and noblue nK2. Thus the bound in Corollary 4 is sharp.

Before we give the proof of Theorem 6, we will prove a technical lemma needed in theproof.

Lemma 3. If a subgraph F of K2n^ has 6kn edges, then F contains a C2k-

Proof. Let A and B be the parts of the bipartite graph F, with \A\ = 2n and \B\ = >Jn.Delete any vertices of A of degree less than 2/c and then delete any vertices of B of degreeless than 2ky/n. Continue to do this until no more vertices can be deleted.

This results in a graph F'. Note that F' is non-empty, since fewer than (2ky/n)y/n+4kn =6kn edges have been deleted, and this is less than the number of edges in F. Let A' and B'be the corresponding parts of F'. Each vertex in A' has degree at least 2/c, and each vertexin B' has degree at least 2ky/n. Select a vertex ft in B\ and let N be the neighborhoodof b in A'. Let N' be a subset of N with 2k^/n vertices, and let G be the subgraph of F'induced by N' U (Br - b).

Thus G is a graph with at most Aky/n vertices and at least Ak2y/n edges. Therefore, bya result of Erdos and Gallai [7], G has a path with at least 2/c vertices. This path (actually2/c — 1 vertices of this path), along with the vertex ft, will give a C2k. This completes theproof of Lemma 3. •

Proof of Theorem 6. Let F be a complete graph on n + 22kyfn vertices whose edges arecolored either red or blue. We will assume that there is no red C2k in i7, and we will showthat there is a blue Hn.

Let L be the vertices of F of red degree at least Iky/n. If the number of vertices inL is as large as y/n, then there is a red bipartite graph with y/n vertices in one part,n + 22ky/n — y/n vertices in the other part, and at least yfn6ky/n = 6kn edges. Then byLemma 3, there must be a red C2k in this bipartite graph, and thus in F. Note that forn sufficiently large, 22ky/n < n, and additional vertices can be added to the large part ofthe bipartite graph to get 2n vertices, and so Lemma 3 applies. Let F' = F — L. Thus, wecan assume that each vertex of F' has red degree less than Iky/n.

Order the vertices of Hn in non-increasing degree order, say hu h2, . . . , hp. For each j ,(1 < j < p), let Sj be the subgraph of Hn induced by the vertices {/zi,/i2,...,/i7}. Assumethat there is an embedding a of Sr into the blue graph of F', but there is no embeddingof 5r+i. Let N be the neighborhood of ht+\ in Sr, so N' = a(N) is the corresponding setof vertices in F'.

From [5] and an unpublished result of Szemeredi we know that there are constants cand d such that

r(C4,Km) < Cm

(logm)2

and for k > 2,

^ ) f f l M ^ (logm)2'

Therefore, since there is no red C2k in F, there is a constant c" such that there is a blue^'V"iog/r This implies that Scn^Xogn can be embedded in the blue subgraph of F , and sor > c"yjnlogn.

We will first consider the case when r < n/2. Each vertex of F' — St is adjacentin red to a vertex of A/7. Therefore, there will be at least (n/2) — yfn red adjacenciesemanating from AT. On the other hand, because of the ordering of the vertices in //„,each vertex of St has degree at least |AT|, so \N'\d'y/nlogn < 2n. This implies that|TV'| < 2yfn/c"\ogn. Since each vertex of F' has red degree at most lky/n, there will beat most (lky/n)(y/n/d'logn) = Ikn/c"\ogn red edges emanating from N'. This implies(n/2) — y/n < Ikn/c" log n, a contradiction for n sufficiently large. This completes the proofof the case when r < n/2.

Next, we consider the case when r > n/2. First observe that, \N'\r < 2n means\N'\ < 4. Since there are at least 2\ky/n vertices in F' — Sr, there are at least 2\ky/nred edges emanating from A/7. This implies there is a vertex of Nf of red degree at least(2\ky/n)/3 = lky/n, which gives a contradiction and completes the proof of Theorem 6.

•

It should be noted that the bound n -f 22k^fn could be improved by more carefulcounting, but this would not give any improvement in Corollary 4, so the additional spaceand effort is not warranted.

Sharper bounds can be obtained for the case k = 2. It is known (see [10]) thatr(C4,K\J}) < n+ 1 + \y/n ], with equality for an infinite number of values of n. However, ifHn is connected and not very 'star like', a sharper bound on r(C4,Hn) can be determined.Using exactly the same techniques and proof structure as in Theorems 5 and 6, thefollowing two theorems can be proved. Due to the similarity to the previous proofs , thedetails will not be given.

Theorem 7. Let Hn be a connected graph of size n and order at most n — \2y/n. If n issufficiently large,

r(C4,Hn) < n + 2.

Theorem 8. Let Hn be a connected graph of size n and A(Hn) < 6^/n. If n is sufficientlylarge, and 6 < 1/8,

r(C4,//n) <rc + 2.

Using Theorems 7 and 8, the proof techniques of these results, and a much moredetailed analysis, the following result can be proved.

Theorem 9. Let Hn be a connected graph of size n and A(Hn) < en. If e is sufficientlysmall and n is sufficiently large,

r(C4,Hn) < n + 2.

If the edges of a Kn+\ are colored such that the red subgraph is a star K\M and theblue subgraph is a complete graph Kn, there is no red C4 and no blue connected graph oforder n + 1. Thus, r(C^Hn) > n + 1 for any tree Hn of size n. Thus, each of the Theorems7, 8, and 9 is sharp.

5. Questions

There are many questions left unanswered. The following density question may be verydifficult, but it is certainly of interest.

Question 1. If every subgraph S of G satisfies q(S) < 2p(S) — 3, is G necessarily Ramseysize linear?

If the answer to the previous question is yes, the minimal graphs £3,3, G5 = K5 — (K\^ UX2), and QT, are Ramsey size linear. Thus, a subquestion of the previous question is thefollowing.

Question 2. Are the graphs K33, G5 = K5 — (K\2 U K2), and Q3 Ramsey size linear?

Trees and complete graphs could play central roles in determining if a graph G isRamsey size linear. In particular, consider the following question.

Question 3. If there is a constant c such that for each integer n, r(G, Tn) < en and r(G, Kn) <en2, is G Ramsey size linear?

In the upper bound on the Ramsey numbers for cycles, only even cycles were considered.Thus, the following questions is of interest.

Question 4. For which constants c is r(C2k+\,Hn) < c(2k+ \)n, where Hn is a graph of size

n without isolated vertices?

A much more difficult problem is to extend the result of Sidorenko and of Corollary 4on triangles to cycles of arbitrary length.

Question 5. Is r(Cm,Hn) < In + [(m — 1)/2J, where m > 3, and Hn is a graph of size n

without isolated vertices?

Question 6. Is there an infinite family of minimal Ramsey size linear graphs, or more specif-

ically, is there a minimal Ramsey size linear graph other than K4?

References

[I] Bollobas, B. (1978) Extremal Graph Theory, Academic Press, London.[2] Chvatal, V. (1977) Tree-Complete Graph Ramsey Numbers. J. Graph Theory 1 93.[3] Erdos, P. (1965) On some Extremal Problems in Graph Theory. Israel J. Math. 3 113-116.[4] Erdos, P. (1965) On an Extremal Problem in Graph Theory. Colloquium Math. 13 251-254.[5] Erdos, P., Faudree, R. J., Rousseau, C. C. and Schelp, R. H. (1978) On Cycle-Complete Graph

Ramsey Numbers. J. Graph Theory 2 53-64.[6] Erdos, P., Faudree, R. J., Rousseau, C. C. and Schelp, R. H. (1987) A Ramsey Problem of

Harary on Graphs with Prescribed Size. Discrete Math 67 227-233.[7] Erdos, P. and Gallai, T. (1959) On Maximal Paths and Circuits of Graphs. Acta Math. Acad.

Sci. Hungar. 10 337-356.[8] Erdos, P. and Lovasz, L. (1973) Problems and Results on 3-Chromatic Hypergraphs and Some

Related Questions. Infinite and Finite Sets 10, Colloquia Mathematica Societatis Janos Bolyai,Keszthely, Hungary 609-628.

[9] Faudree, R. J. (1983) On a Class of Degenerate Extremal Graph Problems. Combinatorica 383-93.

[10] Parsons, T. D. (1975) Ramsey Graphs and Block Designs. Trans. Amer. Math. Soc. 209 33-44.[II] Lorimer, P. (1984) The Ramsey Numbers for Stripes and One Complete Graph. J. Graph

Theory 8 177-184.[12] Sidorenko, A. F. (manuscript) The Ramsey Number of an N-Edge Graph Versus Triangle is at

Most 2N + 1.[13] Simonovits, M. (1983) Extremal Graph Theory. In: Beineke, L. W. and Wilson, R. J. (eds.)

Selected Topics in Graph Theory II, Academic Press, New York 161-200.[14] Spencer, J. (1952) Asymptotic Lower Bounds for Ramsey Functions. Discrete Math. 20 69-76.

Turan-Ramsey Theorems and Kp-IndependenceNumbers

P. ERDOS, A. HAJNALt, M. SIMONOVITS+, V. T. SOS+

and E. SZEMEREDI*

Mathematical Institute of the Hungarian Academy of Sciences, Budapest.

Let the /^-independence number ap{G) of a graph G be the maximum order of an inducedsubgraph in G that contains no Kp. (So /^-independence number is just the maximum sizeof an independent set.) For given integers r,p,m > 0 and graphs Li , . . . ,L r , we define thecorresponding Turan-Ramsey function RTp(n,L\,...,Lr,m) to be the maximum numberof edges in a graph Gn of order n such that onp{Gn) < m and there is an edge-colouring ofG with r colours such that the / h colour class contains no copy of L ;, for j = 1,..., r.In this continuation of [11] and [12], we will investigate the problem where, instead ofoc(Gn) = o(n), we assume (for some fixed p > 2) the stronger condition that ccp(Gn) = o{n).The first part of the paper contains multicoloured Turan-Ramsey theorems for graphs Gn

of order n with small Xp-independence number ccp(Gn). Some structure theorems are givenfor the case ap{Gn) = o{n), showing that there are graphs with fairly simple structure thatare within o(n2) of the extremal size; the structure is described in terms of the edge densitiesbetween certain sets of vertices.

The second part of the paper is devoted to the case r = 1, i.e., to the problem of determiningthe asymptotic value of

RTp(n,Kq,o(n))6p(Kq) = lim —p——f ,

(2)

tfor p < q. Several results are proved, and some other problems and conjectures are stated.

0. Notation

In this paper we will consider graphs without loops and multiple edges. Given a graphG, e(G) will denote the number of edges, v(G) the number of vertices, x(G) the chromaticnumber, and a(G) the maximum cardinality of an independent set in G. More generally,given an integer p > 1, ap(G) denotes the p-independence number of G: the maximumcardinality of a set S such that the subgraph of G spanned by S contains no Kp. Given a

t Supported by GRANT 'OTKA 1909'.

+ This notation, where we put o(n) in place of f(n) is slightly imprecise. It means that any function f(n) = o{n)and will be clarified in Section 2.

254 P. Erdos, A. Hajnal, M. Simonovits, V. T. Sos, and E. Szemeredi

graph, the (first) subscript will mostly denote the number of vertices: Gn, Sn, will alwaysdenote graphs on n vertices. For given graphs Li , . . . ,Lr , K(Li,...,Lr) will denote theusual Ramsey number, that is, the minimum t such that for every edge-colouring of Kt

in r colours, for some v the vth colour contains an L v ' . If we partition n vertices into qclasses as equally as possible and join two vertices iff they belong to different classes, weobtain the so-called Turan graph on n vertices and k classes, denoted by Tn^. This graphis the (unique) /c-chromatic graph on n vertices with the maximum number of edges.

For a set Q, we will use \Q\ to denote its cardinality. Given two disjoint vertex sets, Xand 7 , in a graph Gn, we use e(X, Y) to denote the number of edges in Gn joining X andy, and d(X, Y) to denote the edge-density between them:

e(X,Y)d(X,Y) =

\X\

Given a graph G and a set U of vertices of G, we use G[U] to denote the subgraph ofG induced (spanned) by U. The number of edges in a subgraph spanned by a set U ofvertices of G will be denoted by e(U). We will say that X is completely joined to Y if everyvertex of X is joined to every vertex of Y.

Given two points x, y in the Euclidean space Eh, we use p(x,y) to denote their ordinarydistance.

1. Introduction

Ramsey's Theorem [23] and Turan's Extremal Theorem [33, 34] are both among the mostwell-known theorems of graph theory. Both served as starting points for whole branchesof graph theory. (For Ramsey Theory, see the book by R. L. Graham, B. L. Rothschildand J. Spencer [21], and for Extremal Graph Theory, see the book by Bollobas [2], orthe survey by Simonovits [29].) In the late 1960's a new theory emerged connecting thesefields. Perhaps the first paper in this field is due to V. T. Sos [30], and quite a few resultshave been found since then.

The 'historical' part of the introduction of this paper is slightly condensed, to avoid toomuch repetition. For some further information see [12]. Some important references canbe found at the end of the paper, see [3, 11, 12, 18, 20, 31].

In [11] P. Erdos, A. Hajnal, V. T Sos, and E. Szemeredi investigated the followingproblem:

Suppose that a so-called forbidden graph L and a function f(n) = o(n) are given.Determine

KT(n,L,/(w)) = max{e(Gn) : L £ Gn and a(Gn) < f(n)}.

They showed that this number depends (in some sense) primarily on the so-calledArboricity of L (which is a slight modification of the usual arboricity of L). In acontinuation [12] of that paper, we started investigating the following problem:

' This is the only case when the (first) subscript is not the number of vertices: i.e. when we speak of theexcluded graphs Lx.

Turdn-Ramsey Theorems and Kp-Independence Numbers 255

Let Gn be a graph on n vertices the edges of which are coloured by r coloursX\,...,Xr, so that the subgraph of colour %v contains no complete subgraph KPx,(v = l,...,r). Let a function f(n) be given, (mostly f(n) = o(n)) and suppose that%(Gn) < f(n). What is the maximum number of edges in Gn under these conditions ?

In this continuation of [11] and [12] we will investigate the problem where, instead ofoc(Gn) = o(n), we assume a stronger independence condition: that the maximum cardinalityof a Kp-fvQQ induced subgraph of Gn is o(n):

<xp(Gn) = o(n).

The concept of OLP(G) was introduced long ago by A. Hajnal, and also investigated byErdos and Rogers, see [16]. (A similar 'independence notion' is investigated for randomgraphs in a paper of Eli Shamir [24], where he generalizes some results on the chromaticnumber of random graphs.)

The general problem

Assume that L\,..., Lr are given graphs, and Gn is a graph on n vertices, the edgesof which are coloured by r colours / i , . . . , Xr, and

for v = 1,..., r the subgraph of colour %v contains no Lv

and (xp(Gn) < m.

What is the maximum ofe(Gn) under these conditions?

The maximum will be denoted by RTp(n,L\,...,Lr,m). The graphs attaining the max-imum in this problem will be called extremal graphs for RTp(n,L\,...,Lr,m). It mayhappen that there exist no graphs satisfying our conditions. Then we will say that themaximum is 0.

Of course, for fixed m and large n - by Ramsey's theorem - there are no graphs with theabove properties: the maximum is taken over the empty set. However, we are interestedmainly in the case m —• oo, m — f(n) = o(n), but m/n —• 0 very slowly.

The existence of graphs satisfying (*) is far from being trivial. We will use a theoremof Erdos and Rogers to prove the existence of such graphs for the case of one colourand when the forbidden graph is a complete graph. We will sketch a constructive proofof the Erdos-Rogers theorem in Section 4, and return to this question in a more generalsetting in the Appendix, where we will characterize the cases when (*) can be satisfied(for 2-connected forbidden graphs). Among others, we will see that (*) can always besatisfied when all the forbidden graphs Lt are complete graphs of more than p verticesand m = nl~c for some small c > 0.

Some motivation Our problems are motivated by the classical Turan and Ramsey The-orems [33, 34, 23], and also (indirectly) by some applications of the Turan Theorem togeometry, analysis (in particular, potential theory) [35, 36, 37, 13, 14, 15], and probabilitytheory (see, for example, Katona, [22], or Sidorenko, [25, 26]), (see also [38]).

In [12] we proved (among others), for the problem of RT2(n,Kfl9...,Kfr,o(n))< the

existence of a sequence of asymptotically extremal graph sequences of relatively simplestructures

Assume now, that oc(Gn) is much smaller than en, for example cc(Gn) < ^Jn. Then weknow (since R(K^,Kk) < /c2/log/c) that for every fixed c > 0, every set of > en verticesof Gn will contain not only an edge, but also a K3. Similarly, if we choose even smallerupper bounds for a(Gn), we can ensure the even stronger conditions that every inducedsubgraph of Gn of at least en vertices contains a larger complete graph Kp. This also leadsto the problems of the present paper, though apart from Theorem 2.1 we will deal onlywith the simplest case f(n) = o(n).

Some basic definitions It is probably hopeless to give an exact description of the maximumin the general problem. Therefore we will try to find an asymptotically extremal sequenceof graphs of relatively simple structure. The definitions listed here are needed to makeprecise what we consider 'relatively simple'.

Notation. For any given function / , let

#£ = #£>PJ/(LI, . -,Lr) = lim sup " and 9pj = lim SE

where the limsup is taken for the r-coloured graphs Gn satisfying (*) with m = ef(n):

for v = 1,..., r the subgraph of colour %v contains no Lv

and (xp(Gn) < e/(n).

(If the limsup is taken over the empty set (of graphs), it is defined to be 0.) Clearly, ife —• 0, the lim sup above will converge, since it is monotone in B. One can easily see thefollowing claim.

Claim 1.1.

r RTp(n9Lu..-,Lr,enf(n)) ^ Q ,_ _ .lim sup —- — < Sp/(Li,..., Lr).

y(b) There exist a sequence s*n —• 0 and an infinite sequence (Sn : n G No) fNo <= NJ of

graphs with the property (*) for m = B*nf(n) where the equality holds in (a).(c) For every sn > &*n, en —• 0,

RTp{n9Li,...,Lr,Enf(n))lim j-r = Spj(Li,...,Lr).MGJNQ I j

Proof. Here (a) is trivial from the definition, (c) is trivial from (a) and (b), by monotonicity,and (b) follows by an easy diagonalization.

Indeed, assume that for k = 1,..., t— 1 we have already fixed Snk. Now we fix B = st = l/tand find an Snt with the following properties: nt > nt-\,

and ap(Sni) < (l/t)/(«,). •

' The definitions can be found below.

Unfortunately, we cannot prove the corresponding assertions for all n > no: we cannotexclude the possibility that

RTp(n,Ll9...9Lr98nf(n))

G)jumps up and down as n —• oo.

We will often speak of the problem of determining RTp(n, L\9..., Lr, o(rc)), meaning thedetermination of 9pj(L\,...,Lr), for /(n) = n. This slightly imprecise notation will causeno problems. Similarly, if f(n) = n, we will often use the notation 9p(L\,...9Lr) insteadof 9pj(L\,...,Lr) . Observe that 9 is monotone: if we replace L\ by an L\ 3 L,-, then9pj(L\9...9Lr) < 9pj(L\9...9Lr). In particular, 9p(Kq) is monotone increasing in q.

Definition 1.2. (Asymptotically extremal graphs) Suppose that the forbidden graphsL\9..., Lr, and the function / are given. An infinite sequence of graphs, (Sn), will be calledan asymptotically extremal sequence (for L\9...9Lr and / ) if the edges of each Sn can ber-coloured so that the vth colour contains no Lv, (v = l , . . . , r) , (xp(Gn) = o(f(n)), and

e(Sn) Q (J , .

In Section 2 we will formulate some theorems asserting that, for any r, there are alwaysasymptotically extremal graph sequences of fairly simple structure. To formulate thesetheorems, we have to introduce the notion of matrix graphs, and matrix graph sequences.

We will say that two disjoint vertex sets X and Y are joined e-regularly in the graph Gif for every subset X* c X and Y* c y satisfying \X*\ > e\X\ and \Y*\ > e\Y|, we have

| d ( x * , y * ) - d ( x , y ) | <£.

In the following A = (atj) will always be a symmetric matrix with all atj e [0,1].

Definition 1.3. (,4-matrix graph sequences) Given a t x t symmetric matrix A = (a/;), agraph sequence (Sn) - defined for infinitely many n but not necessarily defined for everyn > no - is said to be an A-matrix graph sequence if the vertices of Sn can be partitionedinto t classes V\,n9..., VUn so that in Sn

— e(Vi,n) = o(n2), for every / = 1,..., t9

— d(Vi,n, Vj,n) = atj + o(l) for every 1 < i < j < t and— the classes V^n and V^n are joined en-regularly for every 1 0}.

The quadratic form will be used to measure the number of edges in the correspondingmatrix graph sequence. The vectors attaining the maximum will be called optimum vectors.(Optimum below will always mean maximum.)

Definition 1.4. (Dense matrices) A matrix A is dense, if for any i deleting the fth row andthe ith column of the matrix A we get an Af with g{A') < g(A).

One can easily see [4] that if A is dense, it has a unique optimum vector and all thecoordinates of this optimum vector are positive. The uniqueness implies that the sym-metries of the matrix leave the optimum vector invariant: the corresponding coordinatesare equal. This means that if a permutation n of l,...,t applied to the rows and to thecolumns of A leaves A invariant, then n applied to the optimum vector also leaves itunchanged. Further, if g(A') < g(A) for some symmetric minor A! of A, there exists an A"obtained from A by deleting just one row and the corresponding column and satisfyingg(A") < g(A). For a more detailed description of this function g(A) see [4, 7].

Definition 1.5. (Asymptotically optimal ^-matrix-graph sequences) Let A be a fixed matrixand u = (u\, . . . , ut) be an optimum vector for A. We will call an A-matrix graph sequence(Sn) asymptotically optimal if the classes VUn can be chosen so that \VUn\/n = ut + o(l), fori = l,...,t.

Clearly, an optimal matrix graph has

l-g{A)n2 + o(n2)

edges. If the matrix A has a submatrix A' such that g{A') — g(A), we can always replacethe matrix graph sequence corresponding to A by the simpler matrix graph sequencecorresponding to A'. This is why we are interested only in dense matrices.

2. Main results

We start with the existence of the limit.

Theorem 2.1. For any p\,... ,pr and for f(n) = n, for any sn —• 0:

(a) Let (Sn) be an extremal graph sequence for RTp(n,KPl,...,KPr,snn). Then

e(S )limsup-J^f < &pJ(KPl,...,KPr). (la)

(b) There exists an e* —• 0 for which on the left-hand side of (la) the limit exists and

^ . . . 9 K P r ) . (Ib)

(c) For every sn —• 0 with en > s*n the same - namely, (lb) - holds.

Here f(n) = n means that we consider the case (xp(Gn) = o(n). The difference betweenthis theorem and Claim 1.1 is that there we regard all possible forbidden graphs, here onlycomplete graphs, and there we assert only the existence of a sparse subset of integers alongwhich a limit exists, (i.e., we assert that the limsup can be obtained in some specific way)here we assert that the actual limit exists.

The meaning of Theorems 2.2 and 2.3 below is that in the general case there areasymptotically extremal graph sequences of fairly simple structure, where 'simple' meansthat the structure depends on n weakly. This is a weak generalization of the Erdos-Stone-Simonovits Theorem (from ordinary extremal graph theory) [17, 19]. The optimal matrixgraph sequences - in some sense - generalize the Turan graphs, while the matrix graphsgeneralize the complete r-partite graphs. (See also [8], and [28]).

Theorem 2.2. For any p\,...,pr let sn —• 0 sufficiently slowly (which means that the conditionof (c) of Theorem 2.1 is satisfied). Then there exists a dense Q x Q matrix A with Q <R(KPl,...,KPr) and an asymptotically extremal sequence (Sn) for RTp(n,KPl,...,KPr,8nri)that is an asymptotically optimal A-matrix graph sequence.

For general L i , . . . , L r we have the following theorem.

Theorem 2.3. Let r forbidden graphs L\,...,Lr be fixed. Let f(n) - • oo (f{n) = O(n)) bean arbitrary function for which for every c G (0,1) there exists an n = n^c > 0 such that

f(cn) > nfj(n).

Then there exist a dense matrix A = (fl//)nxQ -for some Q < R(L\,...,Lr), and an asymp-totically extremal sequence (Sn,) (for L\,...,Lr and j ' , for some subsequence of integers)that is an asymptotically optimal A-matrix graph sequence.

This means that the structure of some asymptotically extremal sequences is simple.The matrix A depends on the function / : for different / ' s we get different extremaldensities. The matrix depends primarily on the sample graphs and on / . However, we areunable to exclude the possibility that A must, even in the simplest case / = n, dependon the actual subsequence of integers as well: that there is no common A for all n > no.The condition f(cn) > Hf,cf(n) is a 'smoothness' condition, which is satisfied in 'all thereasonable cases'.

Remark 2.4. We are primarily interested in functions of type f(n) = ny. By the quantitativeRamsey Theorem, for every family L\,...,Lr we can fix a F > 0 so that if oc(Gn) < f(n) —nr, then every r-colouring of Gn contains an Lv of colour v for some v < r (since itcontains a large clique of colour v): no graphs satisfy (*).

Remark 2.5. When we assert the existence of a matrix A in Theorems 2.2 and 2.3, wedo not know too much about this A. The only thing we know is that it is dense and(therefore, by Lemma 3.3) its off-diagonal entries are all positive.

Unfortunately, most of the non-trivial results for the Xp-free case (p > 2) are related tothe special case when all the forbidden subgraphs Lv are complete graphs. So in Sections4-6 we will assume that the graphs L, are complete graphs. In Section 4 we will provesome general upper and lower bounds for the case of one colour (r = 1). The followingresult is a direct generalization of the Erdos-Sos Theorem from [18].

Theorem 2.6.(a) For any integers p > 1 and q > p we have

P\ H/ — ^

(b) For every k, for q= pk + \ this is sharp:

To get the lower bound in Theorem 2.6 (i.e., Theorem 2.6(b)) we will use a geometric con-struction of Erdos and Rogers [16]. Here we formulate their theorem, but the verificationis postponed to Section 4.

Erdos-Rogers Theorem. Let p > 2 be an integer. There are a constant c = cp > 0 and anno(p,c), such that for every n > no(p,c), there exists a graph Qn not containing Kp+\, butsatisfying Gcp(Qn) < nl~c.

Construction 2.7. Let q = pk + 1. Take k vertex-disjoint Erdos-Rogers graphs of size(n/k) + o(n) (described in the previous theorem) and join each vertex to all the vertices inthe other graphs. (We will sometimes describe this as putting (p, 8)-Erdos-Rogers graphsinto each class of a Tn^.) Thus we get a graph sequence (Sn) with <xp(Sn) < kn{~c for somec> 0 and Kpk+{ £ Sn.

This proves the lower bound in Theorem 2.6. For q =/? + / , / = 2,3,4,5 we can improvethe upper bound of Theorem 2.6, see Theorem 2.11.

Remark 2.8. Now, for p > 2 fixed, we know the value of every pih &p(Kq). Perhaps theother values have a 'pseudo-periodical' behaviour similar to that of 92(Kq): the extremalstructure is determined by the residue of q mod p. The situation is analogous to that inthe Erdos-Hajnal-Sos-Szemeredi [11] Theorem, where the case of odd values of q wasmuch simpler (and also much simpler to prove) than the case of even g's.

In Section 5, we investigate some special cases that seem to be interesting, because theysuggest some conjectures for the general case. Perhaps the following conjecture holds.

Conjecture 2.9. The asymptotically extremal graphs for RTp(n,Kq,o(n)) have the followingstructure: Let q = pk + t\ (Y = l,2,...,p). Then n vertices are partitioned into k + 1classes Vb>w,..., V^n- For each pair {i9j} ^ {0,1}, V^n is almost completely joined to V^n inthe sense that every x G V^n is joined to every y G Vjjfl with a possible exception of o(n2)pairs xy. Further, d(V^m V\yn) = ((£ — \)/p) + 0(1) (as n —• co), and Votn, V\jn are joinedo(\)-regularly. Finally, e(V^n) = o(n2), i — l,...,fc.

Remark 2.10. For graphs of this kind the optimal sizes of the classes Vt can easily becomputed: the optimal class-sizes are as follows. The edges in G[Vt] can be neglected,

\Vt\ = T^n + o(n) for i = 0,12 + ( / c l ) ( 2 ^ )

and

From this, e(Sn) can easily be calculated: if Sn is the graph described in the conjecture, itis almost regular, and the degrees in Vi are n — | V2I- Hence

We describe some cases below, where we can prove the upper bound in Conjecture 2.9.

Theorem 2.11. Let ( = 2,3,4 or 5 and ( 0. Here one of the mainproblems is:

Problem 2.12. For fixed p determine the minimum tf for which

$P(KP+,) > 0.

In particular, is Qp(Kp+2) > 0 or not? If 9p(Kp+/) > 0, how large is it?

Theorem 2.13. For any p>2, Sp(K2p) > - .o

It is worth observing that replacing K2P by K2P+\ we get by Theorem 2.6(b), for any

For p = 2 Theorem 2.13 is sharp: #2(K4) < 1/8 was proved by Szemeredi [31] and> 1/8 was settled by Bollobas and Erdos in [3], via a high-dimensional geometric

construction. In a slightly different form, Bollobas and Erdos did the following. Fixa high-dimensional sphere Sh and partition it into n/2 domains Di, . . . ,Dn / 2 , of equalmeasure and diameter (1/2)^, with \i — e/y/h. Choose a vertex x, e Dt and an yt e Dt

for i = 1,...,n/2 and put X = {xi,...,xn/2} and Y = {y\,...,yn/2}- Let X U Y be thevertex-set of our Sn, and

join an x G X to a y e Y if p(x,y) < yjl — \i\join an x £ X to a xf G X if p(x,xf) > 2 — \i\join a 3; G 7 to a / e 7 if p(y, yr) > 2 — /x.

For p > 3, our result follows from a generalization of this construction. Theorem 2.13may also be sharp for p > 3, but we cannot prove it. Let p = 3. Our results show onlythat

0 < UK5) < ^

and

^ < h(K6) < l-.

One of the most intriguing problems is to determine the values and some asymptoticallyextremal graphs for RT3(n,K5,o(n)) and RT^(n,K^o(n)). Unfortunately, this task seemsto be too difficult. We do not know the answer to the simplest subproblem if ST,(K5) > 0.

The last section contains some further open problems.The basic proof techniques include primarily the application of Szemeredi's Regularity

Lemma, [32], a modification of Zykov's symmetrization method, [39] and multigraphextremal-graph results [4, 5, 6, 7] (in the background).

Remark 2.14. It is difficult to find the places in this paper that would distinguish betweenthe conditions '(+) Sn contains no L,-' and '(++) Sn can be coloured in r colours so thatthe vth colour contains no Lv\ The reason for this is that the limit constants are the onesthat are different: we have the existence theorems in the same generality for the moregeneral case (++).

3. Proofs of Theorems 2.1-2.3

The aim of this section is to prove Theorems 2.1-2.3. We will start with the simplerTheorem 2.1, move on to the proof of Theorem 2.3 and then return to the proof ofTheorem 2.2.

Proof of Theorem 2.1. Again, as in the 'proof of Claim 1.1, (a) is trivial, (c) follows from(a) and (b) and the only thing to be proved is that if the forbidden graphs are completegraphs and we have an infinite sequence (Smt), as described in Claim 1.1 (b), then we canextend this sequence to every n > no.

First fix an s > 0. Assume that Smt is an extremal graph for RTp(mt,KPl,...,KPr,smt).We may choose this sequence so that

So Smt has an r-colouring in which the vth colour contains no KPv and otp(Smt) < smt ift > to(e). Below, we will sometimes abbreviate mt to m. Let h be an arbitrary integerand put Zmh = Sm® h, that is, let Zw/, be the graph obtained from Sm by replacing eachvertex by h independent vertices and joining two new vertices in colour v iff the originalvertices have been joined in colour v ' . Since the forbidden graphs are complete graphs,

' Here Ih means the complementary graph of X/,.

the r-coloured Zmh will contain no KPv (either) in the vth colour. Further, trivially,

e(Zmh) = e(Sm)

(mh)2 m2 '

andKp(Zmh) _ ocp(Sm)

mh m(Indeed, each Xp-independent set increases by a factor h, and each Xp-independent set Xof Zmh induces a Xp-independent set of Sm of size at least (l/h)\X\.)

As described in the proof of Claim 1.1, we may choose a sequence Smt with et = (1/t),ocp(Smt) < etmti and (for f = n and 3 = 9pj = lim,9£)

Now, for every n > m2 choose the largest mt <i ^Jn. Then choose h = \n/mt] and deleten — mth vertices of Zmth to get a graph S*. Clearly, mt —• oo. Since we have deleted at mostmth — n = o(n) vertices from Zmth, we obtain a sequence S* with ap(5*) = o(n) and

•(As mt -> oo, we cannot get all the integers in the form hmt. Therefore we must

approximate some n's by hmt > n: to delete < h = o(mth) vertices from some of the

One of the basic methods we use to handle Turan-Ramsey type problems is theRegularity Lemma [32].

The Regularity Lemma The regularity condition means that the edges behave (in someweak sense) as if they were random. The Regularity Lemma asserts that the vertices ofthe graph can be partitioned into a bounded number of classes Fo, . . . , Vk such that almostevery pair is e-regular.The Regularity Lemma. (See, for example, [32].) For every s > 0 and integer K thereexists a feo(e, K) such that every graph Gn, the vertex set V(Gn) can be partitioned into setsVo9V\,...,Vk -for some K < k < ko(s,K) - so that \VQ\ < en, \Vi\ = m (is the same) forevery i > 0, and for all but at most e^) pairs (i,j),for every X <= y{ and Y <= y- satisfying\X\, | Y | > em, we have

\d(X9Y)-d(VhVj)\<e.

Remark 3.1. The role of Vo is purely technical: it makes it possible for all the other classesto have exactly the same cardinality. Indeed, having a K and choosing K' > K,S~2 andapplying the Regularity Lemma with this K, one can distribute the vertices of Vo evenlyamong the other classes so that \Vt\ « \Vj\ and the e-regularity will be preserved with a

264 P. Erdos, A. Hajnal, M. Simonovits, V. T Sos, and E. Szemeredi

slightly larger £. So from now on (for the sake of simplicity) we will assume that Vo = 0.The role of K is to make the classes Vt sufficiently small, so that the number of edgesinside those classes are negligible. The partitions described in the Regularity Lemma, orhere, will be called Regular Partitions of Gn.

Now we turn to the second tool used in our proof: the application of matrix graphsequences.

Dense matrices, matrix graph sequences

Lemma 3.2. Let A be a symmetric matrix, x and n, x =/= n be given integers, and let ax^n = 0.Then deleting either

— the xth row and column, or

— the 7rth row and column

we get a matrix Af with

g(A') = g(A).

This implies

Lemma 3.3. If a symmetric matrix A is dense, then all its off-diagonal entries are positive.

The lemma is a variant of Zykov's symmetrization [39], and its proof can be found, forexample, in [4]. Hence we only sketch its proof here'.

Proof of Lemma 3.2. (Sketched) Let u be an optimum vector for A, i.e.,

g(v4) = max|iL4ur : ux > 0 (i = 1,...,/) and ^ T u , = l | .

We define u(/z) to be the vector where the ith coordinate of the optimum vector u isdecreased by h and the nih is increased by h. Clearly,

cp(h) = u(h)Au(h)T = (fljt>w + aT,T)/i2 + cxh + c2

for some constants c\,ci (because aXiJt = a^ = 0). For any interval, such functions attaintheir maximum at some end of the interval (and maybe, inside as well). Hence we maychoose either h = ux or h = — un and still get the same maximum g(A). But now one ofthe coordinates is 0, therefore the value of g(A) is the same as if we had deleted the ith

or 7ith row and column: g(A) = g{A'). •

Lemma 3.4. Assume that f(n) satisfies the condition of Theorem 2.3. Then for every sequencesn —> 0 we can find a sequence fin —• 0 such that

/OM) > ^f(n). (2)

' A. Sidorenko [27] has found a generalization of this lemma, providing a necessary and sufficient conditionfor being dense.

Proof. Let t be an integer and /? = 1/r. Then /(/?n) > rjf^f(n). If n > nt then en < rjjp.Thus f(/3n) > ft^f(n). We may assume nt+\ > nt. Define pn = \/t for n e [nt,nt+\),t := 1,2,3... Then pn -» 0 and (2) holds. D

Proof of Theorem 2.3. For every fixed e > 0, for some infinite set of integers N£, for everyn e Ne, we may fix an Sn satisfying

for v = 1,..., r the subgraph of colour Xv contains no Lv

\ and ap(Sn) < ef(n),

and

(ii) Se = l im n e N E 4R

Apply the Regularity Lemma to this sequence (Sn) with this £ and K = l/s (where K is thelower bound on the number of classes). Thus we get a fco = fco(e) such that the vertices ofSn can be partitioned into the classes 7i>w,..., V^n for some K < k < ko so that

(hi) all but E Q pairs are £-regular, (k = k(n).) '

Using a diagonalization, we may find an infinite set of integers N* and for each n £ H*an r-coloured graph Sn, with a Regular Partition {V\fn,..., Vk(n),n}, satisfying

for v = 1,..., r the subgraph of colour Xv contains no Lv

and <xp(Sn) < enf(n\

with some sn —• 0, and

(if) 9 = l i m n 6 N - ^ , and

(iii*) all but ^ Q pairs are£n-regular in the corresponding Regular Partition.

Here sn usually tends to 0 very slowly, but still it tends to 0! We may assume that & > 0.Next, delete the edges (x,y) : x e V^y e V^n if

(a) either (Vitn, VjtH) is nonregular, or(b) d(VUn,Vj^<2sn.

Thus we have deleted by (a) at most snQ(n/k)2 < (l/2)snn2 edges and by (b) at most

2en(n/k)2{k2) edges. In this way we have ensured that all the pairs (Viin, Vj,n) are £n-regular.

The number of edges has been changed by at most (3/2)enn2. Denote the resulting graph

by Tn.There is a matrix A = An of k < ko(sn) rows (and columns), corresponding to this graph

Tn (and its en-regular partition), where a,; = d(V^m Vj^n) (this value being the densityin Tn). Clearly, if e is the k-dimensional vector each coordinate of which is n/k, then

t VQ = 0 is assumed, by Remark 3.1.

(l/2)eAeT counts the edges between the classes (but it does not count the edges withinthe classes) and

e(Tn) < -eAeT + snn2 < -g(A)n2 + snn

2.

Thus

(and therefore

g(A) >2$- 8en.

In the following m = \V\\, M = | U,-G/ Vt\. We will find a subgraph HM of Tn, equallydense (but possibly much smaller), spanned by the union of some Q = Q.n < R(L\,..., Lr)classes V^n. (This makes the problem bounded in some sense.) For any subset {J^n : i £ 1}of {Vi,n} we have a symmetrical minor (submatrix) A' of A and a corresponding numberg{A'). We will choose an / for which g(Af) > g(A) and |/| is the minimum. (Sinceg{A') < g(A), we will actually have g{Ar) — g(A).) By Lemma 3.2, all the densities betweenthese classes are positive in Tn, and therefore are at least 2s. Further, the resulting matrixA' is dense.

So, if we end up with Q classes, any two of which are joined by density > 2an, then,by a very standard application of the Regularity Lemma, Tn => KQ '. (See, for example,[11]) Hence Q < R = R(L\,...,Lr). In other words, we end up with a bounded numberof classes (independently of n and e).

Originally, when n —• oo, we have sn —• 0, and the number of classes in the RegularPartition could have tended to oo and the entries atj to 0. Now the situation is nicer, thenumbers of rows and columns in the matrices A' are bounded, independently of e and n.So we can take a convergent subsequence of these matrices, while n —• oo: we may assumethat the matrices A'n converge to a matrix A*. Still, it can happen that A* is not dense. Inthat case we can take a dense submatrix AQ of A*. (Otherwise AQ = A*.)

Now we have a (mostly very sparse) sequence of integers nt and the correspondinggraphs Snt with their Regular Partitions (described in the Regularity Lemma) and thecorresponding matrices Ant with their dense submatrices A'nt converging to A*. We consideronly the dense submatrix A$ of A*. Let A® be an Q x Q matrix. It has an optimum vectoru and each coordinate of u is positive, say at least y > 0. So we can fix the correspondingQ < R(L\,...,Lr) classes, say V\,..., FQ, and the corresponding w,m vertices in them, thusgetting an optimal Ao-matrix graph sequence

I IKL/<Q

Since each class of Wt := ViC\V(Hm) of Hm has at least ym vertices, the Wfs will be joinedto each other (l/y)eM-regularly: they will induce an optimal ^o-matrix graph sequence.

We have to prove four things:

' Here we need that e is small in terms of the Ramsey number R{L\,...,Lr).

(a) The corresponding graphs can be coloured in r colours so that the vth colour containsno Lv;

(P) ap{Hm) = o(f(m)\(y) this matrix graph sequence has enough edges to be asymptotically extremal.(S) e(Wi) = o(m2).

(a) This is trivial, since Hmt c Snt and the Sn's have this colouring property.(jS) Up to this point we have used one fixed sequence ew. Replacing this sequence by

another e'n > en tending to 0, everything above remains valid (with the same regularpartition). Given the original sequence ew, we fix a sequence fin as described in Lemma3.4. For any fixed e the upper bound fco of the Regularity Lemma is a constant.So we may find an e^ —• 0 (very slowly) for which, for s^ and K = 1/eJJ we haveko(K,e^) < l/Pn- If %n = max{/s^,/?„,££[}, then with this en —• 0 we have for everyinduced subgraph Hm c Sn of at least n/k vertices

*P(Hm) < JTnf(n) < f(n/k) < f(m).

(y) This follows by a simple computation: we have g{A'nt) > 9 — 8st. Hence g(Ao) > 9. Sofor an v4o-graph Hm, we would know that e(Hm) > (l/2)g(Ao)m2. Now the subgraphof Snt spanned by the selected classes V^nt : i € Int is only a 'nearly'-zlo-graph: theentries in A'nt tend to the corresponding entries of AQ, but they are not equal. Thuswe have only

e(Hm)>Sl\-o(mz).

However, this is enough to ensure that (Hm) is an asymptotically extremal graphsequence.

(S) In principle, some classes of Hm could contain too many edges (in terms of m). Nowwe exclude this. By the construction, g(Ao) = $pj(Li,...,Lr) = S. Hence, on the onehand, for Wt = VUn n V(Hm\

e(Hm) > l- (g(A0) - o(l))m2 + J ^ W ) = \ (S - o(l))m2 + ^e(Wi).

On the other hand,

e(Hm) < l-$m2 + o(m2).

Thus YJe(Wi) = o(m2).

D

Remark 3.5. This remark is aimed primarily at those who know the Zykov symmetriza-tion. Here we try to explain something of the background of the above proof. In con-structing (finding) the 'good" subgraph Hm c sn, we have basically used a modification ofZykov's 'symmetrization' method [39]. The original Zykov type symmetrization means that(instead of deleting vertices) we change the edges incident with some vertices, obtaining agraph with the same number of vertices, but of simpler, more symmetric structure. Thismethod breaks down because the symmetrization may increase the independence number

(a(Gn) or oip(Gn)\ and that is not allowed here. Further, symmetrization can introduceunwanted subgraphs: it may happen for example that Gn contains no Ki(10, 10,10) butafter several symmetrizations it will. Deleting vertices, we can replace the original methodof symmetrization: unless we delete too many of them, ocp(Gn) — o(f(n)) will be preserved,and of course, no new subgraphs occur. At the same time, the structure becomes simplerand, in some very vague sense, more symmetric.

Proof of Theorem 2.2. We know that there is a sequence of graphs (described in the proofof Theorem 2.3) that is for some fixed matrix A an optimal ^-matrix graph sequence. Weneed to show that for each n > no the same matrix A can be used. As in the proof ofTheorem 2.1, we will blow up some good graphs Smr

If we have an infinite sequence (Smt) and a fixed matrix A such that (Smt) is anoptimal ^-matrix graph sequence, and asymptotically extremal for some st —• 0, forRTp(mt,KPl,... ,KPr,etmt), then Zmth = Smt ®h will also be optimal ^-matrix graphs.

Hence, fix the matrix A obtained in the proof of Theorem 2.3 for a sequence et —• 0and some sequence mt. For every n, take the largest mt < ^Jn, then put h = \n/mt] anddelete (hmt — n) vertices of Zmth = Smt ® //,. The resulting ^-matrix graph sequence (S*)proves Theorem 2.2. •

4. Quantitative results for one colour

In this section we obtain various estimates for 9p(Kq).

Proof of Theorem 2.6. In the following, the constants CQ9C\,C2, ••• are positive and inde-pendent of n, m. Assume indirectly that there exist a constant Co > 0 and infinitely manygraphs Gn not containing Kq, satisfying ocp(Gn) = o(n) and yet having many edges:

By a standard argument, for some constants c\, ci> 0, there exist subgraphs Hm c Gn

with minimum degree

dmm(Hm) > 11 ^-y +cijm, m> c2n and ocp(Hm) = o{m). (3)

By a 'saturation argument', we may assume that Hm ID Kq-\: if not, add edges to it oneby one, until it does. Clearly, (3) remains valid. Fix a Kq-\ ^ Hm. Now

e(Kq-l9Hm - Kq-i) >(q-p-l+a)(m-q + l).

Therefore, for some c^ > 0, there exists a set U of c^m vertices of Hm — Kq-\, each joinedto the same q — p vertices of this fixed Kq-\. By the assumption, ocp(Gn) = o(n\ if n (andtherefore m) is sufficiently large, then there is a Kp a U. This Kp, together with the fixedq — p vertices of Kq~\ forms a Kq c Hm ^ Gn. This contradiction proves (a). As we havementioned, Construction 2.7 provides the lower bound, i.e. (b). •

For q = p + 1, Theorem 2.6 reduces to the following claim.

Claim 4.1. For any p> 1, 9p(Kp+{) = 0.

This also has a trivial direct proof.

Proof. (Direct) Suppose that (Gn) is a graph sequence with

Kp+l <£ Gn and ap(G«) = o(n).

If x is an arbitrary vertex, then its neighbourhood N(x) contains no Kp. Therefored(x) = \N(x)\ < ocp(Gn) = o(n). Hence e(Gn) < nap(Gn) = o(n2). •

Now we can return to the proof of Theorem 2.11, which improves Theorem 2.6 in somespecial cases. We will need the following two lemmas.

Lemma 4.2. For any integers p > 2 and 0 < y < p, and constant c > 0, there exists aconstant MPyC with the following properties. Let e > 0 be fixed and n > Mp,ce. Supposeccp(Hn) = o(n) and Be ^ Hn be a bipartite graph with colour classes V\ and V2 that arejoined e-regularly. Let \V\\ = |F2I > en and d{Vu Vi) > (y/p) 4- rj and n > no(p,c,n). Then

Obviously, we are thinking of the case when we apply the Regularity Lemma to alarge graph and V\, V2 are two classes in the resulting partition connected to each otherregularly and with a sufficiently high density.

Proof. For n large enough, all but at most en vertices of V\ are joined to at least ((y/p) +(1/2)^)1 K21 vertices of Vi. Hence V\ contains a Kp joined with at least (y + (l/2)pfy)| V2Iedges to \Vi\. Thus (for some fixed constant c\ > 0) V2 contains at least c\n vertices joinedto the same y + 1 vertices of this Kp a V\. They form a Ky+\ c Kp c V\. The c\n verticesin V2 contain a Kp completely joined to Ky+i c yx \ Kp+y+\ c Hn. D

Lemma 4.3. For any integers p,/c > 2, and 0 < y 0 there existsa constant MPyC^ with the following properties. Let e > 0 be fixed and n > Mp^e. Letap(Hn) = o(n) andVu...,Vk^ V(Hn), Vt n Vj = 0, | Vt\ > en. Assume that for every 1 n. If d(V\, Vi) > (y/p) + r]and n > no(p,c,n), then Hn 3 Kp+y+k-{.

Proof. For j = k,k — 1,...,3 we fix, recursively, a vertex Xj £ Vj, so that they forma complete k — j + 1-graph and are joined completely to some sets Vy ^ Vt (i < j)and Vtj > c*n for some constant c* > 0. For j = 3 we get a complete (k — 2)-graphjoined completely to some sets V{ c yx and V{ c y2, \Vf\, \V{ \ > en, for some constantc* > 0. (We use n > Mp^e to ensure that all the sets Vtj above are large enoughto apply the a-regularity of the Regularity Lemma iteratively.) Applying Lemma 4.2

to the corresponding bipartite graph /J(Kj\K2*) (with rj — ks instead of rj), we get aKp+y+M-2 = Kp+y+zc-i c H(V{ U V2 U ... U Vk). D

Proof of Theorem 2.11.

(a) Let cCp(Gn) = o(n) and Kp+s (£ Gn. Fix an e > 0 and put rj = MPtCjcS. We apply theRegularity Lemma to Gn, with this e. Thus we get a partition Vi,...,Vk of the verticesinto k < ko(s,K) sets of size « n/k (see Remark 3.1 on Vo).

(b) For any graph G let

*(G) =

We apply symmetrization in the sense described in the proof of Theorem 2.3: we finda subset of the classes F,, say V\,...,Vt so that the density between any two of themis at least 2rj and the density for the obtained GM = G [U,<t Vt] is high:

There is a unique integer y such that for these t classes the largest density occuring is> (y/p) + rj but < ((y + l)/p) H- ?/. The density O(GM) = e(GM)/(™) can be estimatedas follows:

Here GM 3 Kp+t+y-i and GM Kp+t- Therefore y < £ — t, so

O(GM) < -z (1 ) ( h ^/) • (4)

Put

For t = 2we get the conjectured density: /i(2,/) = (*f — l)/4p. What we have to proveis that for ^ = 2,3,4 and 5, h(tj) < h(2J):

p - 4 p '

which follows from

•Proof of Theorem 2A3 In proving the lower bound on RTi{n,K4, o(n)), Bollobas andErdos used a geometric, or more precisely, an 'isoperimetric' theorem. Theorem 2.13 isa generalization of the Bollobas-Erdos result. So it is natural to prove Theorem 2.13using a generalization of the original Isoperimetric Inequality. This generalization wasconjectured by Erdos and proved by Bollobas [1].

We need the following definition.

Definition 4.4. ([1]) For k > 2 define the k-diameter of a set A in a metric space by

dk(A) = sup minp(x,,x;).

(In other words, this is the /cth 'packing constant' of A.)

A spherical cap is the intersection of an /z-dimensional sphere Sh and a halfspace IT.Bollobas Theorem. ([l])Le£ A be a nonempty subset of the h-dimensional sphere Sh of outermeasure n*(Ay , and let C be a spherical cap of the same measure. Then dk(A) > dk(C) forevery k > 2.

In the following, whenever we speak of 'measure', we will always consider relativemeasure, which is the measure of the set on the sphere Sh divided by the measure of thewhole sphere.

Denote by S = Sp the diameter of a p-simplex. (S2 = 2, (53 = y/39...)Corollary 1 of Bollobas Theorem. Let the integer p and two small constants e and n > 0be fixed. Then for h > ho(p,s,n), if A is a measurable subset ofSh of relative measure > s,there exist p points x\,...,xp e A such that all d(x,,Xj) > Sp — rj.

Proof. Indeed, if A does not contain such a p-tuple, its p-diameter is at most 5P — n.Hence - by the Bollobas theorem - the outer measure of A is at most as large as that ofa spherical cap of p-diameter Sp — rj. For some constant cPyt] > 0 the ordinary diameter ofsuch a cap is at most 2 — cM, independent of the dimension h. Hence the relative measureof such a spherical cap is at most (QPJn)

h for some constant 0 < Q M < 1 and so therelative measure of A is at most {Qp,n)

h < e if /i > ho(p,e,n), a contradiction. •

Corollary 2 of Bollobas Theorem. (Erdos-Rogers Theorem) For any integer p, there existsa sequence (Sn) of graphs with Kp+\ £ Sn but ccp(Sn) = O(n{~c) for some c > 0.

Proof of the Erdos-Rogers Theorem. Let Sp be the edge-length of the regular p-simplexin S^"1 c RP~l:

(5)v p~i

t Clearly, Sp \ yjl.For a given s > 0, we fix a sufficiently high-dimensional sphere S*1 and fix an n > h.

We partition the surface of S into n domains Dt (i = l , . . . ,n) of equal measure and ofdiameter

Op — Op+\<

4(This can be done if n is sufficiently large.) Then we choose n vertices x, e Dt (/ = !, . . . , n).

' We will only use 'nice sets', but Bollobas formulated his result in this generality. The reader can replace 'outermeasure' by 'measure'.

i (5) is taken from [16], and will be obtained (as a by-product) in the proof of Theorem 2.13.

They will be the vertices of our graph Qn. We join x,- and Xj if

Trivially, Kp+\ ^ Qn. If we choose en vertices x/ of Qn and A is the union of thecorresponding D,'s, then the relative measure of A is at least e, and - by BollobasTheorem - A contains some wi,...,wp with p(w,-,w7-) > dp, (1 (1/2)(<5P + <5p+i),i.e. we have found a Kp in the subgraph induced by these nl~c vertices: ccp(Qn) =< m. Asa —• 0, the dimension h -» oo and ap(Qn) = o(n). Using a more careful calculation, we getocp(Qn) = O(nl-C). •

Proof of Theorem 2.13. We will use a Bollobas-Erdos type construction (see [3]) to geta graph sequence (Bn) to prove Theorem 2.13. Fix a high-dimensional sphere Sh andpartition it into n/2 domains Di,...,Dn/2, of equal measure and diameter (l/2)ju, withfi = e/y/h. This can always be done if e > 0 is first fixed, h is then chosen to be sufficientlylarge, and, finally, n > no(e, h).

Choose a vertex x, e Dt and a yt e Dt (for i = l , . . . ,n/2), and put X = {xi,...,xn/2}and Y = {y\,...,yn/2}- Let X U 7 be the vertex-set of our Bn and

join a n x e X t o a y G Y if p{x,y) < y/2 — \i\join an x G I to a x' G I if p(x,xr) > bp — \i\join a y € 7 to a / G 7 if p(y, / ) > Sp — \i.

(a) First we show that ocp(Bn) = o(n). To show this, choose en vertices of Bn. At least(l/2)en vertices belong to (say) X and the union of the corresponding D,'s has relativemeasure > (l/2)a. Denote by A the union of the D,-'s corresponding to these x,'s. ByBollobas Theorem, if dp(A) < (1/2)(5P + 5p+i), then /i(A) < e, provided that h > h0. Sowe may choose wi,...,wp G A such that for each i ^ j , p(wI-,w_/-) > ^p — (l/2)/i, andtherefore p(x,,x;) > <5P — /z, yielding a Kp in the subgraph of £„ spanned by these envertices.

(b)Now we show that the resulting graph Bn contains no K2P. Clearly, if 2p verticesform a Kip c Bn, then p of them must be in X and the other p in 7 , since - forsufficiently small £ - neither X nor 7 contains a Kp+i. Suppose that a i , . . . , a p € l andbi,...,bp G 7 form a X2P. In the following, a,-'s and b/s are unit vectors and points ofthe sphere at the same time. The idea of the proof is as follows. We will show that theexistence of such a Kip implies that (Y^at — ]C^/)2 < 0, which is a contradiction. Toget this, we will estimate ^ a,a;, and ^ fc,b; from above, and J ] fljb; from below.Let d = Sp — \i and t = ^2 — \x. Now, |a,| = 1, \bj\ = 1, and |a, — a;| > rf. Therefore

2 E a^= E (^+a2j)-(ai-ajf)

The same holds for the b,'s. Hence

2

Let us now turn to the mixed terms. By \at — bj\ < t, we have

2 E E a,bj = ± ±(aj + bj)-± ±(at - bj)2 > p2(2 - t2).i=\ ;=1 i=\ j=\ i=\ j=\

This implies that

) = E a < + E ^ + 2 E (aiaJ + bibj)-2J2J2a'bJi=l ;=1 j i j \<i<j<p 1=1 7=1

< 2p + 2(p2-p)-(p2-p)d2-(2p2-p2t2)

= p2t2 - (p2 - p)d2 = P2( V2 - M)2 - (P2 - p)(Sp - n)2

= (2p2 - (p2 - p)S2p) - 2 (V2p2 - (p2 - p)Sp) n + p/z2.

To avoid clumsy calculations involving 5P, observe that in all the above formulas wehave equality if e = 0, /i = 0, that is, a,'s are the vertices of a regular p-simplex andb/s are the vertices of another. Indeed, in this case J3 a, = 0 and ]T bj = 0. Hence2p2 - (p2 - p)(52 = 0, that is,

Returning to the \i > 0 case, we get

provided that [i is sufficiently small, is a contradiction. This shows that Bn ^ K^v.(c) Each vertex has degree (n/4)-fo(n), since each a, is joined to the fr,'s on an 'approximate

half-sphere' and thus the the surface considered has measure > (1/2) — O(e) and thenumber of vertices bj is proportional to this measure. So

j ~ O(sn2) < e(Bn) <j+ O(sn2).

This completes the proof.

D

5. Two special cases

The last problems we discuss here are:How large are 33(K8) and

Conjecture 2.9 asserts that ^(Kg) = 3/11 and 9i(K9) = 3/10. The conjectured extremalstructures (described in Conjecture 2.9) in both cases have 3 classes and are as follows.Put x = (3n/ll) + o(n) vertices in the classes V\, Vi and y = (5n/ll) + o(n) vertices into

V$. Then join V\ and Vi with d{V\, Vi) — 1/3, o(l)-regularly, and join K3 completely tothe other two classes. The classes Vt contain some edges to ensure oi^{Gn) = o(n). However,the problem is that we are unable to find such graphs.

One reason that we cannot prove Conjecture 2.9 (even for p = 3, q = 8,9) is that weare unable to construct bipartite graphs analogous to the Bollobas-Erdos [3], or Erdos-Rogers graph [16], but with density 1/3 (or 2/3) instead of 1/2. Here the 'analogous'means that we fix, for some t, a t x t matrix D = (dij) of positive elements, and on ahigh-dimensional sphere S \ we choose some sets X\,...,Xt, each uniformly distributedon the sphere in some sense, and join two vertices u € AT,,and v e Xj if their Euclideandistance p(u,v) « dy, or p(u,v) > dtj, ...

So we have only an upper bound on the number of edges.

Theorem 5.1. h(Ks) < -^-.

In the proofs of this and the next theorem we need some case-distinction. In many caseswe know that the graph structure considered is dense, and we can easily calculate theedge-densities by solving a small system of linear equations. Here we formulate a lemma,which covers most of the cases we need. (It has a more general form as well.)

Lemma 5.2. Let A = Ahjcx<p,p be a symmetric (h + k) x (h + k) matrix satisfying

U if l<i<j<h,aUj = \ <P lf h<i<j <h + k,

{fi else.

If A is dense, its optimum vector w has coordinates

Rfc _ (n^b — 1)

Wi = 2phk-<ph(k-l))-Wh-l) {i ~

= PhkMhl)(kl)g[ } iphk - cph(k - 1)) - Xk{h - 1) ' l }

and

The density is

Proof. Assume that Hn is an optimal matrix graph corresponding to A. Let the classes ofHn be Ki,..., Vh+k- Then |K,| « wtn. When counting the sizes of the classes in an optimalmatrix graph, it is enough to take into account that the degrees must be asymptoticallyequal - provided that the matrix is dense' (see, for example [4]).Let the first h coordinatesof the optimum vector be x, the others yt. Now the vertices in the first h classes will have

' For dense matrices this condition is necessary and sufficient.

+ Because of the symmetry, the first h class sizes will be asymptotically the same, and the same holds for theother k classes.

degree (k(h — l)x + fiky)n, while in the last k classes the degrees will be (fihx + (p(k — l)y)n.Furthermore, hx + ky = 1. Solving this system of linear equations, we get (6a) and (6b).Now, g(A) is the common degree divided by n, (the edge-density is half of this). Thisproves (7)'. •

Remark 5.3. These formulas become much simpler if, for example, h = 1 or k = 1. Fork = 1, cp drops out and we get

Proof of Theorem 5.1. Let us fix an r\ as described in Lemma 4.3. Using the argument ofthe proof of Theorem 2.11, we get some sets V\,..., Vt, and we define y to be an integerfor which the largest density between these classes is between (y/p) + rj and ((y + l)/p) + rj.By (4), applied with p = 3, t = 5, we have

if t > 3 and rj is small enough. Therefore we may assume that t < 3.With t = 2 the maximum density is 1/4 < 3/11. So we may suppose that t > 3, that is,

t = 3.

(i) If the classes are V\, V2, V3 and d(V\, V2) < (1/3) + rj, then the density is the maximumif the other two densities are 1, i.e. (by (8) applied with k = 1/3 and /? = 1, h = 2) themaximum is at most (3/11) + O(n) and we are home,

(ii) If, for example, d(V3, V{) > (2/3) + rj, and d(V3, V2) > (2/3) + n, then we are home:we may choose a X3 in F3 and a subset V[ c y{ of c\n vertices in both other classes,completely joined to this X3. By Lemma 4.2, we find a K5 in V[ U V^ and we arehome again,

(iii) In the remaining case there is a class adjacent to the other 2 classes with density< (2/3) + rj. We may assume that d(V3, Vx) < (2/3) + rj, and d(V3, V2) < (2/3) + v.By (8) (applied with h = 2, /? = (2/3) + rj, k = 1) the edge-density is at most(4/15) + Ofa)< 3/11.

•

Theorem 5.4. S3(X9) < ^-.

We know that 33(K9) > 2/7 because we may fix 3 classes V\, V2, K3 of sizes 2w/7, 2w/7,3w/7, join F3 to V\ U V2 completely and build a graph on V\ U V2 as described in theproof of Theorem 2.13. Put an Erdos-Rogers graph into F3. The resulting graph containsno Kg, since K3 contains no K4 and G[V\ U V2] contains no K§.

"I" Of course, the proof can be given entirely in the language of Linear Algebra without mentioning graphs.

Proof of Theorem 5.4. (Sketched.) Again, as above, we have to end up with at least t > 3classes after the symmetrization, and if we have t > 5, then, by (4), the density is smallerthan 3/10. So we may assume that t < 4.

The case of 3 classes is easy. Now at least one of the 3 densities is at most (2/3) + 2rj,otherwise we have a Kg c Gn. So the density is at most (3/10) + O(rj) (by (8), appliedwith I = (2/3) + rj, /? = 1, h = 2), and we are home. Hence we may assume that t = 4.

We will distinguish 3 types of connections between Vt and Vj:

— if d(Vu Vj) < (1/3) + rj, we will call (Vu Vj) a (l/3)-pair;— if (1/3) + Y\ < d(VtVj) < (2/3) + rj, we will call (Vu Vj) a (2/3)-pair;— if diYiVj) > (2/3) + rj, we will call (Vu Vj) a 1-pair.

We may assume that there is at least one 1-pair, otherwise the density could be estimated

How many 1-pairs can we have on 4 classes? If we have two adjacent 1-pairs, (Va,Vb)and (Va, Vc), then (Vb, Vc) must be a (l/3)-pair: otherwise - by the proof of Theorem 5.1- we could find Kg ^ Va U Vb U Vc, extendable into a Kg.

This immediately implies that we may have at most 4 1-pairs. If we have exactly 41-pairs, they form a 4-cycle and the remaining 2 densities are 1/3. Applying Lemma5.2 with h = k = 2, A = q> = 1/3 and /? = 1 we get that the edge-density is at most7/24 < 3/10.

Here, unfortunately, we have to distinguish some cases.

(i) If t = 4 and there are 3 1-pairs meeting in one class, the other 3 pairs form a (1/3)-triangle. Applying (8) with h = 3, X = (1/3) + rj, fi = 1 we get that the edge-density isat most 9/32 < 3/10, and we are home again.

(ii) Suppose that we have on 4 classes 3 1-pairs that do not meet. Now they form a path,say VxViV^V^. The density is the highest when

and

3An easy calculation shows that the optimal weights (for rj = 0) are 1/6, 1/3, 1/3, 1/6,the density is 5/18 < 3/10. (Or we can reduce this case to the case when the 1-edgesform a C4.)

(iii) We have settled the case when the number of 1-pairs is 4 or 3. The case of one 1-pairor when we have 2 independent 1-pairs can be majorized by the case when we have2 independent 1-pairs and all the other pairs are (2/3)-pairs. By Lemma 5.2, appliedwith /c = /i = 2, /l = (p = l,/? = 2/3we again get that the edge-density is smaller than(7/24) + O(i/) < 3/10.

(iv) The only remaining case to be settled is when we have 2 adjacent 1-pairs, say (VuVi)and (Vu V3). Now we know that we get the maximum density if d(Vi> V$) = (1/3) + rj

Turan-Ramsey Theorems and Kp-Independence Numbers 211

and d(Vt, V4) = (2/3) + n. One can easily check (by determining the optimum vectorof this structure) that the maximum density is (11/39) + O(rj) < 3/10.

•

6. Open problems

Various open problems are stated in [12] and we have already stated the above Problem2.12. Here we list some others. The first two of these are the simplest special cases ofConjecture 2.9, where we got stuck.

Problem 6.1. How large is

Problem 6.2. How large is

Conjecture 2.9 states that 33(^n) = 11/32 and 33(Ki4) = 8/21.

Problem 6.3. Can one always find a matrix A such that one has a graph sequence (Sn :n > no) obeying the partition rules of the matrix A and being asymptotically extremal forRTp(n,L\,...,Lr9o(n)) (and not only for an infinite sequence of integers n^)?

The answer to this problem is very probably YES. (If it were not, it would probablymean that the extremal structure sharply depends on some parameters such as, forexample, the divisibility properties of n, which are not really graph theoretic properties.)

Problem 6.4. Is there a finite algorithm to find the limit

We have shown in our previous paper that there is a finite algorithm for findingi,. . . , Lr) if the sample graphs L, are complete graphs. A paper of Brown, Erdos and

Simonovits [7] shows that for the digraph extremal problems without parallel arcs (whichseem to be very near to the Turan-Ramsey problems) there is an algorithmic solution,though far from being trivial. What is the situation in case of 9P(L\,..., Lr) ?

Some hypergraph problems (and results) on Turan-Ramsey problems can be found in[18, 20].

Appendix A. Are there graphs satisfying (*)?

In the above, the forbidden graphs were complete graphs, here we discuss the generalcase, where Li , . . . ,L r are arbitrary graphs.

We are interested in two strongly connected problems. Given either a family if or rfamilies of excluded graphs, JS?i,...,JS?r and a graph sequence (Gn) with ap(Gn) = o(n).Under what conditions on if or the families S£\ can we assert that there exists a graphsequence (Gn) such that

(i) Gn contains an L € S£ for n> no; or

(ii) there is an r-colouring of Gn so that for no colour v is there a v-coloured L e J^v?

The case p = 2 is easy. In both problems, if no L is a tree, such graphs exist. On theother hand, if (in each S£\) some L is a tree, those graphs do not exist. Indeed, in [9]Erdos has proved that for every t there exist a c = Q (0 < Q < 1) and an nt such thatfor every n> n<? there exist graphs Sn with girth greater than t and independence numbera(Sn) < O(nl~c). This implies that if none of the graphs L E if is a tree or a forest,and £ — maxLeJs? v(L), the above graphs Sn will contain no L's and cc(Sn) < O(nl~c). Thisanswers (i) and (ii) also, since cc(Gn) = o(n) implies that for all r-colourings of Gn somecolours contain all the trees of at most £ vertices for n > n^. For p > 2 the situationis similar, but somewhat more complicated. First we will solve the problem (i). We startwith some definitions.

Definition A.I. A graph T is a p-forest if

(a) it is the union of complete graphs of order p, having no common edges and(b)for every integer t > 1, the union of any t of these Kp's has at least pt — t+1 vertices;

or(c) it is a subgraph of a graph described in (a) and (b).

Definition A.2. (Girth)

(1) We will say that the girth of a p-uniform hypergraph H is at least t if the union ofany t < £ hyperedges has at least pt — t + 1 vertices.

(2) We will say that the p-girth of a graph G is at least { if every subgraph of G of fewerthan f vertices is a p-forest.

Clearly, the 2-forests are exactly the ordinary forests and the 2-girth of a graph is theordinary girth.

Erdos-Hajnal Theorem. ([10, Theorem 13.3]) For every given p, and / and suitableconstants c\,c> 0 (for n > no(p,£,c9c\)) there exist p-uniform hypergraphs Hn for which

— any two hyperedges intersect in at most one vertex (such hypergraphs are sometimes

called linear hypergraphs),

— any set of c\nl~c vertices contains a hyperedge, and

— the union of any t < / hyperedges has at least pt — t + 1 vertices. (In other words, thep-girth of Hn is at least £.)

The proof used random hypergraphs.Let us call a graph Un the shadow of a /7-uniform hypergraph Hn if Hn and Un have

the same vertex-sets, and (x,y) is an edge of Un iff there is a hyperedge in Hn containingboth x and y. We will call the shadow Sn of Hn of [10, Theorem 13.3] the Erdos-HajnalRandom Graph.

As for the shadow, one can easily see that if the girth of Hn is at least 4, (which impliesalso that Hn is a 'linear hypergraph'), then Hn can easily and uniquely be reconstructedfrom Un. The following claim is an immediate consequence of Theorem 13.3 of [10].

Claim A.3. There exist a constant c = cpj > 0 and an integer npj such that for everyn > npj there exist graphs Sn with p-girth greater than £ and independence number ccp(Sn) =

Indeed, the Erdos-Hajnal Random graph (Sn) proves Claim A.3. This implies thefollowing claim.

Claim A.4. If no L € $£ is a p-forest, then there exist graph sequences (Sn) with (xp(Sn) =O(nl~c) (for some c> 0) and with L £ Sn (Le Z£).

This is sharp:

Claim A.5. If (Sn) is a graph sequence with the property that ocp(Sn) = o(n) and L is ap-forest, then L ^ Sn for n > no.

The case of many colours In the following, we will use the notation R(J?\,..., i?r) in theobvious way. Clearly, if ocp(Gn) — o(n) and n > no, then Kp c Gn

Since ocp(Sn) = o(n) implies Kp c Sn, if p > R(i?i , . . . , J£?r), then any r-colouring of Sn

has for some v an L e ££v of colour v.This trivial assertion is sharp for 2-connected excluded graphs.

Theorem A.6. Assume that the excluded graphs in all the J£v's (v = l,...,r) are 2-

connected and p < R(J?{,...,J?r). Then there exist graph sequences (Gn) with ocp(Gn) =

O(nl~c) (for some constant c > 0) such that the graphs Gn are r-colourable such that no

monochromatic copies of any L e J£v in the vth colour occurs (v = l,...,r).

Proof. Let

£ > max v(L).

We can take the Erdos-Hajnal Random graph Gn = Sn with p-girth larger than *f,and edge-colour each Kp ^ Sn in r colours without monochromatic L's, since p <R(£?\,...,S£r). If L c Sn is 2-connected, L G ifv is in a uniquely defined Kp ^ Sn andtherefore cannot be monochromatic, of colour v. •

Some similar results can be formulated for the case when the 2-connectedness of thegraphs Lv is dropped. In fact, one can define a p-tree W& of size K(p,/) such that ifall the excluded graphs are of order at most {, then (*) can be satisfied iff WK can becoloured in v colours without having L e 5£v in the vth colour. The details are easy andomitted here.


Acknowledgement

We would like to thank the referee for many helpful suggestions.

References

[I] Bollobas, B. (1989) An extension of the isoperimetric inequality on the sphere. Elemente derMath. 44 121-124.

[2] Bollobas, B. (1978) Extremal graph theory, Academic Press, London.[3] Bollobas, B. and Erdos, P. (1976) On a Ramsey-Turan type problem. Journal of Combinatorial

Theory B 21 166-168.[4] Brown, W. G., Erdos, P. and Simonovits, M. (1973) Extremal problems for directed graphs.

Journal of Combinatorial Theory B 15 (1) 77-93.[5] Brown, W. G., Erdos, P. and Simonovits, M. (1978) On multigraph extremal problems. In:

Bermond, J. et al. (ed.) Problemes Combinatoires et Theorie des Graphes, (Proc. Conf. Orsay1976), CNRS Paris 63-66.

[6] Brown, W. G.? Erdos, P. and Simonovits, M. (1985) Inverse extremal digraph problems. Finiteand Infinite Sets, Eger (Hungary) 1981. Colloq. Math. Soc. J. Bolyai 37, Akad. Kiado, Budapest119-156.

[7] Brown, W. G., Erdos, P. and Simonovits, M. (1985) Algorithmic Solution of Extremal DigraphProblems. Transactions of the American Math Soc. 292/2 421^49.

[8] Erdos, P. (1968) On some new inequalities concerning extremal properties of graphs. In: Erdos,P. and Katona, G. (ed. ) Theory of Graphs (Proc. Coll. Tihany, Hungary, 1966), Acad. Press N.Y. 77-81.

[9] Erdos, P. (1961) Graph Theory and Probability, II. Canad. Journal of Math. 13 346-352.[10] Erdos, P. and Hajnal, A. (1966) On chromatic number of graphs and set-systems. Acta Math.

Acad. Sci. Hung. 17 61-99.[II] Erdos, P., Hajnal, A., Sos, V. T. and Szemeredi, E. (1983) More results on Ramsey-Turan type

problems. Combinatorica 3 (1) 69-82.[12] Erdos, P., Hajnal, A., Simonovits, M., Sos, V. T. and Szemeredi, E. (1993) Turan-Ramsey

theorems and simple asymptotically extremal structures. Combinatorica 13 31-56.[13] Erdos, P., Meir, A., Sos, V. T. and Turan, P. (1972) On some applications of graph theory I.

Discrete Math. 2 (3) 207-228.[14] Erdos, P., Meir, A., Sos, V. T. and Turan, P. (1971) On some applications of graph theory II.

Studies in Pure Mathematics (presented to R. Rado), Academic Press, London 89-99.[15] Erdos, P., Meir, A., Sos, V. T. and Turan, P. (1972) On some applications of graph theory III.

Canadian Math. Bulletin 15 27-32.[16] Erdos, P. and Rogers, C. A. (1962) The construction of certain graphs. Canadian Journal of

Math 702-707. (Reprinted in Art of Counting, MIT PRESS.)[17] Erdos, P. and Simonovits, M. (1966) A limit theorem in graph theory. Studia Sci. Math. Hungar.

1 51-57.[18] Erdos, P. and Sos, V. T. (1969) Some remarks on Ramsey's and Turan's theorems. In: Erdos, P.

et al (eds.) Combin. Theory and Appl. Mathem. Coll. Soc. J. Bolyai 4, Balatonfured 395^04.[19] Erdos, P. and Stone, A. H. (1946) On the structure of linear graphs. Bull. Amer. Math. Soc. 52

1089-1091.[20] Frankl, P. and Rodl, V. (1988) Some Ramsey-Turan type results for hypergraphs. Combinatorica

8 (4) 323-332.[21] Graham, R. L., Rothschild, B. L. and Spencer, J. (1980) Ramsey Theory, Wiley Interscience,

Ser. in Discrete Math.[22] Katona, G. (1985) Probabilistic inequalities from extremal graph results (a survey). Annals of

Discrete Math. 28 159-170.


[23] Ramsey, F. P. (1930) On a problem of formal logic. Proc. London Math. Soc, 2nd Series 30264-286.

[24] Shamir, E. (1988) Generalized stability and chromatic numbers of random graphs (preprint, underpublication).

[25] Sidorenko, A. F. (1980) Klasszi gipergrafov i verojatnosztynije nyeravensztva, Dokladi 254/3,[26] Sidorenko, A. F. (1983) (Translation) Extremal estimates of probability measures and their

combinatorial nature. Math. USSR - Izv 20 N3 503-533 MR 84d: 60031. (Original: Izvest.Acad. Nauk SSSR. ser. matem. 46 N3 535-568.)

[27] Sidorenko, A. F. (1989) Asymptotic solution for a new class of forbidden r-graphs. Combina-torica 9 (2) 207-215.

[28] Simonovits, M. (1968) A method for solving extremal problems in graph theory. In: Erdos, P.and Katona, G. (ed. ) Theory of Graphs (Proc. Coll. Tihany, Hungary, 1966), Acad. Press N. Y.279-319.

[29] Simonovits, M. (1983) Extremal Graph Theory. In: Beineke and Wilson (ed.) Selected Topicsin Graph Theory, Academic Press, London, New York, San Francisco 161-200.

[30] Sos, V. T. (1969) On extremal problems in graph theory. Proc. Calgary International Conf onCombinatorial Structures and their Application 407^10.

[31] Szemeredi, E. (1972) On graphs containing no complete subgraphs with 4 vertices (in Hungar-ian). Mat. Lapok 23 111-116.

[32] Szemeredi, E. (1978) On regular partitions of graphs. In: Bermond, J. et al. (ed.) ProblemesCombinatoires et Theorie des Graphes, (Proc. Conf. Orsay 1976), CNRS Paris 399-401.

[33] Turan, P. (1941) On an extremal problem in graph theory (in Hungarian). Matematikai Lapok48 436-452.

[34] Turan, P. (1954) On the theory of graphs. Colloq. Math. 3 19-30.[35] Turan, P. (1969) Applications of graph theory to geometry and potential theory. In: Proc.

Calgary International Conf. on Combinatorial Structures and their Application 423-434.[36] Turan, P. (1972) Constructive theory of functions. Proc. Internat. Conference in Varna, Bulgaria,

1970, Izdat. Bolgar Akad. Nauk, Sofia.[37] Turan, P. (1970) A general inequality of potential theory. Proc. Naval Research Laboratory,

Washington 137-141.[38] Turan, P. (1989) Collected papers of Paul Turan Vol 1-3, Akademiai Kiado, Budapest.[2-9] Zykov, A. A. (1949) On some properties of linear complexes. Mat Sbornik 24 163-188. (Amer.

Math. Soc. Translations 79 (1952)).

Nearly Equal Distances in the Plane

PAUL ERDOSt, ENDRE MAKAI* and JANOS PACH«f Mathematical Institute of the Hungarian Academy of Sciences

t Department of Computer Science, City College, New York, and Mathematical Institute of the HungarianAcademy of Sciences

For any positive integer k and e > 0, there exist nk e, ck e > 0 with the following property.Given any system of n > nk (J points in the plane with minimal distance at least 1 and any tv

f2, ..., tk ^ 1, the number of those pairs of points whose distance is between t( and /,- + fA. ,. \ nfor some 1 < / < fc, is at most (n2/2) (1 — l/(k+ l) + e). This bound is asymptotically tight.

1. Introduction

Almost fifty years ago the senior author [1] raised the following problem: given n points inthe plane, what can be said about the distribution of the Q) distances determined by them?In particular, what is the maximum number of pairs of points that determine the samedistance? Although a lot of progress has been made in this area, we are still very far fromhaving satisfactory answers to the above questions (cf [4], [6], [7] for recent surveys).

Two distances are said to be nearly the same if they differ by at most 1. If all points ofa set are close to each other, all distances determined by them are nearly the same (nearlyzero). Therefore, throughout this paper we shall consider only separated point sets P, i.e.,we shall assume that the minimal distance between two elements of P is at least 1. In [3]we have shown that the maximum number of times that nearly the same distance can occuramong n separated points in the plane is [n2/4\, provided that n is sufficiently large. In fact,a straightforward generalization of our argument gives the following.

Theorem 1. There exists c1 > 0 and nx such that, for any set {px, p2, ..., pn) c: R2 (n n^

with minimal distance at least 1 and for any real /, the number of pairs {/?,., p}) whose distance

j)£[t, t + cx V«] is at most [n2/4]. (Evidently, the statement is false with, say, c\ = 2.)

Research supported by NSF grant CCR-91-22103 and OTKA-1907, 4269 and 326 04 13.

284 P. Erdos, E. Makai and J. Pach

The aim of the present note is to establish the following result.

Theorem 2. Given any positive integer k and e > 0, one can find a function c{n) tending toinfinity and an integer n0 satisfying the following condition: for any set {p19 p2, ..., pn} <= U2

(n ^ n0) with minimal distance at least 1 and for any reals t19 t2, ..., tk, the number of pairs{/?., pd) whose distance

is at most (n2/2) (l-\/(k+l) + e).

To see that this bound is asymptotically tight, let P = {(iNJ): 0 ^ / ^ k,1 ^ 7 ^ n/(k + 1)}, where TV is a very large constant. Now |P| ^ n, and the distance betweenany two points of P with different x-coordinates is nearly /TV for some I ^i^k. Hence,there are at least (n2/2)(l — \/(k + l)H-o(l)) point pairs such that all distances determinedby them belong to the union of the intervals [/TV, /TV4-1], 1 ^ / ^ k.

Let K^2 denote a (k 4- 2)-uniform hypergraph whose vertex set can be partitioned intok + 2 parts V(Kk

m+

]2) =F 1 UF 2 U. . .U Vk+2, \Vt\ = m (1 ^ / ^ k + 2), and K^2 consists of all

(k + 2)-tuples containing exactly one point from each V{. Our proof is based on thefollowing two well-known facts from extremal (hyper)graph theory.

Theorem A. [5, Ch. 10, Ex. 40]. Any graph with n vertices and (n2/2) (1 - \/{k + 1) 4- e) edgeshas at least e((k+ \)\/{k+ l)k+1)nk+2 complete subgraphs on k + 2 vertices.

Theorem B. [2] For n ^ (k + 2)m, any (k + 2)-uniform hypergraph with n vertices and at leastnk+2-a/m) +1 hypere(}ges contains a subhypergraph isomorphic to Kk%.

In the final section, we will show that Theorem 2 is valid with c{n) = cke \/n for a suitableconstant cke > 0. Our main tool will be a straightforward generalization of Szemeredi'sRegularity Lemma. Given a graph G whose edges are coloured by k colours, and twodisjoint subsets F1? V2 c F(G), let er(Vx, V2) denote the number of edges of colour r with oneendpoint in Vx and the other in V2. The pair {Vx, V2) is called S-regular if

< S for every 1 ^ r ^ k,

and for every V[ c K15 V2 c V2 such that \V[\ ^ *|KX|, \V2\ ^ S\V2\. We say that the sizesof V1 and V2 are almost equal if 11 Vx\ - \ V2\ \ ^ 1.

Theorem C. [8] Given any S > 0 and any positive integers k,f there exist F = F(S,k,f) andn0 = no(S,k,f) with the property that the vertex set of every graph G with \V(G)\ > n0, whoseedges are coloured by k colours, can be partitioned into almost equal classes V19 V2,..., Vg suchthat / ^ g ^ F and all but at most Sg2 pairs {V^ Vj} are S-regular.

Nearly Equal Distances in the Plane 285


The proof is by induction on k. For k = 1 the assertion is true (Theorem 1), so we canassume that k ^ 2, e > 0, and that we have already proved the theorem for k— 1 with anappropriate function ck_x e{n) -> oo.

Fix a set P = {px,p2, •••5/?n} — ^2 with minimal distance at least 1, and suppose that there

are reals tx, t2, ...,tk such that the number of pairs {p0Pj} with

is at least (n2/2)(l — \/(k+ l) + e). We are going to show that one can specify the functionc(n) ^ ck_x e(ri) tending to infinity so as to obtain a contradiction if n is sufficiently large.

Lemma 2.1. If c(n) = o{s/n), then min / r /v«-> oo as n tends to infinity.

Proof. Assume that, for example, tk ^ C \/n. For any pt, the number of points pj withd(Pi,Pj) e [tk, tk + c(n)] is at most 100 (tk + c(n)) c(n). Hence the number of point pairs whosedistances belong to \JrZl[tr, tr + ck_ltE(^)] ^ \}k

rZ{[tr<>tr + c(ri)] is at least

provided that n is sufficiently large. This contradicts the induction hypothesis. •

Lemma 2.2. Suppose c(n) = o(\^n). Then one can choose disjoint subsets P} c= P(1 ^ / ^ k + 2) such that \Pt\ > bk e(\ogn)1/{k+1) for a suitable constant bk e > 0, and thefollowing condition holds: for any 1 ^ / =1= j ^ k + 2, there exists 1 r(ij) ^ k such thatr(ij) = r(j\ i) and

dipvpjelt^^t^ + cin)] for all p^PêPy

Proof. Let G denote the graph with vertex set P, whose two vertices are connected by anedge if and only if their distance belongs to \Jk

r=1[tr,tr + c(n)]. By Theorem A (in theIntroduction), we know that G contains at least e(n/(k + 2))k+2 complete subgraphs Kk+.1 onk + 2 vertices. Since for a random partition {Px,..., Pk+2\ of P the number of the above A .+./smeeting each Pt in one point is at least d(k)e(n/(k + 2))k+2, where d(k) > 0 we can supposethis inequality for a fixed partition {Px, ...,PA.+2} of P.

Let Kk+2 be such a subgraph with vertices ps ,ps, ...,ps (pêP^. Then for anyl</=|=y<fe + 2, there exists 1 r(ij) ^ k such that d(ps.,ps)e\tr{ii),tr{in + c(n)\ Thesymmetric array (r(ij'))1î:¥j<k+2 is said to be the type of Kk+2. Since the number ofdifferent types is at most k( *~\ we can choose at least d(k)(e/k{k+2)')nk+2 complete subgraphsKk+2 having the same type. Applying Theorem B to the (A: + 2)-uniform hypergraph Hformed by the vertex sets of these d(k) (e/k{k+2)')nk+2 complete subgraphs, we can show thatH contains a subhypergraph isomorphic to K(

kl\ with m ^ b{, ((\ogn)lnk+1\ for a suitableconstant bk( > 0. From this the assertion readily follows. •

286 P. Erdos, E. Makai and J. Pack

In what follows, we shall analyze the relative positions of the sets P{(\ ^ / ^ & + 2)

described in Lemma 2.2. Consider two sets (Px and P2 say), and assume that all distances

between them belong to the interval [tl9 tx + c(n)]. For any /?, p' e Pl9 all elements of P2 must

lie in the intersection of two annuli centred at/? a n d / / . If d(p,p') < 2/1? c(n) = o(\ n), then

(by Lemma 2.1) the area of this intersection set is at most

and, using the notation m(n) = bk e(\ogn)1/ik+1\ we have

50tlc\n)m(n)

Assuming that c(n) = o(\rn(nf), this immediately implies that d(p,p/)/t1 is either close to0 or close to 2. More exactly,

2 -m(n)

for any /?, p' ePx, provided that n is large enough.Now pick any point qeP2. Px must be entirely contained in the annulus around q whose

inner and outer radii are tx and t1 + c(n), respectively. Thus, if Px has two elements withd(p,pf) ^ (2 — 50c2(n)/m(n)) tx, all other points of Px must lie in the union of the two circlesof radius (50c2(n)/m(n)) /1 centred at/? and/?'. In any case, there is an at least m(«)/2-elementsubset P\ ^ px whose diameter

m(n)

Repeating this argument (A:+ 2 times), we obtain the following.

Lemma 2.3. Let m(n) = bke (log n)1/ik+1\ c(n) = o{\m(nj). Then one can choose disjointsubsets Qi ^ P, |2/| ^ m(n)/2 (1 ^ / ^ k + 2) such that the following conditions are satisfied:(/) For any 1 ^ / =j=/ ^ A'+ 2, r/ r<p exists 1 ^ r(/,/) = r(/ , /) ^ A ^wc/z that

j { J ) { f J ) for all pteQt, p}eQ^

(//) For any 1 ^ / ^ / r + 2,

diam Qi = o( 1) min ^ ( . ; ) ;

(Hi) There is a line f such that the angle between f and any line pipj (p^eQ^ PjeQr i +j)is o(l).

Proof. We only have to prove part (iii). Fix two subsets Qt and Qj (i =)=/). By (ii),

max (diam Qt, diam Qj) = o(\)tni jv

so the angle between any two lines /?,/?. and p\p] (p^p-eQr, p.pp]eQ}\ i 4=7) is o(\).


Let qi and q{ be two elements of Qt whose distance is maximal. Clearly, for any j + /,

——— ^ d(q,, # •) = diam Q, ^ o(\) trU u.10

It is sufficient to show that for axvy p^eQp the lines qtqt and qipj are almost perpendicular.Indeed,

2d(q,,p))d(ql,q'i)

c(n)(2tr(tJ) + 2cD

We need the following key property of the sets Qi constructed above.

Lemma 2.4. Suppose c(n) = o{\/n). Let s ^ 3 be fixed, and suppose that

diam(Qx U Q2 U ... U Qs) = d(p1,p2) for some p1eQ1, p2eQ2.

Then for any 1 ^ / 4=7r s, r{ij) = r(l,2) if and only if {i,j} = {1,2}.

Proof. Suppose, in order to obtain a contradiction, that there are two points p\ e Qf, p] e Qr

2 ^ i =|= j ^ s such that

By Lemma 2.1 and Lemma 2.3 (iii), all points of Q., U Q:i U ... U Q, lie in a small sector (ofangle o{\)) of the annulus around px whose inner and outer radii are \n and dip^p.J.respectively. Obviously, the diameter of this sector is d(u,v), where u (resp. v) is theintersection of one (the other) boundary ray with the inner (outer) circle of the annulus. Butthen we have

d(pl,p2)-d(p'i,p-) 2* dip^p^-diu.v) = d(pvu)-d(u,v)

_ 2d(px, u) d(pv v) cos (L upx v) - d\p19 u)

>d(p^ u) cos ( L upx v) - Pl'U

^ \ n(\ -o(\))-—"—>—-> c(n),

which is the desired contradiction. •

Now we can easily complete the proof of Theorem 2. For the sake of simplicity weassume the intervals are disjoint, but the same arguments work in the general case as well.Assume, without loss of generality, that the diameter of Q = Qx U Q2 U ... U Qk+2 is attainedbetween a point of Qx and a point of Qj, for some j \ > 1. By Lemma 2.4, no distance

determined by the set Q = Q2 U Q3 U ... U Qk+2 belongs to the interval [tr(1J )9 tr{lj } + c(n)].Suppose that the diameter of Q' is attained between a point of Q2 and a point of QJ2J2 > 2.Applying the considerations of the lemma again, we obtain that none of the distancesd e t e r m i n e d b y Q " = Q3 U Q 4 U ... U Qk+2 is in [tr(2j),tr{2j)) + c(n)], w h e r e r(2J2) 4= r ( l , j \ ) .Proceeding like this, we can conclude that no distance determined by Qk+1 U Qk+2 belongsto

where {r(/,y?.): 1 < / ^ A:} = {1,2,..., k}. In other words, there exists no integer r(k + 1, k + 2)satisfying the condition in Lemma 2.3(i). This contradiction completes the proof ofTheorem 2 for any function c(n) = o((\ogn)1/i2k+2)). In fact, our argument also shows thatthere is a small constant ck e > 0 such that the theorem is true with c(n) = ck e(\ogn)1/{2k+2).

•3. Strengthening of Theorem 2

In this section we are going to modify the above arguments to show that Theorem 2 isvalid for any function c(n) = o(\/n). Notice that in the previous section we have not reallyused the fact that all distances between Qt and Qj (in Lemma 2.3) belong to the interval[tr{i j)9 tr(i j} + c(n)]. It is sufficient to require that many distances have this property, and thereare much larger subsets Q{ (1 ^ / ^ k + 2) satisfying this weaker condition. As a matter offact, we can assume that \Qt\ ^ m(n) = b* tn for a suitable constant b* b > 0, and followessentially the same argument as before for any c(n) = o(\/m(n)) = o^n).

In the following we shall assume that k, e < 1 and 3 < (e/\00k)k+b are fixed, c(n) = o(\n),and n is very large, and again we will argue by contradiction. We want to apply TheoremC (in the Introduction) to the graph G on the vertex set P whose two points /?, p' areconnected by an edge of colour r whenever

and r is minimal with this property. Then Lemma 2.3 can be replaced by the following.

Lemma 3.1. There is a constant b = b(k,e,S) such that there exist disjoint subsets Qf <= P,\Qt\ > bn (1 ^ i ^ k + 2) satisfying the following conditions.

(/) For any 1 ^ / 4=7 ^ k + 2, one can find 1 ^ r(ij) = r(j\ i) ^ k such that

\Q,\-\Qi\ 20/c'

(//') For any 1 i ^ k + 2,

diam Qi = 6>(l)min tr{i jyj * i

(Hi) There is a line £ such that the angle between £ and any line ptp^ (pt e Qt, Pj e Qj) is o(l).

Proof. Consider a partition V(G) = P = Vl[] V2[j ... [) Vg meeting the requirements ofTheorem C w i t h / = [10/el. Let G* denote the graph with vertex set V(G*) = {K15 V2,..., Vg},where Vt and Vj are joined by an edge if {Vi9 V^ is a ^-regular pair and

(1):|-|K;| 10A:

for some 1 r(ij) = r(j\ i) ^ k. Clearly,

\E(G*)\ + 8f^%)^\'i\+gMf\

whence

By Theorem A (or by Turan's theorem [T]), this implies that G* has a complete subgraphon k + 2 vertices, say, V19 V2,..., Vk+2.

Assume, without loss of generality, that r(l,2) = 1, t1 = mmj + 1traj), and let Gr denotethe subgraph of G consisting of all edges of colour r. By (1), at least (e/lOA:) | ^ | • | K2| edgesof Gx run between V± and V2. Therefore, we can pick a point p2e V2 connected to allelements of a subset Px c F19 \PX\ ^ (e/10/c) | Vx\. Clearly, Px lies in an annulus centred at p2

with inner radius tx and outer radius tx + c(n). Using the fact that {Vv V2} is a ^-regular pair,it can be shown by routine calculations that there are (e/100&)4 |Z\|2 pairs {p^p^} <= Px suchthat px and p[ have at least (e/100A:)2 \V2\ ^ (\/F(8,kJ)) (e/\00k)2n common neighboursin G r As in the proof of Lemma 2.3, we can argue that, for any such pair,

p'1) = (2-o(l))t1.

Hence, we can find a point px e P1 such that

= 0(1)^)1;

or

Kp'reP^diPvp'J = (2-o(l))/

Let Qx ^ px denote the larger of these two sets. Then

1 / P \ 5

n, (2)ioo^;' 1] \

and diam Qx = o(\) tx. Repeating the same argument for every Vt (1 ^ / ^ A:+ 2), we obtaing^ ^ ^ satisfying conditions (i) and (ii).

To establish (iii), notice that the angle between any two lines pipj and p\p\ (p^p^eQ^;pp p\eQ^ i + / ) is o(\). Using the fact that {F19 Vj} is ^-regular for all 2 ^ 7 < A: + 2, one


can recursively pick pj e Q^ so that

J for all 2^j^

J^ F(S,k,f){l00k)

Thus, two elements of this set, ql and q[ (say), are relatively far away from each other:

This in turn implies, in the same way as in the proof of Lemma 2.3 (iii), that

i.e., every linepxp} (pleQ1,p}eQjJ 4= 1) is almost perpendicular to the line q1q[. Applyingthe same argument for Q2, Q3,... (instead of Qx), we obtain (iii). •

Using Lemma 3.1, (i) and \Qt\ > (e/XOOkflV^ ^ S\V{\ we can see (using induction), thatthere are const(k,e,S)nk+2 (& + 2)-tuples (qx, ...,qk+2), with qteQt, such that

(For details cf. [7].) Fix one of them. Then repeating the considerations of Lemma 2.4 forthis (/c + 2)-tuple only, we get that the assertion of Lemma 2.4 is valid also now. Then theproof of Theorem 2 can be completed in exactly the same way as in the previous sectionwith any function c(n) = o(\H). As a matter of fact, in order to apply our argument, it issufficient to assume that c(n) ^ ck( \7i for a suitable constant ckt > 0. •

References

[1] Erdos, P. (1946) On sets of distances of n points. Amer. Math. Monthly 53, 248-250.[2] Erdos, P. (1965) On extremal problems for graphs and generalized graphs. Israel J. Math. 2,

183-190.[3] Erdos, P., Makai, E., Pach, J. and Spencer, J. (1991) Gaps in difference sets and the graph of

nearly equal distances. In: Gritzmann, P. and Sturmfels, B. (eds.) Applied Geometry and DiscreteMathematics, the Victor Klee Festschrift, DIMACS Series 4, AMS-ACM, 265-273.

[4] Erdos, P. and Purdy, G. (to appear) Some extremal problems in combinatorial geometry. In:Handbook of Combinatorics, Springer-Verlag.

[5] Lovasz, L. (1979) Combinatorial problems and exercises, Akad. Kiado, Budapest, NorthHolland, Amsterdam-New York-Oxford.

[6] Moser, W. and Pach, J. (1993) Recent developments in combinatorial geometry. In: Pach, J.(ed.) New Trends in Discrete and Computational Geometry, Springer-Verlag, Berlin 281-302.

[7] Pach, J. and Agarwal, P, K. (to appear) Combinatorial Geometry, J. Wiley, New York.[8] Szemeredi, E. (1978) Regular partitions of graphs. In: Problemes Combinatoires et The'orie de

Graphes, Proc. Colloq. Internat. CNRS, Paris 399-401.[9] Turan, P. (1941) Eine Extremalaufgabe aus der Graphentheorie. Mat. Fiz. Lapok 48, 436-452.

(Hungarian, German summary.)

Clique Partitions of Chordal Graphs

PAUL ERDOS*, EDWARD T. ORDMAN5

and YECHEZKEL ZALCSTEINU

* Mathematical Institute, Hungarian Academy of Sciences ,

$ Memphis State University, Memphis, TN 38152 U.S.A.

Division of Computer and Computation Research, National Science Foundation,

Washington, D.C. 20550, U.S.A.

To partition the edges of a chordal graph on n vertices into cliques may require as manyas n2/6 cliques; there is an example requiring this many, which is also a threshold graphand a split graph. It is unknown whether this many cliques will always suffice. We are ableto show that (1 — c)n~/4 cliques will suffice for some c > 0.

1. Introduction

We consider undirected graphs without loops or multiple edges. The graph Kn on nvertices for which every pair of distinct vertices induces an edge is called a complete graphor a clique on n vertices. If G is any graph, we call any complete subgraph of G a cliqueof G (we do not require that it be a maximal complete subgraph). A clique covering ofG is a set of cliques of G that together contain each edge of G at least once; if eachedge is covered exactly once we call it a clique partition. The clique covering number cc(G)and clique partition number cp(G) are the smallest cardinalities of, respectively, a cliquecovering and a clique partition of G.

The question of calculating these numbers was raised by Orlin [13] in 1977. DeBruijnand Erdos [6] had already proved, in 1948, that partitioning Kn into smaller cliquesrequired at least n cliques. Some more recent studies motivating the current paper include[11, 14, 2, 7,9].

It is widely known that a graph on n vertices can always be covered or partitionedby no more than n2/4 cliques; the complete bipartite graph actually requires this many.

' This work was done at Memphis State University.

§ Partially supported by U.S. National Science Foundation Grant DCR-8503922.

™ Partially supported by U.S. National Science Foundation Grant DCR-8602319 at Memphis State University.

292 P. Erdos, E. T. Ordman and Y. Zalcstein

Turan's theorem states that if G has more than n2/4 edges, it must contain a clique X3;if it has more than n2(c — 2)/(2c — 2) edges it must contain a Kc. (For a more precisestatement and proof, see e.g. [3, Chapter 11].)

A subgraph H of a graph G is an induced subgraph if for any pair of vertices a and bof H, ab is an edge of H if and only if it is an edge of G.

Two classes of graphs we shall refer to here are chordal graphs and threshold graphs.A graph is chordal (or often triangulated; [10, Chapter 4]) if every cycle of size greaterthan 3 has a chord (no set of more than 3 vertices induces a cycle). A graph G is threshold([10, Chapter 10; 4; 5; 12]) if there exists a way of labelling each vertex A of G with anonnegative integer f(A) and there is another nonnegative integer t (the threshold) suchthat a set of vertices of G induces at least one edge if and only if the sum of their labelsexceeds t.

A graph is split if its vertices can be partitioned into two sets A and B such that thevertices A form a clique and the vertices B induce no edges. (Two vertices, of which oneis in A and one is in B, may or may not induce an edge.)

All threshold graphs are split and all split graphs are chordal. In a sense, most chordalgraphs are split [1]. Induced subgraphs of chordal graphs are chordal; similar results holdfor split graphs and threshold graphs.

2. Preliminary results on split graphs

A complete matching in a graph G is a set of edges such that each vertex of G lies onexactly one edge in the set. It is well known that the t(2t — 1) edges of K2t can beedge-partitioned by a set of 2t — 1 matchings, each of t edges. By the join of two graphsG and //, we mean the graph made by taking the disjoint union of the two graphs andadding all edges of the form g/i, where g is a vertex of G and h is a vertex of H.

By the graph Kn — Km, for n > m, we mean a graph made by taking Kn and deletingall the edges induced by some particular m of the vertices. Equivalently, this is the join ofKn_m with the complement of Km (a collection of m isolated vertices).

Lemma 2.1. Let G = K4t - K2t. Then cp(G) < t(2t + 1).

Proof. Think of G as a complete graph A = K2t joined completely to an empty graph Con 2t vertices. Partition A into 2t—l disjoint matchings; join each matching to a differentvertex in C, each matching yielding t triangles. The remaining vertex in C lies on 2t singleedges to A. Thus we partition G by t(2t — 1) triangles and 2t single edges, a total of2(2t + 1) cliques. •

In fact, cp(G) = t(2t + 1). See, for example, [7].

Lemma 2.2. In the graph G of the previous lemma, suppose r edges are deleted. Then thisnew graph has clique partition number not exceeding t(2t + 1) + r.

Clique Partitions of Chordal Graphs 293

Proof. Start with the same partition as above. Each edge deletion at worst demolishesone triangle, requiring it to be replaced in the partition by two edges. •

3. Preliminary results on chordal graphs

We will rely heavily on the following lemma of Bender, Richmond, and Wormald, whichgives a means of constructing an arbitrary chordal graph.

Lemma 3.1. [1, Lemma l.J For each chordal graph G and each clique R of G there is asequence

R = G r , G r + i , . ..,Gn = G

of graphs such that G,-+i is obtained from G, by adjoining a new vertex to one of its cliques.

Corollary 3.2. If G is a chordal graph on n vertices with largest clique of size r, then G canbe covered by at most n — r -\- \ cliques.

It is easy to see that the bound in the corollary cannot be improved; Kn — Kn-r+\ is anexample requiring n — r + 1 cliques to cover.

Covering G may require less than n — r + 1 cliques. If G consists of two copies of Kt

with a single vertex in each identified, G has 2t — 1 vertices, the largest clique is of size f,this corollary produces a covering by (2t — 1) — t + 1 = t cliques, but obviously there is acovering (and for that matter a partition) by two cliques.

We now utilize this construction with one additional specialization: we begin witha clique of maximum possible size in G. Supposing this clique to be of size r, eachsubsequently added vertex will add, at the time it is adjoined, at most r — 1 edges (or itwould form a clique of more than r vertices).

Corollary 3.3. A chordal graph on n vertices with a largest clique having r vertices has atmost (n-r)(r-l) edges outside that clique.

Theorem 3.4. Let G be a chordal graph on n vertices and 1/4 > d > 0. Suppose G has atleast dn2 edges. Then G contains a clique with at least (1 — y/\ — 2d)n > dn vertices.

Proof. If the largest clique in G contains en vertices, then that clique contains cn(cn— l)/2edges and each of the remaining n — en vertices of G can be added to G adding at mostcn—\ edges at each stage. Hence the total number of edges of G is at least dn2 and at mostcn(cn - l)/2 + (en - \)(n - en), so dn < (2c - c2)(n/2) + (c - 2)/2 and dn < (2c - c2)(n/2)since c < 1. Hence d < (2c — c2)/2 and c > 1 — y/\ — 2d > d as needed. •

The result of this theorem turns out to be essentially best possible, not only for chordalgraphs, but for split graphs and threshold graphs as well.

Example 3.5. Let 0 < c < 1. Consider the graph Kn — K^ where k = n — en + 1, thatis, the base clique has en — 1 vertices and forms a clique on en vertices with each other


vertex. Clearly there are

(en - \)(cn - 2)/2 + (n - en + \)(cn - 1) = (c - c2/2)n2 - (1 - c/2)w

edges. So a graph can be threshold (hence split and chordal) and have almost (c — e2/2)n2

edges and no clique on more than en vertices.

4. Clique partitions of chordal graphs

An arbitrary graph on n vertices may require n2/4 cliques to cover or partition it [8].We saw above that a chordal graph on n vertices may always be covered by fewer thann cliques. It may, however, still require a large number of cliques to partition it. Theexamples in [7] with high clique partition numbers are chordal graphs.

Example 4.1. [7] The graph Kn — K2n/3 requires n2/6 + n/6 cliques to partition it and2n/3 cliques to cover it. Thus for a chordal graph, both cp(G) and cp(G) — cc(G) can beapproximately n2/6.

We note that for a different example, the ratio of cp(G) to ec(G) may be larger.

Example 4.2. [7] The graph Gn composed of 3 cliques Kn/^ with all vertices of the firstclique attached by edges to all vertices of the second and third, is a chordal graph (butnot a split graph or threshold graph). As n increases, cp(Gn)/cc(Gn) grows at least as fastas en2 for some c > 0.

We do not know if cp(G) can significantly exceed n2/6 for a chordal graph, or even fora split graph or a threshold graph.

Conjecture 1. The clique partition number of a chordal graph, split graph, or threshold graphon n vertices cannot exceed n2/6 (except by a term linear in n).

It is even possible that Kn — Kin/i is literally the best example. (Some very minoradjustments to n2/6 + n/6 may be needed because of round-off error). However, it isunclear how one would go about proving the following:

Conjecture 2. No chordal, threshold, or split graph on n vertices requires more thancp( Kn — K2n/3) cliques to partition it.

For chordal graphs in general, we are very far from proving that n2/6 cliques will sufficefor a partition. In fact, we can improve only slightly on n2/4.

Theorem 4.3. There is a constant c > 0 such that if G is a chordal graph with n vertices, Gmay be partitioned into no more than (1 — c)n2/4 cliques.

Proof. As the details are messy, we first give an outline; we follow this by some indicationof more precise calculations, which the reader may choose to ignore, and a few numericindications. As the result is clear for n < 5, we assume n > 5 in the proof. Let the largestclique in G have (1 + a)n/2 vertices (a may be negative). Pick such a clique and call it A.Let C denote the subgraph of G induced by those vertices not in A; the set of edges notin A or C will be denoted B.

In case 1, the large clique is larger or smaller than half the vertices by a reasonableamount (a2 > c). By Corollary 3.3, there are so few edges outside A that we can coverthem by single edges. In case 2, A has close to half the vertices, and C has a significantnumber of edges. By Theorem 3.4, C contains a large clique C'; we can cover by A, C",and single edges. In case 3, A has close to half the vertices and C has few edges; in thiscase the graph must be very similar in form to Kn — Kn/2 and Lemma 2.2 can be used toconstruct a partition with 'little more than' H 2 /8 triangles and edges.

We now give somewhat more precise calculations.

1 If a2 > c, we can cover A with one clique and each edge not in A by a single edge.The number of edges outside A is at most

(1 - a)(n/2)((\ + a)n/2 - 1) < (1 - a2)n2/4 < (1 - c)n2/4

as desired. Hereafter, we suppose a2 < c.2 If C has very many edges, we can cover A with a clique, the largest clique in C

with a clique, and all other edges singly. Suppose C has dn2 edges. Then, since Cis an induced subgraph of G, it is a chordal graph with v = (1 — a)n/2 vertices anddn2 = (dn2/((\ —a)n/2)2)v2 edges; so by Theorem 3.4 it contains a clique with at least(dn2/((\ - a)n/2)2)v = 2dn/(\ - a) vertices and (2(dn)2 - dn(\ - a))/{\ - a)2 edges.Covering this clique by itself, A by a clique, and each remaining edge with an edge,we get a number of cliques guaranteed to be less than

2 + (1 - a2)n2/4 - (1 - a)n/2 - (2(dn)2 - dn{\ - a))/(I - a)2

= (\-a2- 8d2/(\ - a)2)n2/4 + 2 - (1 - a)n/2 + dn/(\ - a)

Now supposing c < .01, \a\ < .1, n > 4, and d < .04, we see that

-a) + a/2< 1/2,

so

2 - ( l -a)n/2 + dn/(\ - a) < 0

and we need only have

1 -a2 - 8 d 2 / 0 - a ) 2 <\-c

to finish, which is clearly true if d2 > (c — a2){\ — a)2/8. If that condition is met,we are done. Hereafter, we assume that d2 < (c — a2)(\ — tf)2/8, and hence thatd2 < c(\ + yjc)2/8. In particular, as c nears 0, so does d.In the remaining case, we will cover the edges in C by single edges, and cover the edgesin B and A by triangles and single edges using the technique of Lemma 2.2. Considerthe number of edges in B. Since B and C together must have at least (1 — c)n2/4

edges and C has no more than dn2 edges, we see that B has at least (1 — c — 4d)n2/4edges. In the 'complete' graph H = Kn — K{i_a)n/2 there are (1 — a2)n2/4 edges inB, so we see that if we can partition H by edges and triangles, we can partition Gwith only a few extra cliques: dn2 for the edges in C and an allowance of at most(1 - a2)(n2/4) - (1 - c - 4d)(n2/4) = (c - a2 + 4d)n2/4 for the 'missing' edges of B.We now set out to clique-partition H. We neglect some constant multiples of n toreduce the bulk of the expressions below. As in Lemma 2.1, partition A = K{i+a)n/2

into (1 + a)(n/2) - 1 matchings of (1 + a)(n/4) edges each (if (1 + a)n/2 is odd, thereis an extra linear factor in n neglected below). We must consider two subcases, a > 0and a < 0.If a > 0, we join {\—a)n/2 of these matchings to distinct points in C to form (1— a2)n21%triangles consuming all the connecting (B) edges of H\ this leaves {2a)(\ + a)n2/8edges of A unused and we cover them with single edges. Thus we partition H with(1 — a2){n2/S) + a(\ + a)(n2/4) triangles and edges. This means we obtain a cliquepartition of G using no more cliques than

(1 - a2)(n2/S) + a{\ + a)(n2/4) + dn2 + (c - a2 + 4d)n2/4

= (n2/4) [(1/2)(1 - a2) + a(l + a) + 4d + (c - a2

But it is easy to see that as c approaches 0 so that a and d also approach 0, thisexpression approaches (n2/4)[l/2 + 0 + 0 + 0], so it can clearly be made less than(n2/4)[\ — c] as required.If a < 0, we are able to join all the (l+a)(n/2) — 1 matchings in A to distinct points in C.The resulting (1 +a)2n2/8 triangles consume all (except a constant multiple of n) of theedges of A but only (\+a)2n2/4 edges of 5, leaving as many as (1— a2)n2/4—(\+a)2n2/4to cover with single edges. Thus we partition H into

(1 + a ) V / 8 ) + (1 - a2)(n2/4) - (1 + a ) V / 4 )

cliques (which approaches (l/2)n2/4 as c approaches 0), and the rest of the argumentgoes exactly as in the prior paragraph.A somewhat more careful calculation suggests that letting c — 1/400 will easily sufficefor n > 5, forcing \a\ < .05 by case 1 and d < .02 by case 2. Unfortunately, linearterms neglected here, such as (1 + a)n/4, complicate the actual calculation of c badlyfor low values of n. •

If we require G to be threshold, or split, the situation simplifies somewhat, since C willcontain no edges and case (2) becomes unnecessary. Still, this method appears to produceonly a marginal improvement in the c in these cases. The first two authors and Guan-TaoChen have made some further progress in the case that G is a split graph, but are stillnot close to n2/6; this will be pursued elsewhere.

References

[1] Bender, E. A., Richmond, L. B. and Wormald, N. C. (1985) Almost all chordal graphs split. J.Austral Math. Soc. (A) 38 214-221.


[2] Caccetta, L., Erdos, P., Ordman, E. and Pullman, N. (1985) The difference between the cliquenumbers of a graph. Ars Combinatoria 19 A 97-106.

[3] Chartrand, G. and Lesniak, L. (1986) Graphs and Digraphs, 2nd. Edition, Wadsworth, Belmont,CA.

[4] Chvatal, V. and Hammer, P. (1973) Set packing and threshold graphs, Univ. of Waterloo ResearchReport CORR 73-21.

[5] Chvatal, V. and Hammer, P. (1977) Aggregation of inequalities in integer programming. Ann.Discrete Math 1 145-162.

[6] DeBruijn, N. G. and Erdos, P. (1948) On a combinatorial problem. Indag. Math. 10 421-423.[7] Erdos, P., Faudree, R. and Ordman, E. (1988) Clique coverings and clique partitions. Discrete

Mathematics 72 93-101.[8] Erdos, P., Goodman, A. W. and Posa, L. (1966) The representation of a graph by set intersec-

tions. Canad. J. Math. 18 106-112.[9] Erdos, P., Gyarfas, A., Ordman, E. T. and Zalcstein, Y. (1989) The size of chordal, interval, and

threshold subgraphs. Combinatorica 9 (3) 245-253.[10] Golumbic, M. (1980) Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York.[11] Gregory, D. A. and Pullman, N. J. (1982) On a clique covering problem of Orlin. Discrete Math.

41 97-99.[12] Henderson, P. and Zalcstein, Y. (1977) A graph theoretic characterization of the PVchunk class

of synchronizing primitives. SI AM J. Comp. 6 88-108.[13] Orlin, J. (1977) Contentment in Graph Theory: covering graphs with cliques. Indag. Math. 39

406-424.[14] Wallis, W. D. (1982) Asymptotic values of clique partition numbers. Combinatorica 2 (1) 99-101.

On Intersecting Chains in Boolean Algebras

PETER L. ERD6Sf, AKOS SERESS* and LASZLO A. SZEKELY^

Centrum voor Wiskunde en Informatica, P.O. Box 4079, 1009 AB Amsterdam, The Netherlands

*The Ohio State University, Columbus, OH 43210

^University of New Mexico, Albuquerque, NM 87131

Analogues of the Erdos-Ko-Rado theorem are proved for the Boolean algebra of allsubsets of {l,...n} and in this algebra truncated by the removal of the empty set and thewhole set.

1. Introduction

One of the basic results in extremal set theory is the Erdos-Ko-Rado (EKR) Theorem [5]:if 3F is an intersecting family of /c-element subsets of [l,n] = {1,2, ...,n} (i.e. every twomembers of 3F have non-empty intersection) and n > 2/c, then \^\ < (£~J) and this boundis attained. We can consider /c-subsets of [l,n] as length-/c chains in the (total) order1 < 2 < ... < n: using this terminology, the EKR theorem is a result about intersecting/c-chains in a special partially ordered set.

Erdos, Faigle, and Kern [3] pointed out that certain results of Deza, Frankl [2, Theorem5.8], and Frankl and Furedi [7] on intersecting sequences of integers may be interpretedas results on intersecting families of chains in some partially ordered sets.

The purpose of this note is to prove analogues of the EKR theorem in two otherpartially ordered sets: in the Boolean algebra &n of all subsets of [l,n] (with A < Bif A a B), and in the truncated Boolean algebra &~ \— @ln \ {0, [l,n]}. We say thatif = (Li,L2,...,L/c) is a k-chain in 38n if Li e @tn for all 1 < i < fe and Li is a propersubset of L|+i for all 1 < / < k — 1. A family 3F of /c-chains in &n is intersecting if anytwo elements of J* have non-empty intersection.

/c-chains and an intersecting family in $~ are defined analogously. Let f(n,k) and

^Research partially supported by NSF Grant CCR-9201303 and NSA Grant MDA904-92-H-3046.^Research partially supported by ONR Grant N-OO14-91-J-1385.

300 P. L. Erdos, A. Seress and L. A. Szekely

f~(n,k) denote the maximum size of intersecting families of k-chains in 38n and 38~,respectively.

Obviously, the family 3F(A) of all /c-chains containing some fixed A e &n is anintersecting family, and the same is true for the family ^~(A) of all /c-chains in &~containing some fixed A e &~. Our main result is the following.

Theorem 1.1. For any /c, n, we have

(i)f(n,k) = \^(0)\ and

Moreover, for 2 < k < n + 1, the only extremal families in <%n are J^(0) and

The most well-known proof techniques for the original EKR Theorem are shifting andthe kernel method. (For a brief introduction to these methods, see e.g., the survey papersof Frankl [6] and Furedi [8].) The kernel method usually ensures short and easy proofs,but rarely gives the exact range of the result. Shifting gives exact (but perhaps slightlymore complicated) proofs.

The situation is very similar in our case: Z. Furedi (personal communication) showed,using only the kernel method, that for n > 6/cln/c Theorem 1.1 (i) holds. In our proof ofTheorem 1.1, we use an analogue of the shifting method and obtain a result without anyrestrictions on the parameters.

We remark, however, that to obtain sharp results in the case of r-intersecting familiesof chains, or the poset obtained by deleting the top m and bottom m levels in @n forsome m < n/2, it seems to be necessary to combine the two methods. Hilton-Milnertype generalizations are also possible. Moreover, we have a common generalization ofthe original EKR theorem and Theorem 1.1. We shall return to these problems in aforthcoming paper.

Let S(p,q) denote the Stirling numbers of the second kind, i.e. S(p,q) is the numberof partitions of a p-element set into q nonempty parts. It is easy to see that |«^"({1})| =k\S(n— 1,/c), since each S£ = (Li,L2,...,L/c) e tF~({\}) corresponds to an ordered partition(L2 \ LUL3 \ L2,...,Lk \ Lfc_i, [In] \ Lk) of [2,n], Similarly, |^(0) | = (k - l)\S(n + 1,/c) =(k — l)\S(n,k — 1) + /c!S(n,/c), the two last terms corresponding to the number of/c-chainsin #X0) containing and not containing [l,w], respectively.

In the proofs, we shall often use the well-known recursion

S(n,k) = S(n - 1,/c - 1) + kS(n - 1,/c)

(see e.g., [9, Chapter 1]). In particular, |^(0) | = (k - l)\S(n + 1,/c).

2. Shifting

In this section we begin the proof of Theorem 1.1. We reduce the problem to theexamination of so-called compressed sets of chains and prove that these satisfy a strongintersection property.

Let 3F be a family of pairwise intersecting /c-chains from @ln or ^~ , and let 1 < / < j < nbe integers. The (ij) chain-shift Stj(&) of the family J* is defined as follows.

On Intersecting Chains in Boolean Algebras 301

For every /c-chain ^ = (Lu... Lk) G &9 let S^) = (L\9...,L£), where

. , _ f L/ \ {7} U {/} if j G L/ and i £ Lu1 \ L\ otherwise.

We say that LJ is the shift of L/. Shifting preserves set containment, so Sij(^) is a /c-chain.The shifted family Sij(^F) is obtained by the following rule: replace every /c-chain !£ G 3Fby Sij(&) if and only if

(1) S l 7 ( i f )^ if and(2) Sij(^) i &.It is clear from the definition that \Sij(!F)\ = \3F\. Moreover, shifting preserves the

intersection property.

Lemma 2.1. If 3F is an intersecting family of k-chains in &n or @~, then Sij(^) is alsointersecting.

Proof. Let !£i, if2 € Stj(P); we have to prove that they contain a common element. Wedistinguish three cases:

Case 1: i f 1, if2 £ ^ - In this case it is obvious that ifi and if2 intersect.

Case 2: û^2 $ &- In this case, there are ^ 3 , ^ 4 € ^ such that S£\ = Si7(i^3) and^2 = Siji&t). Let M e îD ^4. Then the shift of M (which may be M itself) is acommon element of if 1 and if 2.

Case 3: <£x$& and 5£2 e P. Then let ^ 3 ^ ^ such that 5£x = ^ ( ^ 3 ) . There may betwo reasons why i^2 was not replaced. If ^2 = ^7(^2) then let M e ^2C\ i f 3. The shiftof M is itself (since i^2 = S0-(JSf2)) so M e Sf2 n S0-(JSf3) = ^ 2 n if 1 as well.

The other reason is that ^£2 ± îjiî) but Sjy-(J&?2) ^ ^ - In this subcase, let M Gif 3 n 5,7(if 2). It is impossible that 7 € M and f ^ M since M is the shift of some elementof if2. Also, it is impossible that i e M and j & M because there is some X G if3 suchthat j G K and i $. K (because 5,7(if3) ^ if 3) and one of K, M must contain the other. SoM is a set containing either both of i,j or neither of i,j. In either case, from M G Sij(^2)we have M G if2 so M G if 1 n J^2.

D

We say that the family 3F of intersecting /c-chains is compressed if J^ is invariant forall chain-shift operations 5,;, 1 < i < j < n. By Lemma 2.1, for any intersecting family 3F,repeated applications of chain-shifts result in a compressed family of the same size.

Compressed families satisfy a strong intersection property. We say that M e &n (orM G J*~) is an initial segment if M = [l,m] for some 1 < m < n or M = 0.

Lemma 2.2. Let ^ be a compressed family of intersecting k-chains. Then for any <£\,!£2 GSF, Z£\ and if2 intersect in an initial segment.

Proof. Suppose that the lemma is not true and let if 1 G #" be a minimal counterexamplein the sense that

(i) there exists if 2 G J^ such that if 1 Pi if 2 contains no initial segment

among all S£\ satisfying (i).

Let M G if i n if 2. Since M is not an initial segment, there exist 1<i < j <n such that* ^ M and y G M. Then Sij{J£\) =fc i f i , so Siy(ifi) is not a counterexample. Therefore,there exists an initial segment X G S//(if i) n if 2- It is impossible that j G X and / ^ X,since K is an initial segment. Also, it is impossible that i G K and j £ K, becauseK,M G J 2> so one of them must contain the other. So X is a set containing both of i,jor neither of ij. In either case K G if i , which is a contradiction. •

In the next two sections, we prove Theorem 1.1 for 3Sn and 3S~, respectively. ByLemma 2.1, it is enough to consider compressed families.

3. Chains in <%n

We prove by induction on n that f(n,k) = (k — l)!S(n+ l,fc). The base case n = 1 is trivial.Suppose we are done for n — 1 (with all values of k) and let $F be a compressed familyof chains in 38n. We distinguish two cases:

Case 1: <F contains a chain 5£ such that the only initial segment in i£ is [l,ri\. Then,since each chain in 3F must intersect i f in an initial segment (see Lemma 2.2), all chainscontain [l,n] and we are done.

Case 2: There is no chain in 3F such that [l,w] is the only initial segment in the chain.Then delete n from each element of each chain. Each chain is transformed into either afc-chain or a (fe — l)-chain and so we obtain an intersecting family %>k-\ of chains of lengthk — 1 in $n-\ and an intersecting family ^ of chains of length k in 3$n-\.

We claim that each (L\,...,Lk-\) G ^ - i can be obtained from < k — 1 chains of 3F.This is true since we have to add the set L, U {n} to the chain for some 1 < i < k — 1and add n to the sets LI+i,...,L^_i. The value of / uniquely determines the chain in <F.Furthermore, / = 0 is impossible, since then the only initial segment would be [l,w].

We also claim that each (Li,...,L/c) G ^ can be obtained from < k chains of J*\ Indeed,we have to add n to the sets starting at some 2 < i < k + 1; the value k + 1 correspondsto the case that n did not occur in any element of the chain in $F. Furthermore, i = 1 isimpossible, since the only initial segment would be [l,n].

Thus

1^1 <{k- l)f(n - 1,/c - 1) + kf(n - 1,/c) = (k - l)\S(n + l,k). (1)

The uniqueness of the extremal systems can also be proved by induction on n. First, weremark that if k = n + 1, every family J^ of /c-chains must contain the empty set, andmaximality implies & = ^ (0) . If k = 2 and \&\ > 4, then & c #-(/!) for some subset ANow, |^(i4)| = 2|i41 + 2"~|A| - 2, which takes its maximum value for \A\ = 0 and \A\ = n.

In the case 3 < k < n, we first consider a compressed family J*\ If $F belongs to Case1 above, then 3F = J^([l,n]); otherwise, in Case 2, we must have equality in (1). Thisimplies that ^k-x and ^ are extremal families in 3$n-\, and, by the induction hypothesis,they must be the i^(0) of (k - l)-chains and /c-chains in @n-\9 respectively. So $F must beidentical with ^ (0 ) in &n.

On Intersecting Chains in Boolean Algebras 303

Finally, we observe that any family whose compressed image is J*(0) or <F([l,n]) isitself one of these families.

We remark that in a preliminary version of the present paper [4], we proved Theorem 1.1for n > kink. Since then, Ahlswede and Cai [1] have also found a proof for Theorem 1.1(i), but their method does not seem to generalize to the truncated case.

4. Chains in 08n

Again, we use induction on n to prove that f~(n,k) = k\S(n — 1,/c). The base case n = 2is trivial. Suppose we are done for n — 1 and let 2F be a compressed family of chains in&~. We distinguish two cases:

Case 1: If there exists a chain i f e 3F such that n — 1 e L\9 then S£ may contain onlyone initial segment, namely [l,n— 1]. Then, since each chain in 3F must intersect if in aninitial segment (see Lemma 2.2), all chains contain [1, w — 1] and we are done.

Li, then, in particular, we never have L\ =£ {n— 1}.Case 2:Define

If each

t =

, ==

¥ has no n

{& e& :

- 1

LH

Deleting n— 1 from each element of each chain of J*o, we obtain a family J% of intersecting/c-chains in the truncated Boolean algebra on the underlying set {1,2,..., n — 2,n}. Byhypothesis, \^Ff

0\ < f~(n — 1,/c). Each (Li,...,L^) G J^Q can be obtained from < k chainsof J o? since n — 1 could have been inserted starting at L2, L3,..., L&, or could have beenan element of [l,n] \ L^.

Deleting n— 1 from every set in every chain in J^ (for any f = 1,2,...,/c— 1), we obtain afamily 3F\ of intersecting (k — l)-chains in the truncated Boolean algebra on the underlyingset {l ,2 , . . . ,n-2,n}. By hypothesis, \&t\ = | ^ | < /" (n - l,fc - 1).

Finally, define ^ by deleting the largest set Lk = {1,2, ...,n — 2, n) from every chain in&k> Observe that 3F'k is a family of intersecting (k — l)-chains in the truncated Booleanalgebra on the underlying set {l,2,...,n — 2,n}, since the set that we dropped is not aninitial segment in the original underlying set. Therefore, by hypothesis, \^k\ = Wk\ ^f-(n-hk-l).

Hence, \&\ < k -k\S(n - 2,/c) + (fc - l)(/c - l)!S(n - 2,fc - 1) + (fe - l)!S(n - 2,/c - 1) =fc!S(n- 1,/c).

This finishes the proof of Theorem 1.1. •

We remark that, analogously to the discussion at the end of Section 3, it can be shownthat the only compressed extremal families in &~ are «^~([1]) and ^~([l,n — 1]). Theextension that the only extremal families are ^~(A) with \A\ = 1 or \A\ = n — 1 is stillmissing.


Acknowledgements

We are indebted to Ulrich Faigle, Zoltan Fiiredi, and Walter Kern for stimulating

conversations on the subject of the paper.

Note added in proof

We have just learned of a research program initiated by Miklos Simonovits and Vera T.

Sos on 'structured intersection theorems' [10, 11], which has a fairly large literature. They

studied the maximum number of graphs on n vertices such that any two intersect in a

prescribed graph, e.g. a path or cycle. The following problem fits into their scheme: given

a graph G what is the maximum number of pairwise intersecting complete /c-subgraphs.

In this paper we have studied the comparison graphs of some partially ordered sets.

References

[I] Ahlswede, R. and Cai, N. (1993) Incomparability and intersection properties of Boolean intervallattices and chain posets, preprint.

[2] Deza, M. and Frankl, P. (1983) Erdos-Ko-Rado theorem - 22 years later, SIAM J. Alg. Disc.Methods 4, 419-431.

[3] Erdos, P. L., Faigle, U. and Kern, W. (1992) A group-theoretic setting for some intersectingSperner families, Combinatorics, Probability and Computing 1, 323-334.

[4] Erdos, P. L., Seress, A. and Szekely, L. A. (1993) On intersecting k-chains in Boolean algebras,Preprint, April 1993.

[5] Erdos, P., Ko, C. and Rado, R. (1961) Intersection theorems for systems of finite sets, Quart. J.Math. Oxford Ser. 2 12, 313-318.

[6] Frankl, P. (1987) The shifting technique in extremal set theory, In: Whitehead, C. (ed.) Surveysin Combinatorics 1987, Cambridge University Press, 81-110.

[7] Frankl, P. and Fiiredi, Z. (1980) The Erdos-Ko-Rado theorem for integer sequences, SIAM J.Alg. Disc. Methods 1, 376-381.

[8] Fiiredi, Z. (1991) Turan type problems, In: Keedwell, A. D. (ed.) Surveys in Combinatorics 1991,Cambridge University Press 253-300.

[9] Lovasz, L. (1977) Combinatorial Problems and Exercises, Akademiai Kiado, Budapest andNorth-Holland, Amsterdam.

[10] Simonovits, M. and Sos, V. T, Intersection theorems for graphs. Problemes Combinatoires etTheorie des Graphes, Coll. Internationaux C.N.R.S. 260 389-391.

[II] Simonovits, M. and Sos, V. T. (1978) Intersection theorems for graphs II. Combinatorics, Coll.Math. Soc. J. Bolyai 18 1017-1030.

On the Maximum Number of Triangles inWheel-Free Graphs

ZOLTAN FUREDI+, MICHEL X. GOEMANS*and DANIEL J. KLEITMAN§

Department of Mathematics,Massachusetts Institute of Technology,

Cambridge, MA 02139

Gallai [1] raised the question of determining t(n), the maximum number of triangles ingraphs of n vertices with acyclic neighborhoods. Here we disprove his conjecture (t(n) ~~n2/8) by exhibiting graphs having n2/7.5 triangles. We improve the upper bound [11] of(n2 — n)/6 to t{n) < n2/7.02 + 0{n). For regular graphs, we further decrease this bound ton2/1.15

1. Introduction

Let WFGn be the class of graphs on n vertices with the property that the neighborhoodof any vertex is acyclic. A graph G is given by its vertex set V(G) and edge set E(G). Thesubgraph induced by X c V(G) is denoted by G[X]. The neighborhood N(v) of vertex vis the set of vertices adjacent to v. Note that v £ N(v). The degree of v G V(G), denotedby dv or dv(G), is the size of the neighborhood: dv = \N(v)\. The maximum (minimum)degree is denoted by A (S), or A(G) (S(G), respectively) to avoid misunderstandings. Amatching M a E(G) is a set of pairwise disjoint edges. A wheel Wt is obtained from acycle C[ by adding a new vertex and edges joining it to all the vertices of the cycle; thenew edges are called the spokes of the wheel (/ > 3, W3 = K4). Therefore, WFGn consistsof all graphs on n vertices containing no wheel. Let t(G) denote the number of triangles

+ Research supported in part by the Hungarian National Science Foundation under grant No. 1812.New address: Dept. Math., Univ. Illinois, Urbana, IL 61801-2917.E-mail: zoltanta^math.uiuc.edu

* Research supported by Air Force contract F49620-92-J-0125 and by DARPA contract NOOO14-89-J-1988.E-mail: goemans(a^math.mit.edu

§ Research supported by Air Force contract F49620-92-J-0125 and by NSF contract 8606225.E-mail: djk(<^math.mit.edu

306 Z. Furedi, M. X. Goemans and D. J. Kleitman

in G and let t(n) be the maximum of t(G) over WFG«. Gallai (see [1]) raised the questionof determining t(n).

Take a complete bipartite graph K2a,n-2a, where a is the closest integer to n/4, and adda maximum matching on the side of size 2a. We obtain a wheel-free graph G*, having[n2/S\ triangles [1]. Gallai and, independently, Zelinka [10] in 1983 conjectured that thisis the maximum possible. However, Zhou [11] recently constructed wheel-free graphshaving (n2 + M)/8 triangles whenever n is of the form 8g + 7. He also found an upperbound, t(n) < (n2 — n)/6. In this paper we improve both bounds.

Theorem 1. There exists a wheel-free graph on n vertices with n2/7.5 + n/15 triangleswhenever n is a multiple of 15, i.e., t(n) > n2/1.5 + n/15.

This theorem is proved by giving a construction, G2, in Section 2. As t(n) is monotone,we get t(n) > n2/1.5 — O(n) for all n. P. Haxell observed that G\ has the additionalproperty that it is locally tree-like, i.e., every neighborhood induces a tree. (More exactly,she improved it so.) Zelinka [10] proved that any locally tree-like graph with n verticeshas at least 2n — 3 edges and posed the question; what is the maximum number of edgesof these graphs? As G2

n has n2/5 + O(n) edges, we got a counterexample for a conjectureof Froncek [5], who believed that a locally tree-like graph on n vertices contains at most[n(3n + 8)/16j edges (for n > 8).

Theorem 2. Every wheel-free graph on n vertices contains at most n2/1.02+ O(n) triangles,more exactly, t(n) < n2/1.02 + 5n for all n.

The main tool of the proof is Proposition 11 (proved in Section 4), which givest(G) < n2/8 + o(n2) for several types of graphs. One example is given by the followingtheorem.

Theorem 3. Let G e WFGn be a wheel-free graph on n vertices, n > 100. IfS> (2/5)n +16/5, then t(G) < n2/8.

Looking at regular wheel-free graphs one can observe that, if n is of the form 4a — 1,the previously mentioned construction, G*, is regular. Hence, there exist regular graphs inWFGn having [n2/S\ triangles. But the graph constructed in Section 2 is not regular. Infact, the upper bound we prove for regular graphs in Section 7 is lower than the lowerbound for general graphs.

Theorem 4. If G is a regular wheel-free graph, t(G) < n2/1.15 + O(n).

We conjecture that Gallai's conjecture holds for regular graphs.

Conjecture 5. If G is a regular wheel-free graph, t(G) < n2/8.

On the Maximum Number of Triangles in Wheel-Free Graphs 307

/ = 1, ..., «/5

k = 1, ..., w/15

7 = 1, . . . , / i /5

Figure 1 A wheel-free graph having rr/1.5 + n/15 triangles.

2. Wheel-free graphs having more than n2/1.5 triangles

Define the graph G2n on n vertices, where n is a multiple of 15 (see Figure 1) as follows. Its

vertex set V{G2n) consists of a,-, ft,-, c/, d,- for i = l , . . . ,n /5 , and ek,fk,gk for fe = l,...,w/15.

Its edge set E(G2n) consists of

— two matchings of size n/5: (ai9bi) and (c/,d,-),— three matchings of size w/15: (ekjk), (fk,gk) and {ek,gk), and— all the edges of types: (a,-,c/)> fe^), (^,g^), (fc,-,d/), (&,•,/*)> (fc/,g*), {cj9ek), (cjjk) and

{dj,ek). (Here, again, 1 < 1,7 < n/5 and 1 < fc < w/15.)

It is easy to verify that this graph belongs to WFGn. For example, the neighborhood ofthe vertex a, consists of the matching {(cj,dj) : 1 < j < n/5} as well as a star rooted at b\with edges {(bhgk) : 1 < k < n/15} and {(bhdj) : 1 < j < n/5}.

Each triangle in G2n contains an edge from the matchings, and its vertices are in three

different classes. An easy calculation shows that r(G^) = n2/1.5 + n/15.

3. Improved upper bounds for t(n)

In this section we prove a series of Lemmas that lead to Theorem 6, an upper bound forall n. This theorem was independently proved by P. Haxell [8].

Theorem 6. Every wheel-free graph on n vertices contains at most (l/7)n2+(9/7)n triangles,1 9

i.e., t(n) < -n2 + -n.

Let G e WFGn be an arbitrary wheel-free graph. By definition, G[N(v)], the neighbor-hood of any vertex v, is acyclic. Hence, the number of edges in N(v) is less than or equalto dv — 1, where dv = \N(v)\. By summing dv — 1 over all vertices of G, we obtain an upper

bound for 3t(G). On the other hand, the summation of the degrees over all vertices isprecisely twice the number of edges of G. Hence,

3t(G)<2\E(G)\-n. (1)

Since 2|£(G)| < wA, where A stands for the maximum degree, (1) gives

= ^ . (2)

Our next aim is to obtain the following upper bound on the number of edges of G:2

j ^. (3)

Together with (1), this gives Zhou's upper bound, t(n) < (n2 — n)/6. The upper bound (3)follows from the following theorem of Erdos and Simonovits [3]: if G does not containa WT> or a W^ then \E(G)\ < [(n2 + w)/4j, whenever n > no- They wrote: '... are easy toprove by induction and can be left for the reader'. For completeness we reconstruct theirargument in a somewhat simplified form.

A graph is called (fcy)-free if \E(G[K])\ < f holds for every /c-element subset K a V(G).Let j(n\kj) be the maximum number of edges of a (/c,/)-free graph with n vertices.Turan's classical theorem determines /(n;/c, (^)), for example, /(w;3,3) = |_ft2/4j. Heproposed the problem of determining /(n;/c,/), but it is still unsolved in a number ofcases. Erdos [2] investigated, first, all the cases k < 5, he also proved that excluding onlyW4 implies \E(G)\ < [n2/4\ + [(n + 1)/2J (for n > n0). For recent accounts, see [6, 7].A wheel-free graph has neither W^ nor W4, so it is (5,8)-free. Thus (3) follows from thefollowing.

Claim 7. If G is a graph with n vertices such that any 5 vertices span at most 7 edges,

\E(G)\ < (n2 + w)/4 holds for n>5.

Proof. The case n = 5 is trivial. Let G be a (5,8)-free graph with n vertices, n > 6.Considering all the n subgraphs G \ x, we have (n — 2)|£(G)| = Y2xey \E(G \ x)|, implying

/(«;5,8) 1. It is easy to see (by induction, distinguishing four subcases

according to the residue of n modulo 4) that sn = [(n/(n — 2))sw_iJ holds for all n > 2. As

/(5;5,8) = 55 = 7, (4) implies the desired upper bound. •

A vertex cover C of the graph H is a subset of vertices with the property that all edgesof H are incident to at least one vertex in C.

Lemma 8. If C is a vertex cover of G[N(w)], where w is a vertex of maximum degree A,then

Proof. Since C is a vertex cover of G[N(w)], the set S = (V — N(w)) U C is a vertex coverof G. Hence, all triangles of G contain at least two vertices belonging to S. Summing(dv — 1) over all vertices v in 5, we obtain an upper bound on twice the number of trianglesin G:

2r(G) < ^2(dv - 1) < (A - 1)|S| = (A - l)(n - A + \C\).veS

•Observe that if \C\ = o(n), the function (l/2)(p — \)(n — p + \C\) is maximized at

p = (l/2)n + o(n). In this case, we get

t(G)<j+o(n2).

On the other hand, if the acyclic graph G[N(w)] has no vertex cover of'small' cardinality,we can use the following two lemmas to isolate two vertices of G that are contained in asmall number of triangles. The total degree of the subset S (in the graph H) means thesum of degrees: J2xes dx(H).

Lemma 9. Let H be an acyclic graph on p vertices with at least one edge, and let b beany positive integer. Then either there exist two adjacent vertices whose total degree is atmost b + 1, or there exists a vertex cover of H of cardinality at most (p — 2)/b.

Proof. Suppose that the size of each cover exceeds (p — 2)/b. As H is a bipartite graph,Konig's theorem implies that one can find a matching M of size m> (p — 2)/b. If we sumup all the degrees of the vertices of UM, we count all edges of H at most once, exceptthe edges included in UM. Hence ]T{dM : u G UM} is at most (p - 1) + (2m - 1). Thisguarantees the existence of a pair (a, b) G M such that da + dt < (p — 2 + 2m)/m, which isless than b + 2. •

Lemma 10. Suppose that G G WFGn and w G V(G). Suppose also that the du(G[N(w)]) +dv(G[N(w)]) < s, where u and v are adjacent vertices in G[N(w)]. Then G has at mostn — dw -\- s — 2 triangles containing at least one of {u,v}.

Proof. We claim that given any two adjacent vertices u and v, the total number of trianglescontaining u or v is at most \N(u) U N(v)\ — 2. Indeed, u is contained in at most \N(u)\ — 1triangles and v is contained in at most |iV(u)| — 1 triangles, while the number of trianglescontaining both u and v is exactly \N(u) Pi N(v)\. Hence, the total number of trianglescontaining u or v is bounded above by \N(u)\ — l+\N(v)\ — l—\N(u)nN(v)\ = \N(u)UN(v)\-2.

The assumption in the Lemma implies that \N(u)UN(v)\ < \V(G)\N(w)\+s. Combiningthis bound with the previous observation completes the proof. •

Proof of Theorem 6. We prove the bound (n2 + 9n)/7 by induction on the number ofvertices n. Let G be a graph in WFGn with maximum degree A, and let w e V(G) be suchthat dw = A. We consider three cases.

310 Z. Filredi, M. X. Goemans and D. J. Kleitman

If A < (3/7)w + 5, i.e. A < (3n + 34)/7, the result follows from the inequality (2).If there exists a vertex cover of G[N(w)] of cardinality less than or equal to A/8, Lemma

8 implies that

In these two cases, we have not used the inductive hypothesis.Finally, assume that A > (3/7)n + 5, and that there is no vertex cover of G[N(w)] of

cardinality less than or equal to A/8. By Lemma 9, there exist two adjacent vertices u andv in N(w) whose total degree in G[N(w)] is at most 9. Lemma 10 now implies that thereare at most n — A + 7 < (4/7)n + 2 triangles containing u or v. By deleting u and v, weobtain a graph in WFGn_2 that contains at most

triangles, by the inductive hypothesis. Hence,

_ (n-2)2 9, „ An ^ n2 9t(G) < 7 + -(n - 2) + y + 2 = - + -n.

•

4. Triangles from a matching

In order to prove the slight additional improvement described in Theorem 2, we firstprove a weaker form of Gallai's conjecture. This result is also crucial in improving theupper bound in the case of regular graphs. Let dm denote \N(u) n N(v)\, the number oftriangles containing the edge uv.

Proposition 11. Let G be a graph that contains no wheel with 3 or 4 spokes, and let M bea matching in it. Then

£ dm, < nj. (5)

Proof. For simplicity, throughout this proof, a triangle always refers to a trianglecontaining an edge in M. Thus, the left-hand side of (5) is the number of triangles in G.Let m denote the cardinality of M, let P denote the set of unmatched vertices, and letp = \P\ — n — 2m.

Observation 1. For two edges (u,v) and (x,y) in M, the induced graph G[u,v,x,y] cancontain at most two triangles (Figure 2).

Indeed, more than two triangles would imply that {u,v,x,y} induces a K4, i.e., a wheelwith 3 spokes.

Consider first the graph H with a vertex uv for each edge (u,v) of M, and with an edge(uv,xy) whenever {u,v,x,y} induces two triangles. Let Q be a maximum matching in H.Let S be the set of vertices in G belonging to edges of M that are saturated by Q, and letR be the remaining vertices in M. Hence (S,R,P) is a partition of V(G). Let q denote the


Figure 2 Observation 1 in the proof of Proposition 11


cardinality of Q, and let r denote the number of unmatched vertices in H, i.e., r — m — 2q.Clearly, \S\ = 4q and \R\ = 2r. Thus, p + 2r + 4q = n.

Observation 2. If (uv,xy) € Q and w is any vertex of G, then {w, v,x, y} and w can connectwith one another in at most 1 triangle (Figure 3).

Indeed, if {u,v,x} is one of the two triangles induced by {u,v,x,y}9 and w forms twotriangles with {u,v,x,y}, then {M,U,X,W} induces a X4.

Observation 3. If (uv,xy) e Q and (a, ft) e M, then {w,i;,x,y} and {a,b} connect with oneanother in at most two triangles (Figure 4).

Assume without loss of generality that (w, x), (v,x) and (v,y) are in E(G). If {u,v,x,y} and{a, b] connect with one another in three or more triangles, we can assume without lossof generality that {u,v} and {a,b} connect with one another in two triangles. Withoutloss of generality, we can assume that (w,a) and (v,b) are in E(G). We consider two cases.If (v,a) e E(G) (see Figure 4.a), then {x,y,a, b) is included in N(v), implying that anytriangle between {x, y} and {a, b} would create a K4. If (u,b) G E(G) (see Figure 4.b), atriangle of the form {a, ft, z} with z E {X, 3;} would create the cycle u — a — z — v — u in theneighborhood of ft, while a triangle of the form {x,y,c} with c e {a, ft} would create thecycle u — c — y — v — uinlhe neighborhood of x.

From Observations 2 and 3, there are at most q(n — 4q) triangles that connect S toV — S. Within S, in addition to the 2q triangles corresponding to the edges in Q, thereare at most (\/2)4q(q — 1) triangles, by Observation 3. Therefore, the number of triangles

z. Furedi, M. X. Goemans and D. J. Kleitman

{a) (b)


containing vertices in S is at most

2q(q - 1 ) 4 - q(n - 4q) + 2q = q(n - 2q). (6)

We now concentrate on the number of triangles induced by V - S. Recall that V - Scorresponds to vertices in P (which are unmatched in M) and to vertices in R (which areincident to edges in M that are unmatched in Q).

Observation 4. If uv and xy are unmatched vertices in Q, then {u,v} and {x,y} can connectwith one another in at most one triangle by the maximality of Q.

Observation 5. Consider a vertex w e P. We say that two edges (u,v) and (x,y) in M areindependent if {u,v,x,y} does not induce any triangle. Since the neighborhood of wcannot contain any triangle, w can induce triangles only with independent edges inM.

Let s be the cardinality of a largest set of independent edges in G[R]. The number oftriangles in G[R] is at most s{r-s) + ( l / 2 ) ( r - s ) ( r - s - 1) < (r2 -s 2 ) /2 by Observation 4.Moreover, the number of triangles between P and R is at most ps by Observation 5.Therefore, there are at most T = ps + (r2 - s2)/2 triangles in G[V - S]. We have thatps + (r2 - 52)/2 < (2r + /?)2/8 for all real r > 5, p > 0 (because this is equivalent to4p(s - r) < (p - 2s)2). Hence, the number of triangles in G[V - S] is at most (n - 4q)2/S.Combined with (6), this implies that the number of triangles containing edges in M is atmost q(n - 2q) + (n - 4q)2/S = n2/8. •

5. Graphs with large minimum degree

In this section we prove Theorem 3.

Partition the edge set of G into two classes. We say that an edge (u,v) of E(G) is a fatedge if at least ^Jn triangles contain it, i.e., duv > Jh. An edge that is not fat is said to belean. In the next two lemmas, we show that the set of fat edges is an ideal candidate toplay the role of the matching M in Proposition 11.

Lemma 12. Let G be a graph in WFGn with du + dv + dw > n + 3^ for every triangleuvw. (For example, S > n/3 + Jn.) Then every triangle of G contains a fat edge.

Proof. Consider a triangle with vertices u, v and w. Observe that N(u) n N(v) n AT(w) = 0.Indeed, a vertex adjacent to u, v and w would have a cycle in its neighborhood. Hence,

n > \N(u) U N(v) U N(w)\ = 4 + 4 + dw - duv - duw - dvw. (7)

Since du + dv +dw >n + 3^Jn, at least one of the quantities duv, duw and dvw is greater thanor equal to ^jn. D

Lemma 13. Every fat edge of G e WFGn belongs to a triangle with two lean edges.

Proof. Let (u,v) be a fat edge. Suppose, on the contrary, that for each x e N(u) n N(v),at least one of the edges (w, x) and (v, x) is fat. This implies, that

] T ((4x - 1) + (dvx - 1)) > \N(u) n N(t;)|(7^ - 1) > n - duv. (8)xeN{u)nN(v)

Consider all the triangles of G of the forms uxy and vxy, where x E N(w) Pi N(y) andy £ N{u)HN(v), y =fc u,v. The number of these triangles is exactly the left-hand side of (8).However, all of these triangles have a distinct third vertex outside (N(u) D N(v)) U {u,v},so their number is at most n — 2 — duv, contradicting (8). Indeed, for example, if uxy andux'y are triangles with x,x' G N(v\ the cycle xyx'v forms a wheel with center u. D

Proof of Theorem 3. Let G be a graph in WFGn with du > (2/5)n+ 17/5 for each vertexu. For n > 96, Lemma 12 implies that each triangle contains a fat edge.

We claim that the fat edges form a matching.Let m be the maximum number of fat edges incident to a vertex of G. We shall prove

that m = 1. Consider a lean edge (v, w). By the above argument, any triangle containing(v, w) must contain a fat edge. Since there are at most 2m fat edges incident to either v orw, we obtain that

dvw < 2m (9)

for any lean edge (v,w).Let u be a vertex with m fat edges incident to it, say (u,v\),(u,v2),...,(u,vm). Since G is

wheel-free, there exist at most m — 1 triangles containing two fat edges incident to u. Bysumming dUVi over i = l,...,m, we count every triangle containing u at most once, exceptfor those containing two fat edges incident to u. Hence,

Vi <du-\+(m-l). (10)

The left-hand side is at least m^/n, hence we get m < ^Jn + 1. For any fat edge (u,v),Lemma 13 implies that there is a triangle uvw with two lean edges. Then (7) and (9) give

duv >du + dv+dw — n — 4m. (11)

314 Z. Fiiredi, M. X. Goemans and D. J. Kleitman

We derive thatm mm

] T dUVi >mdu + Y^ dVi + ^2 d«> -nm-Am1. (12)

Comparing (10) and (12), we get m — 2 > (3m — \)S — nm — Am2. If 2 < m < ^Jn + 1,n > 500, d > (2n + 16)/5, which leads to a contradiction. If 100 < n < 500, we can usedvw < yjn instead of (9) to get a contradiction in exactly the same way. Therefore m mustbe equal to 1, i.e., the fat edges form a matching.

Every triangle contains a fat edge, so, by Proposition 11, there are at most n2/S trianglesinG. •

6. Proof of the upper bound

Here we prove Theorem 2 by induction on n. If n < 9126, it follows from Theorem 6. Letc = 1/2457= 1/7-1/7.02.

If A > ((3/7) + 4c) n the proof is similar to the proof of Theorem 6. Either thereexists a vertex cover of G[N(\v)] of cardinality less than or equal to A/9, in which caset(G) < 9/64n2 < (1/7 — c)n2, or there exist two adjacent vertices u and v in N(w) whosetotal degree in G[N(w)] is at most 10. In the latter case, Lemma 10 implies that we destroyless than

X4 *- - 4c )n + 8

triangles by deleting u and v. The claim therefore follows by induction.If \E(G)\ < (3/2) ((1/7) - c) n2, the result follows from (1).Assume now that

Assumption 1. A < ( | + 4c) n,

Assumption 2. \E(G)\ > | ( | - c) n2.

If G satisfies the hypotheses of Theorem 3, we are done. Otherwise, we must haveS < (2/5)n+ 16/5. Consider the graph G obtained from G by repeatedly deleting a vertexof minimum degree until

Let vo9v\,...,vt-\ be the sequence of vertices that we delete, and let G, be the graphobtained from G by deleting {uo,...,ty-i}. In particular, Go = G and Gt = G'. Bydefinition, we have that

and

4(G)<^-0 + y +//,where / , denotes the number of edges in G joining v\ to {uo,..., t?,-i}. Let s = t/n. In orderto give an upper bound on e, we consider the number of edges of G. Using Assumptions 1

and 2, we derive

7 " vev

i J \ l<

_ 1 2 Z ^+ 20£ + 20n'

where we have used (3), i.e. that the number of edges of G[{vo,...,vt-\}] is at most(en)2/4 + en/4. For n sufficiently large, say larger than no = 9126, and given the value ofc, the above inequality can be seen to imply e < 0.12. Since G satisfies the hypotheses ofTheorem 3, we have t{G) < (n2/8)(l - e)2. Moreover,

4. 1 * 1

. 11\ 2 , 1 , , 12

Therefore,

\9 ^ \ 9 ?[i £i +• £ 1 M ~\~ Fn <^. n ~\~ Sn

DRemark. The result can be improved to t(n) < n2/1.03 + O(n), as shown below. Letc = 1/1540. We execute the first two steps of the previous proof, so from now on we maysuppose that Assumptions 1 and 2 hold. We delete from G a vertex x ifdx < (n — i)/3 + yJn.Lemma 13 implies that we obtain a graph where each triangle contains a fat edge. Deletea fat-lean-lean triangle, uvw, if du + dv+dw < l.2(n — i) +12. We get that for each fat edge,duv > 0.2(n — i) + 0(1). If there exists two adjacent fat edges, (u,x\) and (w,X2), for somefat-lean-lean triangles ux\w{ and ux2w2 we get that du+dXl +dX2+dW[ +dW2 < 2(w—j)+O(l).Delete these 5 points and repeat these steps.

The upper bound for the number of edges implies e < .235. At each of the above steps,by deleting £ vertices (1 < £ < 5) we have destroyed only £(n — i)/3 + 0(1) triangles atmost. We get that t(G)/n2 < (1/8)(1 - s)2 + (l/3)s - (l/6)e2 + o(l). •

7. Upper bound for regular graphs

In this last section, we prove Theorem 4, the upper bound for d-regular graphs.Ifd< (12/31)n, equation (2) implies that t(G) < n2/U5.If d > (2/5)n + 16/5, then Theorem 3 implies that t(G) < rc2/8, (for n > 100).Assume now that (12/31)rc < d < (2/5)n + 16/5. From (12) we can deduce that any

vertex of G is incident to at most two fat edges. (The details are left to the reader.) If novertex is incident to two fat edges, the result follows from Proposition 11. Otherwise, letr be the maximum, over all vertices w, of the number of triangles containing a fat edge

that is incident to w, and let w be a vertex attaining this maximum. From the definitionof r, we must have duv < r for all edges (u,v) e E. Moreover, since a fat edge is containedin at least 3d — n — 8 triangles, by (11), and since there exists a vertex incident to two fatedges, r must be at least 6d - In - 17. Let S = V - {w} - N(w), and let Tt for i = 0,..., 3be the number of triangles having exactly i vertices of S. Clearly, To < d — 1, since anysuch triangle must involve w. Moreover, by summing dv — 1 over all vertices v € S, weobserve that 3T3 + 2T2 + T\ <(d-l)(n- d). Finally, we claim that T{ < r(d - r) + 0{n).To see this, observe that the number of lean edges contained in N(w) is at least r — 1 if wis incident to just one fat edge, or r — 2 if w is incident to two fat edges. Thus the numberof fat edges contained in N(w) is at most d — r + 1. To compute an upper bound on T\,we sum duv — 1 over all edges contained in N(w) (the —1 term comes from the fact thatwe do not need to count triangles involving vertex w):

Tx < 3(r - 2) + (r - l)(d - r + 1) = r(d - r) + 0{n\

since lean edges are contained in at most 4 triangles by (9), while fat edges are containedin at most r triangles by definition of r. Therefore,

t(G) < To + ^(Ti + 272 + 3T3) + l-Tx < l-{d(n - d) + r(d - r)) + O(n).

When r > 6d — In — 17 (> d/2), the right-hand side is maximized for r = 6d — 2n — 0(1),giving t(G) < (l/2)(d(n-d) + (6d-2n)(2n-5d)) + O(n). Under the constraint (12/31)w < d,this is in turn maximized for d = (12/31)n + 0(1), proving that t(G) < n2/1.75 + 0(n). D

8. Wheel-free triple systems

A family of 3-element sets is called wheel-free if it contains no k triples isomorphic to{{0,1,2},{0,2,3},...,{0,i,(i + l)},...{0,fc,l}}, where k > 3 is an arbitrary integer. Forexample, the vertex sets of the triangles in a wheel-free graph form such a system. Butthe general case is different. Let ex(n,W) denote the largest cardinality of a wheel-free triple system on an n-element set. V. T Sos, Erdos and Brown [9] proved thatlim^ooex(n;^)/n2 = 1/3.

For further problems of this type, see, for example, [4] and the references therein.Another interesting question is whether (and how) our results can be extended to3, M^J-free graphs, or even more generally for (5,8)-free graphs.

9. Acknowledgements

The authors are indebted to P. Haxell for helpful remarks and for improvements to theconstruction in Section 2. We are also grateful for the referees' conscientious reading.

References

[1] Erdos, P. (1988) Problems and results in combinatorial analysis and graph theory. DiscreteMath., 72 81-92.


[2] Erdos, P. (1965) Extremal problems in graph theory. In: Fiedler, M. (ed.) Theory of Graphs andits AppL, (Proc. Symp. Smolenice, 1963), Academic Press, New York 29-36.

[3] Erdos, P. and Simonovits, M. (1966) A limit theorem in graph theory. Studia Sci. Math. Hungar.1 51-57.

[4] Frankl, P. and Rodl, V. (1988) Some Ramsey-Turan type results for hypergraphs. Combinatorica8 323-332.

[5] Froncek, D. (1990) On one problem of B. Zelinka (manuscript).[6] Golberg, A. I. and Gurvich, V. A. (1987) On the maximum number of edges for a graph with

n vertices in which every subgraph with k vertices has at most / edges. Soviet Math. Doklady35 255-260.

[7] Griggs, J. R., Simonovits, M. and Thomas, G. R. (1993) Maximum size graphs in which every/c-subgraph is missing several edges (manuscript).

[8] Haxell, P. (1993) PhD Thesis, University of Cambridge, Cambridge, England.[9] Sos, V. T, Erdos, P. and Brown, W. G. (1973) On the existence of triangulated spheres in

3-graphs, and related problems. Periodica Math. Hungar. 3 221-228.[10] Zelinka, B. (1983) Locally tree-like graphs. Cas. pest. mat. 108 230-238.[11] Zhou, B. (to appear) A counter example to a conjecture of Gallai. Discrete Math.

Blocking Sets in SQS(2v)

MARIO GIONFRIDDOf, SALVATORE MILICI+

andZSOLT TUZA*+ Dipartimento di Matematica, Citta Universitaria, Viale A, Doria 6, 95125 Catania, Italy.

^Computer and Automation Institute, Hungarian Academy of Sciences,H - l l l l Budapest, Kende u. 13-17, Hungary

A Steiner quadruple system SQS{v) of order r is a family $ of 4-element subsets of a^-element set V such that each 3-element subset of V is contained in precisely one B G M.We prove that if T n B ± 0 for all B e M {i.e., if T is a transversal), then \T\ > r/2, andif T is a transversal of cardinality exactly r/2, then V \ T is a transversal as well {i.e., Tis a blocking set). Also, in respect of the so-called 'doubling construction' that producesSQS{2v) from two copies of SQS{v), we give a necessary and sufficient condition for thisoperation to yield a Steiner quadruple system with blocking sets.

1. Introduction

A hypergraph Jf is a pair (V,$\ where V is a finite y-set (= a set of v elements) and& =/= 0 is a family of nonempty subsets B ^ V such that |J BE.#B = K. The integer r = | V\is the orJ^r of Jtf; the elements of V and ^ are called the vertices (or points) and theb/oc/cs (or edges) of the hypergraph, respectively. If \B\ — r(Jf) = r for each block B G .^,then Jf is called an r-uniform hypergraph, or a uniform hypergraph of rank r(Jf).

Given a hypergraph Jf = {V,M) and a nonempty subset W ^ V, <CW^> denotes thesubhypergraph whose blocks are the blocks of Jf contained in W (i.e., subsets B of Wsuch that B e ^ ) . The points of < H / > are those contained in the blocks of < W > . Thehypergraph <CW^> is called the subhypergraph of Jf induced by W.

Given the complete graph Kv on v vertices (v even), a 1-factor F, of Kv is a set of r/2pairwise disjoint edges of Kv. A factorization (also called a 1-factorization) J^ of Xr is aset of v — 1 1-factors of Xr such that each edge occurs in precisely one F{. A factorization

+ Research supported by MURST and GNSAGA, CNR.* Research supported in part by the OTKA Research Fund of the Hungarian Academy of Sciences, grant no.

2569, and in part by C.N.R. Italia while the author visited Universita di Catania.

320 M. Gionfriddo, S. Milici and Z. Tuza

3F of the complete bipartite graph Kv# of order 2v (v arbitrary) is a set of v 1-factors (ofv edges each) in Kv,v containing no edge twice.

A t — (v,k,X) design is a uniform hypergraph (V93$) of rank k such that every r-subsetof 7 is contained in exactly X blocks of $&. If A = 1, a t — (u,fc, 1) design is also calleda Steiner system. For f = 2 and fc = 3, a Steiner system is called a Steiner triple system,abbreviated by STS(v). For t = 3 and fc = 4, a Steiner system is called a Steiner quadruplesystem, SQS(v) for short. Sometimes a hypergraph will be denoted by S> or by £f (insteadof Jf) if we want to emphasize that it is a design or a Steiner system, respectively.

Concerning the construction of Steiner systems, Hanani [5] proved in 1960 that anSQS(v) exists if and only if v = 2 or 4 (mod 6), while it is well known that an STS{v)exists if and only if v = 1 or 3 (mod 6).

Given a hypergraph JT = (V,@), a transversal of Jf is a set 7 c 7 such that T n £ ^ 0for each £ G ^ . The transversal number i(jf) of J*f is defined as the minimum numberof points in a transversal. Moreover, a blocking set is a set T c 7 such that T and 7 \ 7are both transversals. Hence, T is a blocking set of Jf = (V,3#) if and only if

5 O T ^ and B\T ^ 0 for each J5 e ^ .

Very few facts are known so far about the existence of blocking sets in Steiner systems.Let us recall three important results. In what follows, ^k(Y) will denote the set of allfc-subsets of a finite nonempty set 7.

Theorem 1.1. (Tallini [ 8 ] , Berardi and Beutelspacher [1]) If 2 = {V,Sf) is a 2 - ( i ,3 ,A)design, there exist blocking sets in & only for v = 4 and & = ^ ( 7 ) . In this case theblocking sets are all the 2-subsets of V.

As an important particular case of Theorem 1.1, for Steiner systems we obtain thefollowing corollary.

Corollary 1.1. There are no blocking sets in Steiner triple systems STS(v).

Theorem 1.2. (Tallini [7], Berardi and Beutelspacher [1]) If 2 = {V,08) is a 3 - ( M , A )design and T is a blocking set in 2, then either Q) has an even order v and \T\ = v/2, or

v = 5, A is even, ® = 0>4(V) and \T\ e {2,3}.

We can see that Theorem 1.2 does not exclude the possibility of the existence of blockingsets in 3 — (U,4,A) designs. On the other hand, a particular case of Theorem 1.2 yields thefollowing corollary.

Corollary 1.2. / / T is a blocking set in a Steiner quadruple system SQS(v), then \T\ = v/2.

There are only a very few known examples of blocking sets in Steiner quadruple systems,or in 3 — (v,4, A) designs for A > 1. There exist blocking sets in the unique SQS(%) (see [7]),as well as in the unique SQS(IO) (see [3]). On the other hand, the four systems SQS{14)have no blocking sets (see [6]).

Blocking Sets in SQS(2v) 321

Regarding the affine Galois spaces AG(r, 2) of dimension r and order 2 (i.e., all lineshaving two points), we obtain Steiner quadruple systems SQS{2r) when considering theplanes of AG(r, 2) as blocks of the system. For these particular designs, denoted AGiir,!)(with the subscript referring to the dimension of each block), the following holds.

Theorem 1.3. (Tallini [8]) If r > 4, then AGi(r, 2) contains no blocking set.

In [4] and [6], the authors study the existence of blocking sets in the systems SQS(v).In particular, in [2], Doyen and Vandensavel give SQS(v) with blocking sets. It seems thatthere are no other known results for the existence of blocking sets in Steiner quadruplesystems.

In this paper we first prove a strong relationship between transversals and blocking setsof a Steiner quadruple system (Theorem 2.1). Then we investigate those systems SQS(v)that can be obtained by the 'doubling construction' (see Section 3 for the definition) andcharacterize the existence of blocking sets in them, giving a method to determine whetheror not SQS(2v) has a blocking set (Section 4). To derive these results, we need to studythe properties of factorizations of complete graphs (Section 3).

2. Transversals and blocking sets in quadruple systems

In this section we prove a result on the relationship between transversals and blockingsets of Steiner quadruple systems. Note that the first part of the following theorem alsoimplies Corollary 1.2, since if T is a blocking set, T and its complement V \T are twodisjoint transversals of the system.

Theorem 2.1. Let <f = (V ,M) be an SQS(v), v > 8.

(i) The transversal number ofSf is at least v/2.(ii) A set T a V, \T\ = v/2 is a blocking set if and only if T is a transversal.

Proof. For any subset W c ]/, let xt(W) denote the number of blocks of & havingexactly / points in common with W :

x i ( W ) = \{B G « : \ B n W \ = i}\.

Let us prove (ii) first. The 'only if part is clear by definition. Considering the conversestatement, let T be a transversal of ff such that \T\ = v/2, and let x, = x,(K \ T). SinceT is a transversal, and since any three points of the complement of T are in a uniqueblock, we obtain

/2\ t>-4/i>/2N

Furthermore, each pair x,y e V is contained in exactly (v — 2)/2 blocks, hence

v - 2+ 3A 2 i ^ • v •> ^ i


Finally, a point x e V is contained in exactly (v — l)(v — 2)/6 blocks; therefore,

(v-l)(v-2)vXi + 2x2 + 3X3 = 7 " X-

6 2Counting the number of blocks that have a nonempty intersection with V \ T, we

obtain

^ 1 + ^ 2 + ^ 3 + ^ 4 — (*1 + 2X2 + 3X3) — (X2 + 3X3) + X3(v-l)(v-2)v v-2fv/2\ v-4fv/2

^ +6 2 2 V 2 / 6 V 2

- 4V3ywhich is equal to the number of blocks of Sf. Thus, V \ T is a transversal, too, and T isa blocking set of £f.

In order to prove (i), one can apply a similar computation, which is just a little morecomplicated than the proof of (ii) above. Now we let x,- = x,(T). Assuming that T is atransversal, we obtain

x0 = 0,(v - l){v - 2 )

6

^1 ^ A 17 — 2 /^|T-|x? + 3x3 + 6x4 =

Consequently,

1 fv

2 V 2

. . = Xi + X2 + X3 + X4

2x2 + 3x3 + 4x4) - (x2 + 3x3 + 6x4) + (x3 + 4x4) - x4

should hold. One can observe that for v > 3 + ^/5 the right-hand side is an increasingfunction of |T| (as its first derivative is l/2(u/2 - \T\)2 4- l/24(i72 - 61; -h 4) > 0), andequality holds if |T| = v/2 and x4 = 0. Thus, if T is a transversal, it has to contain atleast v/2 points. •

3. Factorizations and separation number

In this section we study 1-factorizations of the complete graph K2m- Our motivation todo so is the following well-known operation on Steiner quadruple systems.

Doubling construction. Let ff' = (V',31') and Sf" = (V'\@l") be Steiner quadruple systemsof the same even order v = 2m, V n V" = 0, and let 3F' = {F[,F'29...F

fv_l} and

.?" = {F[\F2,...,F"_l} be arbitrary factorizations of the complete graph of order 2m

on vertex sets V and V", respectively. Define the quadruple system 5^ = (V\M) withV = V U V" and

& = &uar U {*>' Ue" : e' G Fl,e" G F[\ 1 < i < v - 1}.

One can see that Sf is a Steiner quadruple system. For this new system, we use the notationSe = $f' x y " . Also, if the factorizations J^, J^" are given, we write £? = (V\@\3?')and £f" = (V",09",!F"). Blocks of the form e' U e" are referred to as crossing blocks.

Turning now to the study of factorizations of complete graphs, let us introduce somedefinitions.

Definition. A factorization ^ = {Fi,...,F2m_i} of a complete graph K2m is called decom-posable if there is a partition X' U X" of the vertex set, \X'\ = \X"\ = m, such that foreach i, 1 1 must be even here inorder to admit \e^X'\ G {0,2} for some F,-.

Definition. An equipartition {X'\X"} separates a. factor F, if there is an edge e G F,- suchthat |e Pi X'| 7 1. (We sometimes say that the set X' separates F,- if the vertex set isunderstood.)

Lemma 3.1. Let 3F be a factorization of K2m. If an equipartition {X',X"} is not a decom-position of 3F, then it separates at least m factors of ^; otherwise it separates preciselym — 1 factors of 3F.

Proof. Since \X'\ = m, each factor has at most m/2 edges in X'. Suppose that {X',X")separates at most m — 1 factors F,-. Each of these factors can contain at most m/2 edgesin X'. Since we have a factorization of K2w, each of the m(m — l)/2 edges of X' has tooccur in some F, (in precisely one of them), and this implies that each F,- should containexactly m/2 (or 0) edges in X\ and in X" as well. In this case, however, the F,- separatedby X' form a factorization of X' and also of X" (because the other m/2 edges of F,- notcontained in X' must be contained in X"), so that {X',X"} is a decomposition of J^whenever it separates just m — 1 factors F,-. •

We note that if {X',X"} separates a factor F of the vertex set, there are at least twoedges e\e" G F such that <?' c X' and e" c X". The reason is that denoting by v! (byrc") the number of edges of F that are in X' (in X"), F contains precisely m — 2n' edgeshaving just one point in X', so that n1 = n" holds (and ri + n" > 0 by the definition ofseparation).

For small values of 2m, the situation is as follows. There is a unique 1-factorization of X4and of X6. The former is (trivially) decomposable; in fact each equipartition of the vertexset is a decomposition. On the other hand, the factorization of K6 is indecomposable (sincem = 3 is odd). We shall see later that the existence or nonexistence of a decomposablefactorization depends on the parity of m only.

The nonexistence of blocking sets in some class of quadruple systems will be proved byapplying the following observation.

Lemma 3.2. For every n = 2m (m> 3), Kn has an indecomposable factorization.

Proof. Let V = {vo,v\,...,vn-\} be the vertex set of Kn. Define a factorization SF —{Fu... Fw_!} as follows: Ft = {v0 vt} U {vi+j vH :l<j< (n/2) - 1} for i = 1,2,..., n - 1,where subscript addition is taken modulo n — 1.

We prove that J* is indecomposable. Suppose on the contrary that {X\ X"} decomposes3F. Assume VQ G X". Then X' has its n/2 vertices in {v\,...,vn-\} so that it contains twoconsecutive elements Vk and Vk+\ for some k\ let Vk f/c+i £ F,. Then F, contains a factor ofX\ i.e. vi+j G X' if and only if f,_; G X' (and i>, ^ X'). Note further that this symmetryproperty holds in each edge class of the factorization of X'.

Consider now the vertex vi 'next' to Vk+\ in Xr. By 'next' we mean that v\ G X' butVk+\ and v\ are the only elements of X' in the set {vk+\,Vk+2, -.-,t?/_i,t?/}. The edge t^i;/ iscontained in some factor, Fj say. Then k — j = j — / (mod rc — 1). Since Fj is a factor inX', Vk+\ is also incident to some edge of Fj. By the symmetry of F;, the edge should beVk+\Vi-\, contradicting the assumption that v\ is next to Vk+\. Thus, $F is not decomposable.

D

Our next objective is to give a more explicit description of quadruple systems, obtainedby a doubling construction, that have a blocking set. The structures of these 5^' x Sf"are determined by the factorizations 3F' and 3F" chosen on the vertex sets V and V",respectively, and by the permutation that tells which factor F[ is paired with which F".

Fix a vertex set V of size 2m and a set X' a V of size m. For a factorization 3* ofX2W, denote by siJF) the separation number of J^, defined as the the number of edgeclasses F, separated by {X\ V \ X'}. We have seen in Lemma 3.1 that s(J^) > m — 1 forall IF. We note that s(J^) = m can also hold: for instance, we can obtain such an 3Fby taking two isomorphic factorizations 3F1 and $*" on m points, say on {v[9...,v'm} and{vf{,... , ^ } , when m is even, together with the factors Ej = {vf

tvf/+j : 0 < i < m — 1}. So far

this factorization has separation number m — 1. Taking two isomorphic edge classes F'and F", F' U F" U £o consists of m/2 cycles of length four, and this union can be modifiedto obtain another factorization of K2m, which then has separation number m.

The following result characterizes the cases when Kim has a decomposable factorization.(For applications to quadruple systems, we shall not need the negative statement for modd.)

Theorem 3.1. A complete graph of order 2m has a decomposable factorization if and onlyif m is even.

Proof. The 'only if part follows from the fact that if {X\ V \ X'} is a decomposition ofsome factorization J*\ then some factors F, G 2F induce factors of X' as well, so \X'\ mustbe even.

To prove existence for m even, let m = 2k and denote the vertices by vo9v\9...9V4k-i,taking the subscripts modulo 4k (i.e., vi±4k = vt for all i). Define 2/c — 1 factors F, asfollows. For 0 < j < 2/c - 1 let F, = {vi-jVi+\+j : 0 < j < 2/c - 1}. One can see that these F,cover (precisely once) those edges vsvt for which 5 — f is odd. Thus, we can partition the

vertex set into two classes X' = {v2i+\ : 0 < / < 2/c - 1} and X" = {v2i : 0 < / < 2/c - 1},with the property that an edge e is contained in some F, (0 < i < 2/c — 1) if and only if\X' C\e\ = 1. Consequently, if {Fo,...,F^-i} is completed to a factorization J* of K2m inany fashion, then s(!F) = m — 1 holds and X' decomposes #". One possible way is to takethe factors of even subscripts constructed in the proof of Lemma 3.2. •

4. Blocking sets in factorized quadruple systems

Given an even positive integer w, let 3F be a factorization of Kw on vertex set W ={1,2,...,w} and let Fi,F2,...,Fw_i be its 1-factors. Let (3 be another factorization of Kw

on vertex set W = {l ' ,2' , . . . ,w'}, where W n W = 0, with 1-factors G1,G2,...,Gw_i.Bearing in mind the doubling construction described at the beginning of Section 3, each

permutation a on {1,2,...,w — 1} yields a family F = F(J r ,^ ,a) of 4-subsets of W U Wsuch that

{x,y,x\y'} € F if and only if {x,y} G F, and {x',j/} G Ga(/)

for some i G {1,...,w — 1}.

The sets in F may then provide the collection of crossing blocks, since if (W,&\) and(W\^i) are two quadruple systems of order w, then the pair (K,^) with V = W U Wand ^ = f , U ^ 2 U r ( ^ , ^ , a) is an

Definition. For v = 0 (mod 4), we say that an SQS(v) Sf = (V,&) is a factorized Steinerquadruple system, briefly an FQS(v), if & contains a family F(J^, ^, a), where a is apermutation on {1,2,...,(v/2) — 1} and J*\ ^ are two factorizations of Kv/1 on twodisjoint sets whose union is V.

Let ¥ = (K,^) be an FQS(i;) containing a family F(J^,^,a), and let

F = | J {x,y} and G= \J {x',y'}

for some arbitrarily chosen F, G J^ and G7 G ^.

Proposition 4.1. T/z hypergraphs < F > anJ < G > are sub-SQS(v/2)'s of Sf.

Proof. If x,y,z are three distinct points of F (respectively of G), there exists exactly oneblock B in ¥ that contains them. Let B = {x,y,z,w} e Si. If w G F, then 5 G < F > asrequired. Suppose that u e G, and let / G {1,2,...,v/2 — 1} be such that {x,y} e F;. Thenthere exists an element uf e G satisfying {W,M'} G Ga(/j and {x,y, M, wr} G ^?. Since ^ is aSteiner system, we obtain u' = z, wr G F, a contradiction. •

Proposition 4.2. ,4/t F2S(t;) ex/sts i/an^ only if v = 4 or 8 (mod 12,).

Proof. By Proposition 1, in every FQS(v) there exist two subsystems 52^(^/2). By theresult of Hanani [5], v/2 = 2 or 4 (mod 6), hence the condition v = 4 or 8 (mod12) is necessary. Conversely, if u = 4 or 8 (mod 12), we can consider two quadruple

systems (W,3?) and (W\$) of order v/2, with W n W = 0. The doubling constructionSQS(w) - • SQS{2w) then yields an FQS(v). •

In what follows, y = ( F ' , ^ ' , ^ ' ) and y " = {V"/$",^") denote two Steiner quadruplesystems of the same order, with V n K" = 0 and with factorizations !Ff and J^" on K',respectively on V". The next assertion follows by definition.

Lemma 4.1. / / T is a blocking set of\9 = &" x <f\ then TnV and Tn V" are blockingsets in £f" and in <f", respectively.

Lemma 4.2. / / Sf = 9" x £f" has a blocking set, then at least one of 3F' and .¥" isdecomposable.

Proof. Let V (T") be a blocking set in ¥' {Sf"\ T = T n V\ T" = T n V" (whereT is a blocking set of &>). If neither {T\ V \ T'} decomposes ,¥' nor {V, V" \ T")decomposes J^/r, then, by Lemma 3.1, V and T" separate at least v/2 factors of 3?f andof J^", respectively. Since \^'\ = \,¥"\ = v — 1, there must exist a subscript / such that 7"separates F,r, and T/r separates F/'. Say, V ^ e' e F\ and T" =) ^ e F" (such e' and e"exist - see the comment after Lemma 3.1). Then e' U e" is a block in £f' x ,9^", but e' U er/

is a subset of 7\ contradicting the assumption that T is a blocking set in y ' x ff". •

Now we are in a position to prove two closely related results. The first presents an infinitefamily of designs without blocking sets, while the second characterizes under precisely whatconditions the doubling construction yields a quadruple system with blocking sets.

Theorem 4.1. Let $f' = (V'9@f) and <f" = (V\@") be Steiner quadruple systems of order

v > 8 on disjoint point sets. Then there are factorizations 3*' and &" of V and of V",respectively, such that the SQS(2v), 6?' x <f" = (V\@\&') x (V",&'\<F") has no blockingset.

Proof. By Lemma 3.2, we can take indecomposable factorizations ,¥' and 3*" on V andon V". Then Lemma 4.2 guarantees that y ' x ^f" has no blocking set. •

The main result of this section is the following characterization theorem. Let us recallthat the cardinality of a blocking set (if it exists) in a Steiner quadruple system £f is halfthe order of y .

Theorem 4.2. Let £f = (V,&) be a factorized quadruple system of order 4v, such that<f = y x / , ¥' = (V, &',&'), <f" = (V\0S\&"). Then a set T a V of cardinality 2vis a blocking set if and only if the following five conditions are all satisfied:

(i) The sets T' = V n T and T" = V" nT are blocking sets in S?' and &"', respectively,(ii) {T\ V \ V) decomposes ^' or {T\ V" \ T"} decomposes <F".(in) The separation numbers satisfy s(J*r/) < v and s(^") < v.

(iv) If s(3F') = s(^F") = v — 1, t/zere is precisely one subscript i, 1 < / < 2v — 1, wit/i f/zeproperty that F[ is not separated by 7" and F" is rcof separated by T", and for everyj G {1,.. .,2v — 1} \ {/}, Fj is separated by T' if and only if F'j is not separated by T".

(v) If s(^F') = v — 1 and s(^F") = v (or vice versa), then for each i, 1 < i < 2v — 1, F- isseparated by T' if and only if F" is not separated by T".

Proof. Sufficiency is easy to see: if the assumptions (i) through (v) are satisfied by SP' andy " , then the blocks of $ and (%" are partitioned according to (i); and any other blockof £f is of the form e' U e" where e' e F[ and e" e F" for some i. Since at least one ofF[ and F" is not separated (by (iv) or by (v), according to the actual values of s(<F') ands(<^")), e' VJe" is indeed split into two nonempty parts by T.

The necessity of (i) and (ii) has already been verified in Lemmas 4.1 and 4.2. To prove(hi), (iv) and (v), recall that J*' and 3F" contain 2v—\— s{^') and 2v — \— s(J*") factorsnot separated by T', respectively, T". Moreover, the definition of 9" x f" implies that ineach of the 2v — 1 pairs {F'^F'-}, at least one of the two factors F- and F" is non-separated.Indeed, otherwise there would exist edges e' E F[ and e" G F" with e' cz T1 and e" a T",and hence the block e'Ue" e y'xSf" would be a subset of T, contradicting the assumptionthat T is a blocking set. Consequently, the number s* of subscripts / (1 < / < 2v — 1) suchthat neither F\ nor F[' is separated, is equal to ((2v-l-s(^')) + (2v-l-s(^")))-(2v-l\i.e., 5* = 2v - 1 - (s(^f) + s{<F")). On the other hand, s(&') > v - 1 and s(Jr//) > i? - 1hold by Lemma 3.1. Thus, 0 < s* < 1, implying the necessity of (iii), (iv), and (v). •

We have to note that Theorem 4.2 gives a necessary and sufficient condition for aspecified T of size 2v to be a blocking set in a factorized SQS(4v). Some factorizationsmay have several decompositions, and our characterization theorem says that at least oneof them should satisfy the requirements (i)-(v) if we wish to obtain a blocking set.

We conclude this paper with a construction that provides an example in which theFQS obtained does have a blocking set, but, for some suitably chosen decompositionof the corresponding factorizations, the requirement that precisely one of F\ and F[' isseparated is satisfied by none of the pairs {F/,F/'}. This structure will be an FQS(\6k) bydoubling some FQ5(8/c), where the ordering of the factors in the originally isomorphicfactorizations will be permuted. The 'twist' in the ordering will admit an 8/c-element setthat separates all factors in the union.

Construction 4.1. Consider a set of 8/c points, divided into four groups X\, Xi, X-$, X4,of cardinality 2/c each. Let 3F\ be a factorization on X\. Taking a one-to-one mappingbetween X\ and X( (i = 2,3,4), we find factorizations J^/ on Xt isomorphic to 3F\. Theisomorphic 1-factors of &\ U <¥i (of 3 U ,#4) provide 1-factors on X' = X\ U X2 (onX" = X^ UI4). Now / 1 U / 2 can be extended to a factorization <¥' of X' by factorizingthe pairs that join X\ with X2 (call these pairs 'X\ — X2 edges'; they form a completebipartite graph on 4/c vertices). The isomorphisms among the Xj extend J^3 U 4 to afactorization ^" of X" such that iF" = dF'. Taking the isomorphic mapping between i^2

and J^3 (but fixing 3F\ and F4 for the moment) we obtain 2/c further factors that coverthe X\ — X} and X2 — X4 edges, and the isomorphism between X\ and X3 (while fixing X2

and X4) covers the remaining edges of X\ UX4 and X2 U X$. Denote this factorization byJ*\ Obviously, both of the partitions (X\ UX2,X^ UX4) and (X\ UX3,X2 UX4) decompose3F (and so does (X\ VJX^X2 UX3), also, but this third partition is not needed for ourpurpose).

We are going to observe that two disjoint isomorphic copies 3F(\) and J*(2) of J* (on16/c points in all) can be mixed in such a way that Xi(l)UXi(2)UA3(l)uX3(2) satisfies therequirement described in (iv), but on the other hand the set X\{\) UXi(2) UX2(1) UX2(2)separates all factors, and therefore this latter union can by no means become a blockingset in the corresponding FQS obtained by doubling. To ensure this, we have to find asuitable permutation that defines a one-to-one correspondence between the factors ofF(\) and &(2).

There are 4/c factors between X\(i) U X2(i) and Xj(z) 11X4(1) (i = 1,2). A permutationof the desired properties for these factors is provided by an isomorphism X$(2) <-• X4(2).Indeed, X\(2) UX3(2) separates precisely those factors whose pairs are not separated byX\(l) UX3(1), while X\(i) U X2(i) does not separate any factor of this type. Next, assignthe 2/c - 1 factors of ^ i ( l ) U #"2(1) (of ^ 3 (1) U 4(1)) to all but one of the factors ofJ^(2) \ (J^i(2) U ^2{2)) (of &"{2) \ (.^3(2) U ^4(2)), and do the same for the factors ofJ^i(2) U ^2{2) and ^ 3 (2) U ^ 4 (2) . There is just one factor left in J^(l) and in #"(2), andthey are assigned to each other.

It can be verified that this permutation of the factors satisfies the properties givenabove.

References

[1] Berardi, L. and Beutelspacher, A. (to appear) On blocking sets in some block designs.[2] Doyen, J. and Vandensavel, M. (1971) Non-isomorphic Steiner quadruple systems. Bull. Soc.

Math. Belg. 23 393-410.[3] Eugeni, F. and Mayer, E. (1988) On blocking sets of index two. Annals of Discrete Math. 37

169-176.[4] Gionfriddo, M. and Micale, B. (1989) Blocking sets in 3-designs. J. of Geometry, 35 75-86.[5] Hanani, H. (1960) On quadruple systems, Canad. J. Math. 12 145-157.[6] Phelps, K. T. and Rosa, A. (1980) 2-chromatic Steiner quadruple systems. European J. Comb. 1

253-258.[7] Tallini, G. (1983) Blocking sets nei sistemi di Steiner e d-blocking sets in PG(r,q). Quaderno n.

3 Sem. Geom. Combinatorie Univ. L'Aquila.[8] Tallini, G. (1988) On blocking sets in finite projective and affine spaces. Annals of Discrete

Math. 37 433-450.

(1,2)-Factorizations of General Eulerian NearlyRegular Graphs

ROLAND HAGGKVIST and ANDERS JOHANSSON

Department of Mathematics, University of Umea, S-901 87 Umea, Sweden

E-mail address: [email protected], [email protected]

Every general graph with degrees 2k and 2k — 2,k^3, with zero or at least two vertices ofdegree 2k —2 in each component, has a A>edge-colouring such that each monochromaticsubgraph has degree 1 or 2 at every vertex.In particular, if T is a triangle in a 6-regular general graph, there exists a 2-factorization ofG such that each factor uses an edge in T if and only if T is non-separating.

1. Introduction

In this paper we will characterize those general graphs with degrees 2k —2 and 2k that canbe decomposed into spanning subgraphs with degrees 1 and 2 everywhere. Before we statethe result, it is perhaps of some interest to review some related problems and their history.

1.1. BackgroundOne of the starting points of graph theory is a classic investigation by the Danishmathematician Julius Petersen who in 1891 published a paper [7]: 'Die Theorie derregularen graphs', which contains a wealth of material on the problem of factorizingregular graphs into graphs of uniform degree k. An excellent source of informationconcerning Julius Petersen and problems spawned by his 1891 paper is the conferencevolume [1].

The motivation for Petersen's work, as given in the first few line£ of his article, came fromHilbert's proof of the finiteness of the system of invariants associated with a binary form.Petersen notes that Hilbert's proof employs a theorem by Gordan, which, among otherthings, implies that for a given n one can construct a finite number of products of the type

O l ~ *2) a 0l ~ XsY(X2 - Xz)7 ' ' • (*»-l - *»)'>

so that all other products of the same type can be built up by multiplying them together,

330 R. Hdggkvist and A. Johansson

the type in question being the property that the exponents are positive integers, possiblyzero, and that the degree in x1?x2, ...,xn is constant for each product. Petersen calls aproduct thus constructed a ground-factor, and sets himself the problem of determining theground-factors for every form of given degree and order. He notes a remarkable differencebetween forms of even or odd degree: the first have all ground-forms of degrees 1 or 2,whereas in the second case there exist infinitely many examples, the smallest with n = 10and of degree 3, found by Sylvester (writes Petersen), for which this is not true.

Returning to graphs, Petersen shows two theorems, namely, every 2/c-regular graphadmits a 2-factorization, and every 3-regular graph with at most one separating edge hasa 1-factor (and consequently a 2-factor). Petersen also set himself the task of determiningwhen two edges in a 4-regular graph always belong to the same 2-factor in the 2-factorization, or when they always belong to different 2-factors. This was to some extentmotivated by a statement in a letter from Sylvester, who erroneously believed that if asimple 4-regular graph admits a Hamilton decomposition, then every pair of edges can beseparated by 2-factors in some 2-factorization. Petersen worked with an auxiliary graph,called the stretched graph, and obtained a slightly cumbersome criterion (see Sabidussi [8]for references and a discussion of this particular problem). In this context, note thefollowing characterization, from Sabidussi, determining when two edges in a 4-edge-connected Eulerian graph have the same or different parity in every Euler tour of the graph:if two edges e a n d / a r e parity equivalent, G — e and G— / a r e both nonbipartite, whileG — {e,f} is bipartite. The similar, easier, problem for diregular digraphs was settledcompletely by a simple lemma in [2] concerning regular bipartite graphs: two given edgesin a A>regular bipartite graph with k > 2 can be separated by a proper edge-colouring if andonly if they do not form a separating set. When k = 2 the condition is obvious: the edgesshould not be of the same parity in a common component.

The above prompts the question determining conditions ensuring that three given edges<?,/and g in a bipartite regular graph belong to different colours in some edge-colouring,or the more general problem of determining when a given partial three edge-colouring canbe completed to all of B. Unfortunately this question has not yet been resolved, despitesome notable efforts, in particular by Hilton and Rodger [3]. Equivalently, we could ask forcriteria ensuring that a diregular digraph admits a 1-difactorization such that three givenedges are completely separated by the 1-difactors (a 1-difactor is a spanning set of cycles).

It is therefore of some interest that the corresponding question for general graphs admitsa solution, as long as the prescribed edges form a triangle. The general question would be:when do three prescribed edges e, /and g in a regular Eulerian graph G lie in three different2-factors in some 2-factorization of G? The answer when the three prescribed edges forma triangle is, as shall be seen here, that such a 2-factorization exists if and only if the triangleis nonseparating and the degree is at least 6. In this theorem, loops and multiple edges areallowed. In fact a more general theorem will be proved, which determines when it is possibleto find a balanced ^-edge-colouring of an Eulerian graph with degrees 2k and 2/: —2everywhere (i.e. every colour must appear at least once and at most twice at each vertex).The condition is that if k = 2, no component has an odd number of vertices of degree 2,and if k > 2, no component contains exactly one vertex of degree 2k —2.

An auxiliary motivation for a resolution of this particular problem comes from a

(1,2)-Factorizations of General Eulerian Nearly Regular Graphs 331

problem about embedding partial Steiner triple systems with multiplicity A on r verticesinto Steiner triple systems on n vertices. It is well known that this problem is TVP-completefor A = 1, and indeed for odd A, but it will be seen elsewhere [6], that, as conjectured byHilton and Rodger in [4], there do exist natural conditions that are necessary and sufficientfor such an embedding when A is even.

1.2. Definitions and the theoremGeneral graphs (sometimes called pseudographs or multigraphs with loops) are considered,i.e., graphs are allowed to have multiple edges and loops. The degree of a vertex v belongingto a graph G, denoted by dG(v), is the number of edges, with loops counted twice, thatcontain v. If all vertices have the same degree r, then G is called r-regular. All graphs arefinite.

An edge-colouring a of a graph is a mapping a: E(H) H> Q of the edges into some set Qof "colours". It is called a fc-edge-colouring if \cr(E(G))\ ^ k, i.e., when at most k coloursare used. In this paper we will, somewhat sloppily, sometimes refer to edge-colourings ascolourings; vertex colourings do not appear.

We use the notation da(v) = dG (v) for the chromatic degree of a vertex v in an (edge-)coloured graph, where Ga is the monochromatic factor Ga = Gla'1^)] (the graph induced bythe edges assigned colour a). For bichromatic factors, the notation Ga/? = G[cr~l({ot, fi})] isused, and so on.

If da{v) = 2, for any colour a and any vertex v, the colouring is said to be a 2-factorization. If

da{v) = 1 or 2, VaeQ \/ve V(G),

we have a (1,2)-factor'ization.A colouring that satisfies

for all pairs of colours a and /? is said to be equalized. Let us call a colouring vertex-balancedif the degree-difference is at most 1, i.e.,

for all pairs a, /? of colours and for all vertices v in the graph.The following theorem states the main result. It was stated as a conjecture in a somewhat

different form by Hilton and Rodger in [4] and [5, Conjecture 2]. These authors are mainlyinterested in the extension-properties of certain partial Steiner triple systems, a problem wewill not attempt to settle in this paper.

Theorem 1. Let G be a connected general graph, such that all vertices have degree 2k or 2k — 2,for some k > 1. Then G admits an equalized (l,2)-factorization if and only if the number ofvertices of degree 2^ — 2 is either 0 or at least 2, and not an odd number ifk = 2.

We note that this immediately implies the following corollary.

Corollary. Let G be a connected 6-regular general graph and let T a G be a triangular


subgraph ofG. A 2-factorization such that all three edges of T are coloured differently existsif and only if T is non-separating.


It is quite evident that the conditions in Theorem 1 are necessary. Each vertex of degree2A: — 2 is the endvertex of exactly two monochromatic paths, and since the other endvertexof one of these paths must have degree 2k — 2, we have, if any, at least 2 vertices of degree2k —2. Also, since G is finite, the monochromatic paths make up a collection of cycles. Ifk = 2, we have only two colours, and the paths along the cycle alternate in colour,changing colours at the 2k — 2-vertices, which clearly means that the number of thesevertices is even.

So, the real issue is then to prove that these conditions are sufficient. We first give somelemmas, and then, ultimately, the proof of Theorem 1.

The problem of finding (l,2)-factorizations in Eulerian graphs is closely related to thetheory of Eulerian trails and Eulerian orientations. An example of this is the proof of thefollowing lemma, which is needed as a starting point in the proof of Theorem 1.

Lemma 1. Theorem 1 is true if the number of vertices of degree 2k —2 is even.

Proof of Lemma 1. Let E = x1x2...xm_1xmx1,xie V be an Eulerian tour, and give theedges in G the corresponding forward orientation. Assume that S = {a19..., a2k} is the setof vertices of degree 2k — 2, and assume, without loss of generality, that they occur in Einthe given order, i.e., starting at ax the first vertex in S, distinct from a19 that occurs is a2,and so on.

The edges on the segments [al9 a2], [a3, a4] , . . . , [a2k_l9 a2k] of E are now given the reverseorientation; the vertices in S thereby obtain the (oriented) degrees:

d+(a2i) = k, d~(a2i) = k-2d+(a2i+i) = k-2, d~(a2i+1) = k.

All other vertices still have out- and in-degree k. If an oriented alternating cycle C on thevertex-set S, with degrees

is added to the graph, the result will be a regular general di-graph with in- and out-degreek. That any such has an (oriented) 1-factorization is a well-known fact. This induces a 2-factorization on the underlying undirected graph, and since the added cycle C isalternating, any two consecutive edges on this must get different colours. Consequently, weobtain a (l,2)-factorization after deleting C. •

2.1. Eulerian 2-colourings. Given a connected Eulerian graph G = (F, E), i.e., a graph withvertices of even degree, we may give the graph an Eulerian 2-edge-colouring as follows: pick

(\,2)-Factor izations of General Eulerian Nearly Regular Graphs 333

any Eulerian tour and colour the edges alternately a and /? along the tour, starting andending at a prescribed vertex x. The resulting 2-edge-colouring satisfies

da(v) = dfo)

at all vertices, except possibly at x (if \E(G)\ is odd), where the difference in the exceptionalcase will be 2.

Suppose now that we have a graph with at least one (and hence at least two) vertices ofodd degree. Since the number of odd vertices is even in any graph, the odd vertices may bepaired off as

By joining each of these paired odd vertices by a subdivided edge ^ .w. j . , for some newvertex wi9 with the possible exception of one pair, xt9yt say, which instead is joined by a newedge xtyt, an Eulerian graph with an even number of edges can be constructed. An Euleriancolouring of the new graph clearly induces a vertex-balanced 2-colouring on the originalgraph. Moreover, this colouring is equalized, since the subdivided edges have one edge ofeach colour. We state this observation as a lemma.

Lemma 2. Let G = (V, E) be a connected graph and assume that the number of edges \E(G)\is even or that G has at least one vertex of odd degree. Then G admits an equalized vertex-balanced 2-colouring. If the degrees of G are all 2, 3 and 4, this colouring is also a (1,2)-factor ization.

Note that if dG(V) c {2,3,4}, a vertex-balanced 3-colouring is immediately a (1,2)-factorization.

Remark. Note that Lemma 2 implies that

a graph has a (l,2)-factorization if and only if it has an equalied one.

This is easily seen, since if we have a (1,2)-factorization, then by applying Lemma 2 on allnon-equalized components of the bichromatic factors Ga/?, we eventually end up with anequalized colouring of all such components, and, by a suitable renaming of the colours ineach component, we obtain an equalized colouring. Hence, in the following, we maydismiss the discussion of the equalized property altogether.

The following technical lemma (which actually is contained in the preceding proof) isneeded in a key step of the proof.

Lemma 3. Let F be a connected graph with two distinct vertices x and s of odd degree, wherethe degree ofx is at least three. Suppose x and s are in the same component of the graph F\xy,for some edge xy,y 4= x, s. There is then a vertex-balanced 2-colouring of F such that themonochromatic factor containing the edge xy has degree one more at x than the other factor.

Proof of Lemma 3. Split the vertex x into two vertices x' and x" in such a way that x hasdegree 1 and is only joined to y by the edge x'y. This may or may not split the graph into twocomponents, but the component containing x" has the odd-degree vertex s and the

component containing x' also has odd vertices, one of which is x' itself. Lemma 2 impliesthat the graph admits a vertex-balanced colouring. Since the degree of x" is even, the edgesincident with x" are equally divided between each factor. Restoring x by identifying thevertices xf and x", we have obtained the sought for colouring of the original graph. •

2.1. Proof of Theorem 1We now turn to the proof of Theorem 1. First a definition: given a coloured graph G, withcolouring a, a recolouring of a subgraph H is a colouring a of G such that (T'\G^H = cr\G\H,i.e., a new colouring that only differs on the edges in H.

The idea behind the proof of the theorem is that if S has an odd number of vertices, wecan at the very least, by Lemma 1, find a colouring that is a (l,2)-factorization except forone vertex. This colouring we then transform by a sequence of Eulerian recolourings ofbichromatic components, so that we eventually can apply Lemma 2 on the (by that timealtered) bichromatic component that misses one colour at a vertex. Implicit in the proof isa polynomial algorithm for the problem of finding a (l,2)-factorization in our type ofgraph.

Proof of Theorem 1. Assuming the theorem to be false, we choose G as a graph that fulfilsthe conditions of Theorem 1, but fails the conclusion that it admits a (l,2)-factorization.Let S be the set of vertices of degree 2k — 2. We may assume that |5 | is odd, and hence atleast 3, since the case with \S\ even is handled by Lemma 1. Consequently, Theorem 1 is truefor k = 2, and therefore we have at least 3 colours.

Pick an S-path P, i.e., a path between two distinct vertices a,beS, with all interiorvertices in the complement of S.

If we add one loop to any prescribed vertex zeS not equal to b, we are again in thesituation covered by Lemma 1, so the resulting graph has a (l,2)-factorization. Thisinduces a /c-edge-colouring cr on G that clearly satisfies

ds(v) = 1 or 2 (A)

for all colours S and all vertices v, except at the unique vertex z = z(a) in S, where onecolour, which we name a, say, is missing.

We have at least three colours, and all colours have degree 1 or 2 at b, so there is alsoat least one colour /?, such that

3. (B)

Let H = H(<T) be the component of Ga/i that contains z. By Lemma 2, we may assume that\E(H)\ is odd and that no vertex in H is of odd degree, since any (1,2)-factorization of Hgives a (l,2)-factorization of G. So, by (B), b is not a vertex of H.

We may assume though, that

V{H) intersects V(P),

since this holds if the chosen vertex z is the endpoint a of P. It is therefore possible toassume that the colouring (and also /?) is chosen such that

the distance between V{H) and b along P is minimal. (C)

(\ ,1)-Factor izations of General Eulerian Nearly Regular Graphs 335

Hence there is a vertex x in V(P) f] V(H) of minimal distance to b along P. Since b is nota vertex of //, x is joined in P to a vertex y 4 V(H) closer to b by an edge xy e P. The colourof xy is called y and is distinct from a and /? (since otherwise >> would be nearer to b thanx is along P). The typical situation is illustrated in Figure 1.

Figure 1. The colours a, /? and y are distinguished by one, two and three crossbars respectively.

We may make some further assumptions on this colouring: we first note that since thecomponent of Ga/? containing b is disjoint from H, we may interchange the colours a and/? in this component, without violating (A), (B) or (C). This observation, together with (B),makes it legal to assume that

dfib) + dy{b) is even. (1)

This means that (B) will still hold after any (l,2)-factorization of the graph Gfir

If x = a, we may also assume that z = a and then interchange the colours /? and yglobally, which is legal since (1) implies that (B) then still holds. However, y is now in thesame component of Ga/jy as z = a, which contradicts (C). It is therefore established that xmust be an interior vertex of P, and hence that

dG{x) = 2k and, by (A), ds(x) = 2 (2)

for all colours S, since P is a S-path.We now fix this colouring and call it <r, and in the rest of the proof we recolour the graph

in three steps. In this process we will only change the colours a, /? and y, and the result willbe a colouring cr\ that satisfies (B) and (A) with the exception of the colour a at somevertex z = z(a'). But this vertex will be in the same a/?-component as the vertex v,contradicting (C).

The first step is to recolour H in colours a and /?, starting at x with colour /?. Since weknow that \E{H)\ is odd, the resulting colouring satisfies (A) for all colours and all vertices,except at x where the discrepancy is 2, which by (2), means that d^x) = 3 and da(x) = 1.

Now, let F be the component of the current G^y that contains the edge xy. The degreeof x in F is 5, by the previous recolouring in a and /? and by (2), but for any other vertexv =t= x the degree is between 2 and 4, since (A) is satisfied at v. The situation is illustratedin Figure 2.

Note also that, as d^(x) = 3 is odd, there is another vertex s of G^-degree 1 joined by a

Figure 2. The graph Ga/3y with / / recoloured.

monochromatic /?-path to x. This vertex is also in H, where dH(s) = 2, since H does notcontain any vertex of degree 3. But, the degree in G is 2k — 2 and we have k colours,dy{s) = 2, and hence

This means that the conditions in Lemma 3 are fulfilled for the edge xy and the verticesx and s, therefore we may recolour Fin such a way that the degree d^x) is still 3. At the sametime, the edge xy is coloured /? instead of y. Since the recolouring is also vertex-balancedon F and at all vertices except x the degree is 2, 3 or 4, (A) is now satisfied for all verticesand colours except at x, where d^x) = 3 and da(x) = 1. The condition (1) ensures moreoverthat: if b should be a vertex of i7, this recolouring has not changed the condition (B).

As the last step of the proof, we consider the component of the current Gajj that containsthe edge xy, and call this graph / / ' . Note that dH,(x) = 4, and for the other vertices it isbetween 2 and 4, since (A) is valid there. If//' has an even number of edges, or some vertexin / / ' is of odd degree, we find a (l,2)-factorization of / / ' by Lemma 2. This contradictsthe choice of G, since we have thus obtained a (l,2)-factorization.

Consequently, H' has an odd number of vertices of degree 2, and the rest have degree4. But, by adding a loop at some vertex z' of degree 2, we can find a recolouring of / / ' ina and /? that satisfies (A), for all vertices except at z\ where da(z') = 0. Moreover, thisrecolouring has not changed (B), as this vertex cannot be in / / ' , since its a/?-degree is 3.

However, the resulting colouring <r' now contradicts (C) since y is in the same componentH' = H{af) as z' — z{cr') is. This proves that, contrary to assumption, the graph G musthave a (l,2)-factorization. •

References

[1] Andersen, L. D. et ai (1992) Special volume to mark the centennial of Julius Petersen's 'DieTheorie der regularen Graphs'. Discrete Mathematics 100 101.

[2] Haggkvist, R. (1976) A solution of the Evans conjecture for Latin squares of large size. In:Proceedings Fifth Hungarian Colloquium, Keszthely 1976, vol. 1. Combinatorics 18.

[3] Hilton, A. J. W. and Rodger, C. A. (1991) Edge-colouring regular bipartite graphs. Graphtheory {Cambridge, 1981) 56. North-Holland, Amsterdam-New York, 139-158.

[4] Hilton, A. J. W. and Rodger, C. A. (1990) Edge-Colouring Graphs and Embedding PartialTriple Systems of Even Index. Cycles and Rays (NATO ASI Series, eds.), Kluwer, 101-112.

(1,2)-Factorizations of General Eulerian Nearly Regular Graphs 337

[5] Hilton, A. J. W. and Rodger, C. A. (1991) The Embedding of Partial Triple Systems when 4Divides lambda. Journal of Combinatorial Theory Series A 56 109-137.

[6] Johansson, A. (1993) A Note on Embedding Partial Triple Systems of Even Index (inpreparation).

[7] Petersen, J. (1891) Die Theorie der regularen Graphs. Ada Mathematica 15 193-220.[8] Sabidussi, G. (1993) Parity Equivalence in Eulerian Graphs (preprint).

Oriented Hamilton Cycles in Oriented Graphs

ROLAND HAGGKVIST1 and ANDREW THOMASON2

department of Mathematics, University of Umea, S-901 87 Umea, Sweden2DPMMS, 16 Mill Lane, Cambridge CB2 1SB, England

We show that, for every e > 0, an oriented graph of order n will contain n-cycles ofevery orientation provided each vertex has indegree and outdegree at least (5/12 + e)n andn > no(e) is sufficiently large.

1. Introduction

Dirac's theorem states that every graph G with minimum degree S(G) ^ |G|/2 has ahamilton cycle. The simplest analogue for digraphs is given by the theorem of Ghouila-Houri [3]. Given a digraph G of order n and a vertex v G G, we denote the outdegree ofv by d+(v) and the indegree by d~(v). We also define d°(v) to be min{d+(v), d~(v)}, andS°(G) to be min{d°(v) : v G G}. Ghouila-Houri's theorem [3] implies that G containsa directed hamilton cycle if S°(G) ^ n/2. Only recently has a constant c < 1/2 beenestablished such that every oriented graph satisfying S°(G) > en has a directed hamiltoncycle; Haggkvist [5] has shown that c = (1/2 — 2~15) will suffice. He also showed that thecondition S°(G) ^ n/3 proposed by Thomassen [9] is inadequate to guarantee a hamiltoncycle, and conjectured that S°(G) ^ 3n/8 is sufficient.

When considering hamilton cycles in digraphs there is no reason to stick to directedcycles only; we might ask for any orientation of an n-cycle. For tournaments G, Thoma-son [8] has shown that G will contain every oriented cycle (except the directed cycle if G isnot strong) regardless of the degrees, provided n is large. For general digraphs, Grant [4]proved that G contains an antidirected hamilton cycle if S°(G) ^ 2n/3 + ^Jnlogn; anantidirected cycle is one in which the edge orientations alternate (of course n has to beeven).

We know of no published result in this vein which covers all oriented ^-cycles. Howeverwe recently proved [6] that any digraph with S°(G) ^ n/2 + n5/6 contains every orientedrc-cycle, provided n is large enough. Our purpose in this paper is to consider the analogousproblem for oriented graphs. We believe that the condition S°(G) ^ (3/8 + e)n will beenough to guarantee all oriented n-cycles in any sufficiently large oriented graph. Here weshall prove that the condition d°(G) ^ (5/12 + e)n is enough.

340 R. Haggkvist and A. Thomason

The proof of our theorem is based on the expansion properties of a graph with largeminimum degree. As such, we expect that the machinery developed (by refining the ideasof [6]) could be used to prove similar results for any directed graph having a suitableexpansion property. In particular, the methods here give an alternative way to prove thedigraph result of [6], though we shall not make this explicit. It is our intention to exploreelsewhere a possible extension to a more general context. We hope also to prove thepresent theorem under the weaker constraint that S°(G) ^ (3/8 + e)|G|. The reason fornot proving this stronger result here is a purely technical one, which we have not yet hadthe opportunity to tackle. The problem is pointed out in section 6.

Let e be some constant which remains fixed throughout the paper. Note that thedefinition of 8° implies e < 1/8 throughout. Several times we shall claim that a statementis true "provided n is sufficiently large". This will mean that there exists an rc0 = no(e) suchthat the statement is true provided n > no. In particular, just how large is "sufficientlylarge" will depend on e only, and not on any other parameters.

Here is some notation. Given an oriented graph G and a vertex v e G we denote byT+(v) the set of out-neighbours of v and by T~(v) the set of in-neighbours. Given anoriented path P, the length of P is denoted l(P); the two paths A(P;k) and Z(P;k) arethe paths spanned by the first k edges and the last k edges of P respectively. If two pathsP and Q are isomorphic we may write P = Q or even P = Q. The path PQ is the pathof length l(P) + l(Q) formed by identifying the end of P with the beginning of Q. Wemay also identify the end of Q with the beginning of P to form a cycle, also denoted PQ;whether PQ denotes a path or a cycle will be clear from the context.

2. A first strategy

The proof of the main theorem and the number of supporting lemmas might appear, atfirst sight, to be a mass of technical details. Indeed, the technical difficulties encountered inimplementing our basic strategy are considerable. Nevertheless the essence of the strategyis very straightforward. Consequently, it is worth devoting a paragraph or two to anoutline of the idea underlying our construction of a given rc-cycle C in an oriented graphG. The reader will thereby be able, later on, to distinguish the wood from the trees.Crucial to the method are two devices for finding collections of paths, namely pipelinesand sorters.

Definition. A pipeline of width s and length t is an oriented graph whose vertex set comprisest+1 subsets So,..., St, each of order s, such that Sj_i n S*•= 0, 1 < i ^ t. It has the propertythat for any s oriented paths Pj = x/,o-X/,i • • • */,* of length t, 1 < j ^ s, there exist vertexdisjoint copies of Pj with Xjti G Si, 1 < j ^ s, 0 ^ i ^ t. (The set So is called the start of thepipeline and the set St the end. Usually the sets Si will be mutually disjoint, for otherwisethe Pj may be realised as trails rather than as paths.)

We will show that a randomly chosen sequence of subsets St c V(G) almost surelyspan a pipeline, provided S°(G) is large. Note that a pipeline guarantees the existence ofgiven paths Pj but does not allow us to specify the end vertices within So and St. If such

Oriented Hamilton Cycles in Oriented Graphs 341

a specification were possible, it would be easy to find C in G, at least if n were a multipleof t, as follows. Partition V(G) into t subsets So = St,Si,...,St forming a pipeline. Let Cconsist of paths PiP2.. .Ps and let So = St = {y\,...,ys}- Then there would be copies ofPj in the pipeline joining y; to y7-+i, 1 ^ 7 < 5 (where ys+i denotes yi), and so we wouldhave a copy of C. It is possible that a randomly chosen collection of subsets S, wouldallow such specification of endvertices, but we do not investigate this here. Instead weachieve a similar effect by using a sorter.

Definition. Let P = xoX\... xt be an oriented path of length t. A sorter for P of widths, or (s,P)-sorter, is an oriented graph whose vertex set comprises t + 1 disjoint subsets$i = {yi,h---9ys,i}> 0 ^ i ^ t, such that for any permutation a o /{ l , . . . , s} there exist svertex disjoint copies Pj = x/,o*/,i • • • xj,t of P, 1 ^ j ^ s, with Xjj 6 S,-, Xj$ = X/,o andxu = y<ru\t> 0 < i < t, 1 ^ j ^ 5.

Note that a sorter is stronger than a pipeline in that it allows specification of endvertices,but weaker in that it requires the paths Pj to be isomorphic. However, if we can buildboth a pipeline and a sorter in G we can still, under suitable circumstances, constructC as follows. Suppose we can write C = P\Q\PiQ2...PsQs> where Pi = ... = PS9

KQi) = ... = /(6s) and both /(Pi) and l(Q\) are not too small. We shall show in section3 that, if S°(G) ^ (3/8 + e)\G\9 then there exists a constant k = k(e) such that we canconstruct, between any two given vertices, a path of length k with any desired orientation.We call such a path a handbuilt path. Write Pj = AJPJZJ, where l(Aj) = l(Zj) = k. Thefollowing plausible construction for C is, in fact, feasible. Construct a sorter for P[ of widths and construct a pipeline of width 5 and length l(Q\). Join the end of the sorter to the startof the pipeline with s handbuilt paths isomorphic to Z\ and join the end of the pipeline tothe start of the sorter with s handbuilt paths isomorphic to A\. Then C can be found in G.

The reason the above strategy fails in general is that it is not possible to find isomorphicpaths P i , . . . ,P s equally spaced around C. We therefore have to accept that the lengthsof the Qj may be unequal. To construct paths of differing lengths from a pipelinerequires some extra manipulation with handbuilt paths. To obtain the elbow room forsuch manipulation we shall use two sorters and another pipeline to find Pi , . . . ,P s , whiledropping the constraint that these paths be isomorphic. But by now it is time to beginthe proof in detail.

3. Expansion properties

In this section we give some elementary lemmas describing how expansion properties yieldshort oriented paths with prescribed endvertices. We first show that graphs with largeminimal degree are expanders.

Lemma 1. Let G be an oriented graph of order n with S°(G) ^ (3/8 + e)n. Let A c V(G)satisfy \A\ ^ 3n/8, and let O^n ^ 1/80. Then

\{yeG; \r~(y) HA\> nn}\ ^ (1/4 + 2e - \\n)n + \A\/2.

In particular \T+(A)\ ^ (1/4 + 2e)n + \A\/2.

342 R. Hdggkvist and A. Thomason

Proof. Let A* = { y e G ; \T~(y)nA\ > nn}, let B = An A* and let C =A\A*. Since theindegrees in C are all at most nn, there are at most \C\rjn edges in C. But each vertex in Ghas total degree at least (3/4 + 2e)n, so there are at least (3n/4 + 2en — (n — \G\))\C\/2 edgesin C. Therefore \C\ ^ n/4+2nn-2en, and so |B| - \A\-\C\ ^ |X|-(l/4+2*j-2e)n ^ n/10.

Within 5 there are at most C ') edges, and the number of edges from B into C isat most \C\rjn. Thus e(£, G \ ,4), the number of edges from B into G\A, is at least\B$8° - \B\/2) - \C\r\n. Writing D for 4* \A we have

|D||B| + (n - \A\ - \D$rjn ^ e(B9G\A)> \B\{3° - \B\/2) - \C\nn,

and since n-\A\- \D\ + \C\ = n - \B\ -\D\^n^ 10\B\ we have |D| + lOrjn ^ 3° - \B\/2.Therefore \A*\ = \B\ + \D\ ^ 3° + \B\/2 - lOnn ^ (1/4 + 2e - llf/Jn + \A\/2. D

Note that Lemma 1 implies that sets of size less than (1/2 + 4e)n expand, in the sensethat |r+(^4)| > \A\. This fact means the graph has small diameter, as we now show.

Lemma 2. Let G be an oriented graph of order n with 3°(G) ^ (3/8 + e)n and let x andy be vertices of G. Let 4[log2(l/e)"| ^ k ^ en/4, and let P be an oriented path of lengthk. Then, if n is large enough, there will be a path from x to y isomorphic to P. Moreoverthere can be found a set of at least en/4k disjoint such paths.

Proof. We show first that a copy of P exists if k = 2[log2(l/e)l and n > (12/e)*.Suppose, for the sake of argument, that the first three edges of P go backwards, forwards

and backwards. Let A = F~(x), let n = e/Yl and define fi by \A\ = (1/2 + 4e — 22r\ — jx)n.Note that \i < 1/2. Let A* = { y e G ; \r~(y) n A\ > nn] and let A** = { y eG ; |F+(y) f) A*\ > nn}. Each vertex of A* can be reached from x by a forward-backward trail in nn ways and, by Lemma 1, \A*\ ^ (1/2 + 4e — 22n — \i/2)n. Likewiseeach vertex of A** can be reached from x by (nn)2 backward-forward-backward trailsand \A**\ (1/2 + 4e — 22n — n/4)n. Hence after \_k/2\ such steps there are at least(1/2 + 3e — 22n)n ^ (1/2 + e)n vertices which can each be reached in (nn)^/2^1 ways bytrails oriented like the first half of P. Likewise there are (l/2 + e)n vertices which can eachbe reached in (nn)^k^2^~l ways by trails oriented like the second half of P. Since 2e ^ rj, xcan reach y by at least (nn)k~l trails oriented like P. At most nk~2 of these trails can beself-intersecting and the rest (a positive quantity if n > (l/n)k~l) must be paths.

To find longer paths we apply induction on /c; an x-y path of length k is found byselecting a suitable neighbour z of y and finding an x-z path of length k — 1. The lowerbound of 4|~log2(l/e)l claimed in the lemma allows for the reduction from e to e/2 in theexpression for 3° as this process is used up to en/4 times.

A set of disjoint paths can be found by repeatedly removing from the graph the internalvertices of any path found, and reapplying the above argument to find another path inthe remaining graph. Even after en/4k applications there still remains a graph G with8°{G') ^ (3/8 + e/2)\Gf\, so at least en/4k paths can be found. •

Definition. Let 0 < X < 1. A pair of disjoint subsets S, T of the vertices of an oriented graphis said to be i-expanding if for every subset A ^ S with 0 < \A\ ^ \S\/2, the inequalities

\T+(A)nT\ ^ (l+X)\A\ and \r~(A)nT\ ^ (1+X)\A\ both hold, and for every subset B^TwithO< \B\ ^ | T | / 2 , the inequalities \T+(B)nS\ ^(l+X)\B\ a n d | r " ( B ) n S | ^ ( 1 + A ) | 5 |both hold.

The next lemma shows that, if ^-expanding pairs of subsets are available, small orientedpaths can be built wherein the choice of the internal vertices is constrained. Paths builtby means of this lemma will be nicknamed handbuilt paths.

Lemma 3. Let G be an oriented graph and let 0 < X < 1. Let So, ...,Sfc be disjoint subsets

of V(G), each of order s, such that each pair Sj_i, S, is X-expanding, 1 ^ i ^ k. Let P be

some oriented path of length k, let vo G So and let Vk G S&. Then there exists a copy vo...Vk

of P joining v$ to Vk in G, such that v\ G S,, 0 ^ i ^ k, provided k ^ 1(4/A) log2 s\.

Proof. We show that the path P exists provided k ^ 2t, where t is the smallest integersuch that (1 + Xf ^ s/2. This is sufficient to prove the lemma, since 2t = 2[(log2s —1)/ log2(l + X)] ^ 2log2 s/ log2(l + X) ^ (4/X) log2 5.

Consider first the case k = 2t. Since the pair So, -Si is ^-expanding there are at least(1 + X) choices for v\ G Si. Let A a S\ be the set of these choices. Since the pair Si, S2

is ^-expanding there are at least (1 + X)2s choices for vi G S2 (that is, there are at least(1 + X)1 vertices b in S2 for which there is a vertex a e A such that the path v$ab isisomorphic to A(P\2)). Continuing in this way we see that there are at least (1 + X)1 ^ s/2vertices w G St such that there exists a path vov\ ...vt = w isomorphic to A(P;t) withvt G Sf, 0 ^ i ^ t. Similarly there are at least s/2 choices of w for which there is a copyw = vtvt+\ ...V2t of Z(P;t) with vt G Si9 t 2t select a vertex Vk-\ G Sk-\ so thatthe edge Vk-\Vk has the orientation required; this can be done because the pair Sfc_i, Skis ^-expanding. The induction hypothesis ensures the existence of a copy of A(P;k — 1)joining vo to Vk-u which extends to the desired copy of P. •

Later we shall see that it is quite easy to find A-expanding pairs, after which Lemma 3will prove very useful. In fact the rate of expansion we can achieve in oriented graphsof high minimum degree is much greater than that required by Lemma 3, and handbuiltpaths of constant length (that is, 0(1/e) independently of \S\) are achievable. Howeverthe notion of A-expansion as defined is one which is customary in the literature and morenatural in other contexts.

4. Sorters

A sorting network of width s with t stages is an undirected graph consisting of disjointsets of vertices So, Si, . . . , Su each of size 5, say S, = {x,-5i,...,Xj>s}. The edges of the fthstage consist of [s/2\ disjoint 4-cycles Xi-ijXjjX/-i,fcXi,fc for various pairs of indices j andk. The sorter has the property that the input vertices So can be joined to the outputvertices St in any prescribed order by s vertex disjoint paths of length t. As is well known,

Ajtai, Komlos and Szemeredi [1] have described a sorting network using only Clogsstages. Batcher [2] has described a simple sorting network using |~(log2s)2] stages, andthis is the one we shall use. Any reasonable sorting network would suffice for the presentapplication.

It is clear that if the 4-cycles Xi-ijXjjXi-ijfcX^ in the sorting network were replacedby disjoint paths x,-_ij = yoy\...ym = xuj and x,-_i^ = zoz{ ...zm = xlVfe, along with twoedges ym-\zm and ymzm-i, then the resultant graph would still function as a sorter. Letm = (mi,...,mt). An (s,m)-sorter will be a sorting network in which the 4-cycles arereplaced in this way, the paths used in the fth stage all having length mt. The length ofthe (5, m)-sorter will then be J2t mi- Suppose now P is an oriented path of the same lengthas the (s, m)-sorter. It is clear that the edges of the (5, m)-sorter can be oriented in sucha way that the s paths from So to St will always be oriented like P. Such an oriented(s, m)-sorter is therefore an (s, P)-sorter.

Theorem 4. Let G be an oriented graph of order n with 5°(G) ^ (3/8+e)n. Let she a naturalnumber with s < n1/2 and let P be an oriented path of length (3[log2(l/e)l + l)|~(log2s)~|2.Then G contains an (s,P)-sorter, provided n is large.

Proof. From the discussion above it can be seen that (s,P )-sorters exist; let us fix ourminds on one such and construct a copy of it in G. Suppose the first few stages of thesorter have been constructed, up to say the class S,_i, so that the length of every stageis k + 1, where k = 3[log2(l/e)~|. We construct the next stage as follows. Let P' be thesubpath of P, of length k + 1, which will need to traverse the gap between S,-_i and S,-.Let Q = A(Pf;k) and let Q* be the path of length 2/c made from two copies of Q byidentifying their terminal vertices. If the sorting network, from which the (s,P )-sorter wasderived, requires a 4-cycle based on x,-_ij and Xj_i,/c, select h = [en/8k\ vertex disjointXi-ij-xt-ik paths each oriented like Q* and which avoid all vertices used so far in theconstruction. These h paths can be found by applying Lemma 2 to the graph consistingof G after the removal of the vertices used so far in the sorter. Let H be the set of the hmidpoints of these paths.

Assume now that the final edge of P' is a forward edge (the argument is very similar ifthe edge is a backward edge). Each vertex of H has at least p = (3/8 + e/2)n neighboursamong the set U of vertices not so far used in the sorter or in the h paths linking x,-_ij toxt-ijc via H. Since w(/ip

2/w) > (2), where u = \U\, at least two vertices of U, call them xtj

and Xij<, will receive edges from the same two vertices of H, called say y and z. From thetwo copies of Q* joining x,_ij to x,-_i}fc via y and z, select copies of Q joining xz-_ij to yand xt-ik to z. Hence we have copies of P' joining x,-_ij to xtj and x/_i^ to x,- , plus thetwo extra edges needed for the sorter. The vertices of St are now created by performingthis operation for all pairs 7, k which are the bases of 4-cycles in the fth stage. •

5. Pipelines

The main tool in our proof is the pipeline, defined earlier. In this section we shall describehow our pipelines are to be constructed. In fact a condition on a sequence of sets So,..., St

which is nearly sufficient to guarantee that a pipeline is formed is that the pairs Sj_i, S,-be ^-expanding. The property of ^-expansion can be used in two ways. One is to enableus to form handbuilt paths, as in Lemma 3. The other is to derive matchings.

Definition. Let S and S' be two subsets of the vertex set of an oriented graph, with \S\ = \Sf\.We say S m a t c h e s Sf if there is a set of \S\ independent edges from S to Sf.

L e m m a 5. Let S and S' be disjoint sets of vertices in an oriented graph with \S\ = \S'\,such that the pair S, Sf is ^-expanding. Then S matches S'.

Proof. The Konig-Hall theorem tells us that if S does not match Sf then there is a set Awith \A\ ^ \S\/2, such that either A c S and \T+(A)\ < \A\ or A a S' and |r~(>4)| < \A\.However, these are both ruled out by the 2-expansion of the pair S, Sf. D

It is clearly necessary that two consecutive sets St-\ and St in a pipeline match eachother. More generally, when trying to find copies of s paths in a pipeline, we may needto extend partial paths from Si-\ to 5, via u forward edges and s — u backward edges.The reader is reminded that the sets 5,- we shall eventually use will be chosen at random.It would be ideal if we were able to guarantee that every w-subset A c St-\ matchedsome w-subset B c Si, but that is not the case. We can, however, ensure that almost everyw-subset A c Si-\ matches almost every w-subset B c St, which will do. Nevertheless, eventhis is too much to hope for if u is small (say u = 1); in practice we will have to extendfrom Si-i to Si via handbuilt paths if u is small. These considerations lead to the nextdefinitions, describing the property we actually need to build pipelines.

Definition. Let 0 < a < 1 and let r be a natural number. A pair of disjoint w-subsets W,W of the vertex set of an oriented graph is said to be (r, a)-wed if for every integer u withu = 0, u = worr^u^w — r, and for each of at least (1 — a)(^) of the u-subsets B a W'',there are at least (1 — a)(^) u-subsets A a W such that both A matches B and W \ Bmatches W \ A.

Definition. Let 0 < X, a < 1 and let r,s be integers. A pair of disjoint s-subsets S, S' ofthe vertex set of an oriented graph is said to be a (X, r, a)-matched pair if the following twoconditions hold:

(I6r/X)\og2s ^ s and (8/A)log2s < I/a, andfor every integer h < (16r//l)log25,/or every (s — h)-subset T <= S and for every (s — h)-subset T' c= S', the pair T, T' is both X-expanding and (r,ct)-wed.

The following theorem shows that matched pairs of sets will form a pipeline.

Theorem 6. Let So,...,Stbea sequence oft+1 subsets forming the vertex set of an orientedgraph, each subset having order s. Suppose that each pair St-i, Si is a (X,r, unmatched pair,1 ^ x t. Then So,...,St form a pipeline of width s and length t, provided t < l/(2a).

Proof. Let k = [(4/1) log2 sj. Observe that k < l(2a) by the definition of a (X, r, a)-matchedpair. Moreover we may assume that t ^ k. For if t < k we may extend the sequence of setsSi by adding extra sets of vertices and edges forming (A, r, a)-matched pairs, so that thelength of the sequence becomes at least k. If now the extended sequence forms a pipeline,so did the original sequence.

Let Pj = Xj#Xj,i... Xjit9 1 ^ j ^ s, be s oriented paths of length t, and let 0 ^ m ^ t. Wesay a labelling of Sm by x;,m, 1 ^ j ^ s, is reachable if there exist vertex disjoint copies ofthe subpaths x/,ox/,i • • • */> with Xjj G S,-, 0 ^ i ^ m. We need to show there is a reachablelabelling of St. Of the q\ labellings of Sm let (1 — 8m)q\ be reachable. Clearly <5o = 0; ifwe show that Sm+k ^ 8m + 2/ca (or (Sf ^ 5m + 2(£ — m)a if t — m < 2k), then St ^ 2fa < 1,which proves the theorem. We shall show then that 8m+i ^ 8m + 2/a, provided k ^l <2k.

For 1 < f ^ / let F, be the set of paths Pj with the edge x7;m+i_ix7;m+, oriented forwardfrom x;?m+,_i to Xjjn+u and let 2?, be the s— \Ft\ other paths. If either Ft or J5t is non-emptybut small we will have to take special care with those paths. We form the set H of pathsneeding care by the following simple algorithm. Initially H = 0; now repeat the followingstep. If, for some i, 0 < \Ft\H\ < r holds, replace H by HuFt. Likewise, if 0 < |B,-\H| < rholds, replace H by H U Bt. If no index / has one of these two properties, stop. Observethat, since 5 4/cr > 2/r, no index i can be used in more than one step of the first / stepsof the algorithm. Therefore the process terminates after at most / steps with \H\ = h,where h ^ 2/cr. We may suppose that the paths in H are Pi,...,P/,.

For each choice fi = (yu...,yh) of a sequence of h vertices from Sm, let there be(l-^)(s-h)\ reachable labellings of Sm with xj/n = yJ9 l^j^h. Then E M ( 1 - ^ ) ( s ~ / z ) ! =(1 — 5m)s\, the sum being over all s(s— 1)... (s — h +1) choices for fi. It follows that /^ ^ 5m

for some choice of \i\ we make such a choice now and keep it fixed for the remainder ofthe proof.

Make a choice v = (zi,...,z/,) of h vertices from Sm+i. By applying Lemma 3 h times(making use of the A-expansion of pairs of subsets of the St) we construct handbuilt copiesof the subpaths xj/n... x ; > + / , 1 ^ j ^ h, with x ; > = yJ9 x ; > + / = Zj and xj/n+i e Sm+i,0 < i < /. Let Tm+i be the set of s — h vertices in Sm+i not used in these subpaths, 0 ^ i ^ /.We will show that (1 — Sm — 2lcc)(s — h)\ of the labellings of Tm+/ are reachable from Tm,via the sets Tm+i, by paths not in H. Therefore (1 — dm — 2h)(s — h)\ of the labellings ofSm+/ will be reachable in such a way that x;?m+/ = z7, 1 ^ j ^ h. This holds true for anyof the s(s — 1)... (5 — h + 1) choices for v, and summing over these choices we see that atleast 5ZV(1 — dm — 2ltx)(s — h)\ = (1 — Sm — 2lot)s\ labellings of Sm+/ are reachable. Thus8m+i 8m + 2/a, as claimed.

To complete the proof, therefore, it is enough to show that (1 — 8m — 2/a)(s — h)\of the labellings of Tm+/ are reachable via Tm,..., Tm+/_i. In fact we shall show that(1— Sm—2cc)(s—h)\ labellings of Tm+\ can be reached via Tm; analogously (l—Sm—4cc)(s—h)\labellings of Tm+2 can be reached, and so on until the desired outcome is achieved. Letu = \F\ \ H\, so \Bi\H\ = s - h - u . B y t h e d e f i n i t i o n o f H, e i t h e r u = 0 or u = s - hor r ^ u ^ s — h — r. For each w-subset B c Tm + 1 there are u\(s — h — u)\ labellings

(not necessarily reachable) of Tm+i so that x;?m+i G B if Pj G F\\H and x ; m + i ^ 5 if

Pj e B\\H; let L B of these labellings be reachable. Likewise, for each u-subset A c 7m,

there are ul(s — h — u)\ labellings of Tm so that x7>m e A if Pj• e F\ \ H and x7> ^ A ifPj e Bi\H; let L^ of these be reachable by paths not in // .

Suppose now that for some choice of A c Tm and # c Tm+i it happens that A matchesB and Tm+i \ B matches Tm \ A. Fix these two matchings. Each of the LA reachablelabellings of Tm can now be extended to a different reachable labelling of Tm+i, soit follows that LB ^ LA. Since (Sw,Sm+i) is a (2, r, a)-matched pair, there are at least(1 — u)(s~h) choices for B for which there are at least (1 — a)(s~h) choices for A such thatLB ^ LA.

The sum of all (s~^ values of LA is at least (1 — Sm)(s — h)!, by choice of \i. Denote thesum of the (1 — a)(s~ ) smallest values of LA by L. Since LA < u\(s — h — u) for every A,we have

Therefore

Y,B LB ^ J2B m a x { ^ : A matches B and Tm+i \ B matches Tm \ A}

(S~uh)~l EB J2{LA : A matches B and Tm+l \ B matches Tm \ A}

^ (1 _ a ) ( i _ Sm _ a ) ( 5 _ / , ) ! ^ ( i _ ^ _ 2a)(5 - / i ) ! .

Notice that the number of labellings of Tm+i which can be reached via Tm is precisely^2B LB, which we now see is at least (1 — Sm — 2a)(s — h)\, as claimed. This completes theproof of the theorem. •

6. Robust pipelines

Our aim in this section is to show how to find a pipeline within an oriented graph. Infact, we shall show that a randomly chosen sequence of sets Si form a pipeline. Thiscannot happen if d°(G) < (3/8 — e)\G\, since examples of such graphs exist which arenot expanders, and in that case a randomly chosen pair of subsets is very unlikely tobe A-expanding. We believe that if S°(G) > (3/8 + e)\G\, in which case the graph is anexpander (by Lemma 1), it is likely that a randomly chosen sequence of sets will yield apipeline. Given such a pipeline, the machinery for proving our main theorem for graphswith this value of d° is all in place.

However, our present proof that randomly chosen sets form matched pairs (and so, byTheorem 6, a pipeline) does not make full use of the expansion properties of G. Rather,we achieve the required effect by using only properties of the neighbourhoods of pairs ofvertices. In consequence, our proof works only for S°(G) > (5/12 + e)|G|. We hope to havethe opportunity to repair this deficiency in the future. In the meantime, we shall need thefollowing definition.

Definition. Let e > 0 and let G be an oriented graph of order n. A set of vertices S a V(G)is said to represent a set X cz V(G) if \S n X\ > (\X\/n - e/2)\S\. The set S is said to be

e-typical if it represents all the following sets:

r-(x), r+(x) u r+oo, r-(x) u

m) = {yeG : | r+(x) UT+OOI > m },

Y-(x,m) = {y£G : | r " ( x ) U T-(y)\ > m },

for all x, y G G and for all integers 0 ^ m ^ n.

To exploit the property of typicality we need a simple estimate for the size of the setsY+(x,m) defined above.

Lemma 7. Let x be a vertex of an oriented graph G and let m ^ d+(x). Then

|7+(x,m)| ^ 4S°(G) + 2d+(x) - Am.

Proof. Let A = F+(x) \ Y+(x,m). If A ^ 0, then some vertex of A sends at least3°(G) - (d+(x) - \A\) - (\A\ - l)/2 = 8°-d+ + (\A\ + l)/2 edges out of r+(x). But by thedefinition of Y+(x, m), no vertex in A can send more than m — d+ edges out of F+(x), so\A\/2 < m — 3°. This inequality remains true even if A = 0.

Let B = V(G) \ (Y+(x,m) U T+(x)). If B ^ 0, then the subgraph induced on B hasminimum total degree at least 23°(G) — (\G\ — \B\), so some vertex in B sends at least3° — (\G\ — \B\)/2 edges to vertices within B. This quantity must be at most m — d+, so\B\/2 < m - d+ - 3° + |G|/2. Once again, this inequality holds even if B = 0.

Adding the two inequalities which have been derived we obtain that \A U B\ ^ Am —A30 - 2d+ + \G\. But Y+(x,m) = V(G) \(AuB) so the proof is complete. •

We can now show that, in a graph with a large value of 3°(G), typicality is a guaranteeof expansion.

Lemma 8. Let G be an oriented graph of order n with 3°(G) ^ (5/12 + e)n. Let S and Sf

be e-typical subsets with \S\ = \Sf\. Then the pair S, S' is e-expanding.

Proof. Let s = \S\ = \Sf and let A cz S with 0 < \A\ = a ^ s/2. Let x e A and letm = [(a/s + e)n\. We will show A n Y+(x,m) ± 0. This is certainly true if m < d+(x),since then Y+(x,m) = V(G). But if m ^ ^+(x) we can make use of the estimate given byLemma 7, and the fact that S represents 7+(x,m), to obtain that

\S H Y+(x,m)| > (\Y+(x, w)|/n - e/2) s ^ (6^° - 4m - e/2)s

^ (5/2 + 3e/2)5 - Aa (1 + 3f/2)s - a > \S \ A\.

We may therefore select yeAnY+(x,m), the possibility that y = x being permitted.Since y e 7+(x,m) we see that | r+(x) U T+(y)\ ^ m + 1 > (a/s + e)n. But Sr is 6-typicaland so represents T+(x) U T+(y). Hence

\S' n (r+(x) U r+(y)) | > ((a/s + e)- e/2)s >(a + es/2) > (1 + e)|A|.

Likewise |F~(^)| ^ (1 + e)\A\, and the same two inequalities hold for subsets A a S'. Itfollows that the pair S, S' is e-expanding. •

To obtain properties of random subsets we need simple bounds on the tail of the hyper-geometric distribution. In fact the usual bounds on the tail of the binomial distributioncan be used. The following lemma is proved in [6], following Janson who proves strongerresults [7].

Lemma 9. From an urn containing pn red balls and (1 — p)n blue balls, j balls are chosenat random and without replacement. Let X be the number of red balls among the j chosen.Then, for h^O,

Qxp(-2jh2) and F{X ^(p + h)j} ^ exp(-2y/z2).

The next lemma shows that, roughly speaking, a randomly chosen set S will be verytypical, as will any small perturbation of it.

Lemma 10. Let G be an oriented graph of order n and let S be a randomly chosen subset ofV(G) of size s ^ (logn)9. Let r > (50/e2)logn be an integer. Then, provided n is sufficientlylarge, with probability at least 1 — 1/n the set S is has the following property:

Given y e S and x e {y} U (F(G) \ S), let S* = (S \ {y}) U {x}. Then S* is e-typicalMoreover, given an h-subset H c f , with h < (logn)3, and an integer u with u = 0,u = s — horr^u^s — h — r, then at least (1 - l/n)(s~h) of the u-subsets of S* \Hare e-typical.

Proof. Let X a V(G) and let J a V(G) be a randomly chosen subset of size \J\ = j ^ r.

By Lemma 9,

P{|J nX\^ (\X\/n - e/4)j} < e~e2j/* < n~6/8.

Therefore the probability that J fails to be (e/2)-typical is at most l/2n4, typicality beingdefined by the representation of at most An1 subsets. Moreover, if \J\ = j ^ (logn)5 andh < (log n)3 then the probability that J is not (e/2)-typical is at most

Now let S be a random s-subset of V(G) and let u ^ r. The expected number ofw-subsets of S failing to be (e/2)-typical is at most ^ Q ) , so with probability at least1 — 1/n2 there are at most ^ Q ) such w-subsets. Moreover, if u ^ (logn)5 and H is aspecific /z-subset of S, the expected number of u subsets of S\H failing to be (e/2)-typicalis at most s~hn~4(s~h). We call a set H bad if S\H contains more than ^(s~h) non-typicalw-subsets. Thus the probability of a given H being bad is at most s~hn~~3. Hence theexpected number of bad H within S is at most n~3, so with probability exceeding 1 — 1/n3

there are no bad H in S. These calculations were all performed for fixed values of uand h\ taking into account all possible values, we conclude that with probability at least1 — 1/n the set S has the following properties: it is itself (6-/2)-typical, for each u ^ r atleast (1 — l/2n)Q) of the w-subsets of S are (e/2)-typical, and for each h < (logrc)3, each

/z-subset H a S and each u > (logn)5 there are at least (1 — l/n)(s~/j) w-subsets of S \ Hwhich are (e*/2)-typical.

It suffices to show that any set S with the properties just described will also have theproperty claimed in the lemma. First of all, it is clear that S* is e-typical because S is(e/2)-typical. Now let y, x and H be as defined in the lemma, and, for a w-subset A a S,let A* = A if y £ A and otherwise let A* = (A\{y})U{x}. Once again, A* will be 6-typicalif A is (<r/2)-typical. Therefore, if u ^ (logn)5 and H is any /z-subset of S* there will be atleast (1 — l/n)(s~/l) w-subsets of S* \ H which are e-typical. It remains only to verify thesame property in the case r ^ u < (logn)5. But in that case 2(h + l)u < s and so

u J \uj \ s — u J s — u 2'

and since (1 — l/2w)Q) w-subsets A* cz S* are e-typical it follows that at least

sets A* a S* \H are e-typical. This completes the proof of the lemma. •

We are now ready to show that a randomly chosen sequence (So,..., St) of sets forms apipeline. In fact, when we finally come to work with the pipeline in the proof of the maintheorem, we shall need to make a few alterations to the pipeline after we have chosen itbut before we make use of it. Naturally, we shall need to know that the alterations wemade have not destroyed the pipeline property; this is the motivation behind the nextdefinition.

Definition. A pipeline (So, Si, . . . , Sf) in an oriented graph G is robust in G if, for any choiceof vertices yt e S/ and for any choice of distinct vertices xt G {.y/}U(F(G)\|Jy=0 Sj), 0 ^ i < t,the sequence (SJ,..., S*) is also a pipeline of width s and length t, where S* = (S,-\ {)>,•} )U{x,-}.

Theorem 11. Let G be an oriented graph of order n with 5°(G) ^ (5/12 + e)n. Then Gcontains a robust pipeline (So,Si,...,Sf) of width s and length t in which each set S/ ise-typical, provided s > (logn)9, 3 ^ t ^ n/s and n is sufficiently large.

Proof. Given a random sequence of t + 1 disjoint s-subsets (So, Si, . . . , Sr), the probabilitythat any set S, fails to have the property stated in Lemma 10 is at most 1/n. Thus theexpected number of sets failing to have the property is less than one, and so G containsa sequence (So,Si,...,Sr) in which every set has the stated property. We claim that sucha sequence forms a robust pipeline. To verify the claim, and so prove the theorem, itsuffices by Theorem 6 to show that any pair of sets S*_x, S* (defined by the property inLemma 10) form an (e, r, 2/n)-matched pair, where r = |~(50/<f2)logH~|.

Let S = S/_i and Sf = S,. Let S* and S'* be defined in the obvious way. Notethat (16r/e)log2s ^ s and (8/e)log2s < n/2, as needed. Let h be an integer withh ^ (16r/e)log2s < (logrc)3. Let H c S* and H1 a S'* be /z-subsets. Let T = S* \ H andT' = S'* \ H'. Then, for any integer u with u = 0, u = s — h or r^u^s — h — r, there are

at least (1 — 2/n)(s~h) w-subsets A a T for which both A and T \A are e-typical, and thesame is true for (1 — 2/n)(s~h) w-subsets B a T'. By Lemmas 8 and 5, for such subsets, Amatches B and V \ B matches T\A. Therefore the pair T, T' is (r, 2/n)-wed. Moreover,the case u = s — h shows that the pair T, V is e-typical and so, by Lemma 8, ^-expanding.This completes the proof of the theorem. •

7. Oriented cycles

We now have all the ingredients needed for the proof of the main theorem.

Theorem 12. Given 0 < e < 1, there exists no = no(e) with the following property. Let Gbe an oriented graph of order n > no with 8°(G) ^ (5/12 + e)n. Let C be an oriented cycleof length n. Then G contains a copy of C.

Proof. As indicated in section 2, we shall decompose C into a collection of paths sochosen that they can be found in G using pipelines and sorters joined by handbuiltpaths. Let s = Lexp{(loglogn)(logn)1/4}J > (logrc)9, let k = [(4/e) log2 s\ + 2 , let / =(3pog2(l/e-)] + l)[(log2s)2] and let t = [nl/2\. Some estimates in the proof will hold onlyif n is large; we will assume without further comment that n is sufficiently large for thoseestimates to be valid.

First, letting p = 21 + 2k + t, split the cycle C into [n/p\ paths each of length p(except for one whose length is between p and 2p), and then from these select [n/p\/2non-incident paths of length p. Since [n/p\/2 > s22/+2/c, we can choose 5 of these pathsP( , . . . ,P ; such that l(P[) = ... = /(P/) = p, A(P[;l + k) ^ ... ^ A(P's\l + k) andZ(P{;/ + fe) = ... =Z(P s ' ; / + fc). Hence C = P|Q/

1P2/g2...P;g;, where l(Q'j) p for each

In order to find the paths Pj in G, we write Pj = AJBJPJCJDJ, 1 ^ y" < S, wherel(Aj) = l(Dj) = /, l(Bj) = l(Cj) = k and l(Pj) = t. Notice that, by the choice of the Pj,we have A\ = ... = As, B\ = ... = Bs, C\ = ... = Cs and D\ = ... = Ds. We shall laterconstruct two sorters of width 5 and length / (one for the Aj and one for the Dj), apipeline of width s and length t (for the Pj) and join them together by handbuilt paths oflength k (the Cj and the Dj), thereby realising the paths Pj.

Unlike the paths Pj, 1 ^ j; ^ 5, the paths Q'j may be of unequal length, so requiringmore care in their construction. Let Q'j = EjQjFj where l(Ej) = l{Fj) = /c, 1 ^ j ' ^ s, sothat our n-cycle can be written

C = P[EXQXFXP^E2Q2F2...P'SESQSFS.

We shall find in G a path Q = Q\TiQ2T2...Qs-\Ts_iQs, where the paths Tj have lengthone (they are just edges); the orientations of the Tj are immaterial. We have therefore

s

l(Q) = s - 1 + ] T l(Qj) = s - l + n-sp- 2sk.

7=1

Let Q be split into consecutive paths i ^ , . . . , ^ where l(Rj) = [l{Q)/s\, 1 < j ^ s, and< s. Finally let R'j = HjRj where l(Hj) = k, 1 < j < 5, and let u = l(Ri) = ... =

352 JR. Hdggkvist and A. Thomason

l(Rs) = lKQ)/s\ -k; thus l(Ro) = l(Q)-(u + k)s. We shall find the cycle C in this way. Theshort path R^ will be handbuilt and the paths R\,...,RS will be obtained from a pipelineof length u. The ends of these paths will then be linked by handbuilt paths H\,...,HS

to form the path Q. By deleting the edges Ti, . . . , Ts_i we obtain Qi,...,<2s. These latterpaths will then be linked to P{, . . . , Ps' via handbuilt paths Ej and Fj, 1 < j ^ s.

The above describes the manner in which the cycle C can be constructed from variouspaths, but the order of the operations is critical. The difficulty is that each of the operations(building a pipeline, a sorter or a handbuilt path) requires a large population of verticesfrom which to choose, and towards the end of the construction of C the availablepopulation becomes very small. It is at this point that the notion of robustness will beneeded. We are now ready to give the exact method by which the cycle is constructed.

First, using Theorem 4 we may find in G an (5,^4i)-sorter 2^. Similarly we may findan (s,Di)-sorter Z^. A copy of the path Ro may be found in the remaining graph viaLemma 2, since 1(RQ) < s < en/4. Partition the remaining vertices into t + u + 2 sets oforder 5, plus a set X of residual vertices; note that

\X\=n-us- l(Ro) - 1 - (r + 2/ + 4)s = 5s(k - 1).

From Theorem 11 it can be seen that the sets of order s can be chosen to form a robustpipeline of length t + u + 1, which we break into two robust pipelines, namely 11/? oflength u and HP of length t. The sets of these pipelines are all ^-typical, as provided byTheorem 11.

Find, in YlR, copies of the paths Ru...,Rs. We are now in need of 5s paths of length/c, namely, the Hj to form Q and hence Q\,...,QS, the Ej to join the end of E/> to thestart of the Qj, the Fj to join the ends of the Qj to the start of TA, the Bj to join theend of HA to the start of Flp, and the Cj to join the end of lip to the start of HD. Toconstruct, say, the path H\ from the endvertex x of RQ to the first vertex y of R\, selectk — 1 consecutive s-sets from the pipeline lip, say Sj+\9...,Sj+k-i. Using the 6-typicalityof these sets, choose an appropriate neighbour x' of x in Sj+i and a neighbour y' of y inSj+k-\- In view of the pairwise e-expansion of these sets proved in Lemma 8, Lemma 3shows that the rest of H\ may be found linking x' to y'. Since t > 5s(k — 1) all the 5spaths required can be handbuilt in this manner without using more than one vertex fromany s-set of lip. Replace the vertices removed from lip with those of X, so forming H'P.

The robustness of lip implies that YlfP is itself a pipeline of width s and length t. We

can therefore find P\9...9PS in lip and extend these via the Bj and Cj to HA and Z/>.Within Z^ and Z/> we can find copies of the Aj and the Dj, so finding copies of P{,. . . , Ps

r

between the start of I,A and the end of Z#. Since Z^ and Z/> are sorters, we can arrangethat the ends of the P- link correctly with the Ej and the Fj to form the cycle C as desired.

•

References

[1] Ajtai, M., KomlosJ. and Szemeredi, E. (1983) Sorting in C logn parallel rounds, Combinatorica3 1-19.

[2] Batcher, K. (1968) Sorting networks and their applications, AFIPS Spring Joint Conf. 32307-314.


[3] Ghouila-Houri, A. (1960) Une condition suffisante d'existence d'un circuit hamiltonien,C.R. Acad. Sci. Paris 251 495-497.

[4] Grant, D. D. (1980) Antidirected Hamiltonian cycles in digraphs, Ars Combinatoria 10 205-209.[5] Haggkvist, R. (1993) Hamilton cycles in oriented graphs, Combinatorics, Probability and Com-

puting 2 25-32.[6] Haggkvist, R. and Thomason, A. Oriented hamilton cycles in digraphs, (to appear in J. Graph

Theory).[7] Janson, S. Large deviation inequalities for sums of indicator variables (preprint).[8] Thomason, A.G. (1986) Paths and cycles in tournaments, Trans. Amer. Math. Soc. 296 167-180.[9] Thomassen, C. (1979) Long cycles in digraphs with constraints on the degrees, in Surveys

in Combinatorics (B. Bollobas, ed.) London Math. Soc. Lecture Notes 38 211-228. CambridgeUniversity Press.

Minimization Problems for Infinite n-ConnectedGraphs

R. HALIN

Mathematisches Seminar der Universitat Hamburg, BundesstraBe 55, D-20146, Hamburg, Germany

A graph G is called «-minimizable if it can be reduced, by deleting a set of its edges, to aminimally ^-connected graph. It is shown that, if ^-connected graphs G and H differ only byfinitely many vertices and edges, then G is «-minimizable if and only if H is ^-minimizable(Theorem 4.12). In the main result, conditions are given that a tree decomposition of an^-connected graph G must satisfy in order to guarantee that the ^-minimizability of eachof the members of this decomposition implies the «-minimizability of the graph G(Theorem 6.5).

1. Introduction

It is an obvious fact that every finite ^-connected graph can be reduced by deleting edgesto a minimally ^-connected (^-minimal for short) graph. However, the situation changescompletely if we consider infinite ^-connected graphs for some n ^ 2. (Throughout thisarticle, n denotes a positive integer ^ 2.) In the finite case we reach an ^-minimal factorsimply by deleting edges successively, with the sole restriction that the ^-connectedness ispreserved at every step. Clearly this method fails in the infinite case. An ^-connected graph

Figure 1. Figure 2.

356 ^- Halin

G is called n-minimizable if there is a set of edges L in G such that G — L is ^-minimal. Eachsuch ^-minimal factor G — L of G is called an ^-minimization of G. The infinite ladder ofFigure 1 is the classical example of a 2-connected graph that is not 2-minimizable, whereasthe locally finite 2-connected graph of Figure 2 has uncountably many pairwise non-isomorphic 2-minimizations. (We get all 2-minimizations by deleting one or two (suitable)edges in each Kx x 2.)While ^-minimal graphs have been thoroughly studied in the literature, there is, to theauthor's knowledge, only one major result on rc-minimizability, namely the followingtheorem of R. Schmidt [9].

Every n-connected graph that is rayless {i.e. that does not contain a one-sided infinite path)is n-minimizable.

The motivation for the present article arose from the search for some of the deeperreasons why an infinite ^-connected graph G is, or is not, «-minimizable. An edge of G iscalled redundant if its deletion does not destroy the ^-connectedness; Rn(G) denotes the setof all redundant edges in G. Clearly, the effect of deleting a redundant edge, or a set ofredundant edges, on the 'redundancy character' of the other edges is of basic importancein this context; it is studied in Sections 3, 4 and 5.

In Section 3, sets of edges whose deletion does not destroy the ^-connectedness arestudied from a 'global' point of view. Theorem 3.6 shows that if Rn(G) is infinite, there isan ^-connected subgraph H of G such that Rn(G) = Rn{H), the order \H\ of H equals\Rn(G)\, and H is ft-minimizable if and only if G is ft-minimizable. From this we see that ourminimization problem is reduced to the case that \G\ = \Rn(G)\; G is then called 'of fullredundance'.

The 'local' aspects of the problem are considered in Section 4. It is shown thatelementary operations (deleting or adding an edge, deleting or adding a vertex of finitedegree) leave ft-minimizability invariant (provided that we stay in the class of ^-connectedgraphs). The key observation for all that follows is Lemma 4.2, which states that deletinga redundant edge of G diminishes Rn(G) only by finitely many edges. (This observation isnot as obvious as one might perhaps expect at first glance.) Theorem 4.7 is seminal for themain result in Section 6. Roughly speaking it says that if the ^-connected graph / is pastedtogether from two ^-connected graphs G and H along a common finite subgraph, thenRn(J) differs from Rn(G) U Rn(H) only by a finite set of edges.

In Section 5 we try to imitate the reduction procedure of successively deleting redundantedges, which leads to an ^-minimization in the finite case. We get certain maximal well-ordered sequences of redundant edges, which will be called reducing sequences. Thesmallest ordinal occurring as the order type of a reducing sequence of an «-minimizablegraph is defined as its minimization type. All ordinals that may occur as the minimizationtypes of graphs with full redundance are characterized. For ^-connected graphs that cannotbe minimized, a structure theorem is proved (see Theorem 5.6), which in the locally finitecase exhibits the presence of an infinite ladder. Section 5 stands by itself, and is not neededelsewhere in this article.

Section 6 investigates ^-connected graphs G that have a tree-decomposition with n-

Minimization Problems for Infinite n-Connected Graphs 357

connected members GA. (Roughly speaking, such a representation arises by a well orderedsequence of pasting operations as considered in Theorem 4.7.) In the main result of thepresent article we show the conditions under which the ft-minimizability of each GA impliesthat G is ^-minimizable also. These conditions are as follows:1. The decomposition tree (associated to the decomposition with the members GA) is

rayless;2. In each GA there are only finitely many vertices with neighbours outside GA.This result may be considered a generalization of Schmidt's theorem, since, by Diestel [9],every rayless ^-connected graph has a tree-decomposition with rayless decomposition treeand finite ^-connected members.

The simplest case not covered by Schmidt's theorem is that of 2-connected graphscontaining a ray but no double ray (two way infinite path). A discussion of this case withrespect to 2-minimization is given in Section 7. In Section 8 a few other relatedminimization problems are considered, and in the final section some open problems arepresented.

2. Prerequisites

The graphs considered in this article are undirected and do not contain loops or multipleedges. In general, we adopt the terminology and notation that has become standard ingraph theory.

If G = (F, E) is a graph, then \G\ denotes the cardinality of Fand ||G|| the cardinality ofE. An edge joining vertices a, b is denoted ab. If T ^ F, by (\) we mean the set of all abwith a 4= b in T; (^) is the edge set of the complete graph KT with vertex set T. For L c= (T

2),we denote the graphs (F, E—L), (F, E U L) by G — L and G u L , respectively. In the specialcase L = {<?}, we write G — e and G[}e instead of G — {<?}, G U {e}. Union and intersection ofgraphs are formed by joining and intersecting the vertex sets and edge sets of the graphsin question. A denotes the symmetric difference for sets. If G, H are graphs, then by GAHwe mean the set (F(G) AF(//)) U (E(G)AE(H)).

We consider a subset of F(G) as a subgraph of G with empty edge set. If H ^ G is asubgraph of G, then G — H denotes the induced subgraph of G having vertex setV(G)- V(H). For T c G, G[T] denotes the induced subgraph of G having vertex set V(T).A factor of G is a subgraph H of G with V(H) = V(G). If L is a set of edges, V(L) denotesthe set of end vertices of all these edges.

If T a V and a, beV, we write a. T. b(G) if a, b$T and each (a, /?)-path in G meets T (or,equivalently, if a, b belong to different components of G— T)\ then T separates a, b in G.

For vertices a =t= b an {a, b)-skein of strength k (k an arbitrary cardinal) is the union ofk (a, bypaths having pairwise nothing but a and b in common. Such a configuration isdenoted by ®k(a, b). /iG(a, b) denotes the greatest k such that, for given vertices a =t= b of thegraph G, there is a &k(a, b) in G. (This maximum always exists.) For non-adjacent a, b, wehave Menger's theorem:

jtiG(a,b) = min(r|there exists a 7with a. T.b(G) and \T\ = t).

The connectivity c(G) of G is the minimum of all jtiG(a, b) with a =N b (if \G\ ^ 2); we putc(Kx) = 0. G is ^-connected if c(G) ^ n.

358 R. Halin

Lemma 2.1. If c{G) ^ n and L c E{G), then c(G-L) ^ n if and only if fiG_L(a,b) ^ n forevery abeL.

(The if-part is easy to prove as follows. Let x,ye V(G), T c y(G)-{x,y} with \T\ < n. Thenthere is an (x, j)-path P in G—T. For each edge e = abeL occurring on P, we have an(#, Z?)-path Pe^ G — L that does not meet r . Replacing every such e by Pg, we find from Pa connected subgraph of G — L containing x, y and avoiding 7".)

Let T c K(G) and ae V(G)-T. An (a, T)-fan of strength n is the union of paths Px,..., Pw

each starting in a and ending in a vertex of T such that for / =t=y the paths />, P; have onlya in common, and, further, each Pt has only its end vertex =j= a with T in common. Wedenote such a configuration by the symbol *¥ n{a, T). G with \G\ ^ # + 1 is ^-connected if andonly if for such a and 7" a *Fw(fl, T) always exists in G.

G is n-minimal if c(G) ^ « and c(G — e) < n for all eeE(G). The following is well known.

Lemma 2.2. L^/ c(G) ^ n. G is n-minimal if and only if /iG(a,b) = n for each e = abeE{G).If G is n-minimal and e = abeE(G), then c(G — e) = n— 1 and for each separating TofG — ewith \T\ = n— 1 we have a.T.b (G — e).

Let G be an ^-connected graph. An edge or vertex e of G is called n-redundant in G if G — eremains ^-connected. Let Rn(G) denote the set of ^-redundant edges of G. ObviouslyabeE(G) is in Rn(G) if and only if fiG{a,b) > n, and H c G implies Rn{H) c 7^(G). Theedges not in Rn(G) are called necessary.

G with c(G) ^ nis n-minimizable if it has an ^-minimal factor H. H then must be of theform G — L, L <= Rn(G). Clearly, if Rn(G) is finite, then G is ^-minimizable.

It is obvious that every ^-connected subgraph of an ^-minimal graph is ^-minimal, too.However, not every ^-connected induced subgraph of an ^-minimizable graph must be n-minimizable again. For instance, if we add, for each 'step' (horizontal edge) e of the ladderin Figure 1, a new vertex ve and two edges from ve to the end vertices of e, we get a2-minimizable graph G (delete the original 'steps'), whereas the original ladder is not2-minimizable.

It is easy to show that the union of a non-empty chain of ^-minimal graphs (with respectto inclusion) is again ^-minimal. Therefore we get by Zorn's Lemma:

Lemma 2.3. If H is an n-minimal subgraph of the n-connected graph G, then H is containedin an inclusion-maximal n-minimal subgraph of G.

Of course the statement is no longer true if'^-minimal' is replaced by 4«-minimizable':every countable 2-connected graph is the union of an ascending chain of finite 2-connected(hence 2-minimizable) graphs.

We use a) to denote the cardinal of the countable sets. For an ordinal a-, W(a) denotesthe set of all ordinals v < a.

Lemma 2.4. Let Gx(Xe W(&)) be a well-ordered family of n-connected graphs such that forevery A > 0 we have

veW(A)

Then LLW«D G A is n-connected.

The proof is routine and left to the reader.The maximal number of disjoint rays (i.e. one-way infinite paths) in G is denoted by

m±(G), see [5]. For a connected G, mx(G) = 1 means that G contains a ray but not a doubleray (i.e. a two-way infinite path); the structure of these graphs is described in [4].

A profound theory of rayless graphs was developed by R. Schmidt [8], [9]: an ordinalo(G) (called its order) is associated with every rayless graph G such that o(G) = 0 if and onlyif G is finite, and for every rayless graph G with o(G) > 0 there exists a finite F in G suchthat for all components C of G — F, o(C) < o(G). This concept allows proofs by transfmiteinduction on o(G). Also, G ^ H always implies o(G) ^ o(H).

3. Sets of redundant edges

In this section some elementary 'global' reductions of infinite ^-connected graphs are given.If r and s are cardinals, we write r = s if either r and s are both finite or r and s are equalinfinite cardinals.

Proposition 3.1. Let G be n-connected and T G. Then there is an n-connected subgraph Hof G with the following properties:

1. H=> T;2. \H\ = \T\if T is infinite, and

\H\^ID if T is finite;3. Rn(H)=Rn(G)0E(H).

Proof. If / is any subgraph of G with | / | ^ 2 , then for a pair a 4= b of V(H) choose a0 s ( f l , i ) c G and a 0f l + 1(fl ,J)cC if abeRn(G); define O(/) as the union of / andall these /7-skeins and (n+ l)-skeins. Then the desired H is obtained in the formr u <S>(T) U <D2(7) U O3(T) U .... (It is no restriction to assume \T\ 2.) •

Corollary 3.2. If c(G) ^ n, then the n-minimizability of G implies that every T c: G with\T\ oj is contained in an n-minimizable H ^ G with \H\ w.

Problem 3.3. Is the inverse implication of Corollary 3.2 true also?A positive answer would reduce the problem of /7-minimization to the countable case.

Theorem 3.4. Let c(G) ^ n and L be an infinite set of edges in G such that G — L remains n-connected. Let P be the set of edges in Rn(G) that become necessary in G — L, i.e.

P = (R,,(G)-L)-Rn(G-L).

360 R. Halin

Then\P\ ^ \L\-

Furthermore, if H is an n-connected subgraph of G — L that contains K(L), then E(H) ^ P.

Proof. By Proposition 3.1, every such H contains an ^-connected subgraph H' withV(H') 3 V(L) and \H' \ = \L\. Assume that there is an edge e = xyeP that does not belongto H'. Then by c{H')^n we have /iG_L_e(a,b) ^ ptH,(a,b) ^ n for each abeL; henceG — L — e would be ^-connected by Lemma 2.1, in contradiction to e$Rn(G — L). So we

and \P\^\H'\ = \L\. •

Corollary 3.5. If c(G) ^ n and G — L is an n-minimization of G, then

m = \Rn(G)i

Proof. By Theorem 3.4 we have \Rn{G) — L\ < |L|, whence the result. •

Theorem 3.6. Let G be n-connected with infinite Rn(G). Then there exists an n-connectedinduced subgraph H of G with \H\ = \Rn(G)\ such that

Rn(G) = Rn(H)9

and for each L <= Rn(G) we have that G — L is n-minimal if and only if H—L is n-minimal.

Proof. Let T= V(Rn(G)). For each e = abeRn(G) and every finite F c Rn(G) choose asHeF 2L ®n(a,b) ^G-e with E(®n(a,b)) n Rn(G) = Fif such an (fl,6)-skein exists; if not, letHe,F = 0- ^ and all these He F form a graph D with \D\ = \T\, and by Proposition 3.1, Dcan be extended to an induced subgraph H of G with \H\ = \T\ and c(H) ^ n. Byconstruction we have E(H) 3 i?n(G) and (by the choice of the He F)Rn(H) = Rn(G).

1. Let L c 7?W(G) with G - L ^-minimal. We claim that H—L is ^-minimal too. To verifyc(H—L) ^ «, by Lemma 2.1, we only have to show fiH_L(a,b) ^ n for every e = abeL.Now for such an edge, by c(G — L) ^ n, we find a ©w(#, &) c G — L. Let

F - ^(0n(a, b)) H /*„((?) ;FOL = 0.

By construction of 7/ we have He F c /f? which is a 0w(a, 6) that also shares exactly Fwith Rn(G), hence avoids L. So we find /tH_L(a,b) ^ «.

2. Now assume that H—L is ^-minimal for an L c Rn(H). Since fiG_L(a, b) ^ fin-ik0-* b)^ nfor every abeL, we conclude (by Lemma 2.1) that G — L is ^-connected. Assume thatthere is an e = abeRn(G) — L such that juG_L(a,b) > n. Then there is a 0w(a,6) inG — L — e. Let Fbe the set of edges which ®n(a, b) has in common with Rn(G). Then byHF e<=H—L — e we would get ftH_L(a,b) ^ « + l , contradicting the ^-minimality ofH—L. Hence juG^L(x,y) = n for each xyeE(G — L), and G — L must be ^-minimal. •

By this theorem, our minimizing problem requires us to consider only such infinite n-

connected graphs with order coinciding with the number of ^-redundant edges. We thenspeak of ^-connected graphs with full redundance.

Theorem 3.7. Ifc(G) ^ n and Rn(G) is infinite, then there exists L c Rn(G) with \L\ = \Rn(G)\such that G — L remains n-connected.

Proof. For each e = abeRn(G) choose a ®n(a9b) ^ G-e as He. Let 0> be the set of allsubsets S of Rn(G) such that no eeS lies in He for an e'eS—{e}. If S^iel) is a chain ofelements of ^ , then, clearly, {Ji€lSf is again in 0>. By Zorn's Lemma, 0* contains a maximalelement L.

Then G — L is ^-connected. Namely if e = abeL, then He c: G — L, hence fiG_L(a,b) ^ ft,and c(G — L) ^ ft follows from Lemma 2.1.

Moreover |L| = |/*n(G)|. Otherwise |L| < |*n(G)| = | / t n(G)-L| and IlLJeeL^JI < I^W(G)|;for e* 6 ^(G) — |Jg 6 L E{He), we would have L U {e*} e 0>, in contradiction to the maximalityof L. •

4. Elementary operations

In this section we study the effect on the fl-minimizability of applying certain elementaryoperations (deleting and adding vertices or edges).

Let G = (K,E) be an ^-connected graph. For distinct a, beV and eeE, call e necessaryfor a,b, if {iG_e(a,b) < n. Let Qn{G\a,b) be the set of all necessary edges for a, b. Thenclearly

Lemma 4.1. Qn(G;a,b) is the intersection of the edge-sets of all ®n{a,b) c= G, and is hencefinite.

The following observation is the key to what follows.

Lemma 4.2. For each e = abeRn(G),

Rn(G)-Rn(G-e)^Qn(G-e',a,b)[){e}.

Hence, if an n-redundant edge e = ab is omitted, only finitely many edges e Rn(G) becomenecessary in G — e {and these must all be necessary for a and b).

Proof. Let e' = xy e Rn(G) — Rn(G — e) with e' 4= e. Since e' is necessary in G — e, we havefiG_e(x,y) = n ; h e n c e t h e r e i s a T c K w i t h | T | = n — \ a n d * . T . y ( G — e — e ) . {G — e — e ) — Thas exactly two components, say Cr and Cy with xe ViCJ, ye V(Cy). e must lead from Cx

to Cy\ for otherwise x.T.y (G — e), contradicting e eRn(G). Hence

a. T.b(G — e — e/),

but /iG_e(a,b) > « - l . So we conclude e'eQn(G — e\a,b), and the proof is complete. •

Our proof also yields the following for edges e 4= ef.

362 R. Halin

Lemma 4.3. e' e Rn(G) - Rn(G -e)=>ee Rn(G) - Rn(G - ef).

So the pairs (e,e')eRn(G)x Rn(G) with e * e' and e'eRn(G)-Rn(G-e) define anirreflexive and symmetric relation on Rn(G), or a graph with vertex set Rn(G). We denotethis graph by t%n(G) and call it the n-redundance graph of G. By Lemma 4.2, Mn(G) is locallyfinite. If eeRn{G), then 0tn(G — e) arises from 0tn(G) by deleting e and all its neighbours, andadding (eventually) finitely many new edges. The local finiteness of &n(G) can also be readin the following way.

Lemma 4.4. For each eeRn(G) there are only finitely many e' = xyeRn(G) such that e isnecessary for x, y in G — e\

Furthermore, we have

Lemma 4.5. If e = abe(^) — E, then in G[je there are only finitely many edges that arenecessary in G but n-redundant in G U e.

(By Lemma 4.2 we have Rn(G U e)-Rn(G) c Qn(G;a,b) U M, which implies the result.)

Lemma 4.6. If G = (V,E) is n-connected and, for an e = abeC2) — E, G U e is n-minimizable\then G is also n-minimizable.

Proof. Let G' U e be an ^-minimization of G U e, where G' is a factor of G. By c(G) ^ n, wefind a Gn(a,b) ^ G; and by Lemma 2.1, Gf \}<dn(a,b) is ^-connected. By Lemma 4.5,G" = G' U e U ©n(#,b) has only finitely many /7-redundant edges and e is one of them. Bydeleting e and then an appropriate finite sequence of ^-redundant edges from G\ we findan /7-minimal graph that is a factor of G. •

Theorem 4.7. Let G, H be n-connected graphs with n ^ \G 0 H\ < oo. Then G U H is n-connected {Lemma 2.4) and we have

Rn(G{jH) = Rn(G){jRn(H){jL

with a finite L.

Proof. Let V(G 0 H) = F. By Lemma 4.5 we have

with finite L19 L2.We only have to show that if e is in Rn(G U / / ) , but not in Rn(G) U ^,X^) ' t h e n e i s i n

Lx U L2 U (2). Assume ^ = xy not in (Q, say xe V(G) — F (without loss of generality).

There exists a 0,, (*,>') g(GU H)-e with the (x, y)-paths Px,..., P,,. If a i> leaves G, it hasa first and a last vertex with Fin common; then we replace the subpath of Pi between thesetwo vertices by the edge e (0 joining them. In this way we get an (x, v)-skein of strength nin (G U (O) — e, and we find that e is /7-redundant in G U (£)• As e is not in Rn(G), it lies inLx, and our proof is complete. D

Corollary 4.8. Let G, H be n-eonnected graphs with nminimizable if and only if G and H are n-minimizable.

\G n H\ < 00. Then GU H is n-

Proof1. Let G', H' be /7-minimizations of G and //, respectively. Then G' U H' is an ^-connected

factor of G U # . By Theorem 4.7, /*n(G' U H) is finite, hence G' U /f' (and therefore alsoG U / / ) is ft-minimizable.

2. Let J — G\] H and let / ' be an ^-minimization of/. Let Gr, / / ' denote the subgraphs of/ ' induced by V{G), V(H) (respectively). With F= V(G) n V(H) put

G* = Gr U

G* and / /* are ^-connected (this follows by an argument similar to that at the end ofthe proof of 4.7). By 4.5, Rn(J' U Q) is finite. Hence also Rn(G*) ^ Rn(J' U Q) is finite,and therefore G* is ^-minimizable. We see that G U (£) is ^-minimizable, and from 4.6it follows that G must be «-minimizable. Analogously, we find that H is ^-minimizable.

•Corollary 4.8 is no longer true if we allow G n H to be infinite. This is shown by

Example 4.9. The graphs G and H of Figure 3 are 2-minimal, but it we take their union andidentify each / with /', we get a 2-connected graph that is not 2-minimizable.

111

4

3

2

O • —

<>

o

—o

We define the operation of adding a vertex of degree A: to a graph G by: add a new vertexx to G, choose a /c-element set F of vertices in G, and join .v to all the vertices of Fby edges.By applying Lemma 4.6 and Corollary 4.8 to G and to the complete graph H with vertexset F U {x} we get

364 R. Halin

Corollary 4.10. If G is n-connected, adding a vertex of finite degreeminimizable graph J if and only if G is n-minimizable.

n leads to an n-

If k is allowed to be infinite, J can be ft-minimizable whereas G is not. This is shown inthe case n = 2 by the graph in Figure 4.

Figure *

Two graphs G, H are called finitely n-related (written G ~ H) if G and H are bothn

^-connected and G can be transformed into H by a finite sequence of the followingoperations:

1. adding a new edge,2. adding a vertex of degree n,3. deleting an ^-redundant edge,4. deleting an ^-redundant vertex of degree n.

Clearly ~ is an equivalence relation in the class of ^-connected graphs, and we have

Lemma 4.11. G ~H if and only if G, H are n-connected and the set GAH is finite.

(Clearly G ~ H implies \GAH\ < oo, and if \GAH\ < oo, by operations 1 and 2 we get a

graph / with G ~ / , H ~ /.) We see that the finite ^-connected graphs form one of the

equivalence-classes induced by ~ .

Summarizing the results of this section we can state:

Theorem 4.12. For every equivalence class <& of finitely n-related graphs, either every element

ofm is n-minimizable, or no element of%> is n-minimizable. Moreover, if G ~ H and G' is ann

n-minimization of G, then there is an n-minimization H' of H with G' ~ H'.

(Notice that the ^-minimization we constructed in the proofs of Lemma 4.6 andCorollary 4.10 from a given one always remains finitely ^-related to the latter.)

Also we can interpret Corollary 4.8 (and its proof) in the following way: if G, H are n-

connected and n ^ \G D H\ < oo, we get all ^-minimizations of G U //modulo ^ as the

unions of ^-minimizations of G and H.

From Theorem 4.12 we immediately conclude

Corollary 4.13. If c(G) ^ n, either G is n-minimizable or for every n-minimal subgraph H ofG we have \G — H\ ^ OJ.

Corollary 4.14. Let e = xy be an edge of an n-connected graph G such that x is of finite degreeand such that the graph H arises from G by contracting e to a single vertex. Then G isn-minimizable if and only if H is n-minimizable.

5. Reducing sequences

In this section G is ^-connected and of infinite order a (where a is identified with the initialordinal of cardinality |G|), and G is supposed to be of full redundance.

Let cr be an ordinal and W be a well-ordering of Rn(G) with order type <r; then we canconsider W as a family (e|P)1)<(7, where the ev are the elements of Rn(G). Such a family(<?,,),, < a is called a reducing sequence of G.

We now define a subset L = Lw of Rn(G) by the following inductive selection procedure(which generalizes the natural method of minimizing a finite ^-connected graph). Let eoeL.Now let 0 < A < a and assume that for all ev with v < A it is decided whether eveL or notand let the set of the selected ev (y < A) be denoted by L(A). Then, if c(G — L(Xj) < n, putL — L{X) and let the procedure be finished. If c(G — L(Xj) ^ n, let eAeL if and only ifeAeRn(G-L(A)).

Lemma 4.5 guarantees that the procedure does not terminate before w, so L is an infiniteset that may be considered as a well-ordered subsequence of W.

Lemma 5.1. G is n-minimizable if and only if there is a reducing sequence W such that G — Lw

remains n-connected.

Proof. If G — L is an ^-minimization of G, choose a well-ordering W of Rn{G) such that Lis an initial segment of W\ then L = Lw.

If W is such that G — Lw is ^-connected, then G — Lw must be ^-minimal; for if itcontained an <?,„ it would have to be selected in the vt\v selection step. •

Now we consider the case that G is ft-minimizable. We define the smallest ordinal r forwhich there is a well-ordering W of Rn(G) of type r such that G — Lw is ^-minimal as theminimization type of G (with respect to ^-connectedness), and denote it by rn{G).

If G — L is an ^-minimization of G, we well-order L according to oc and put behind it awell-ordering of Rn(G) — L of type ^ a ; we see that rn(G) ^ a + a.

Assume that rn{G) = CL + ft with ft < oc. Let W be a corresponding reducing sequence(eA)A<a+/j and L = Lw, so that G — L is ^-minimal. Let Lf be the subsequence of the eA withA ^ a and eAeL. Put L before e0, and re-order the set L" of elements el,eRn(G) — L withv ^ a according to the initial ordinal y of \L"\\ clearly y ^ /?. The new well-ordering Wobtained in this way is of type a + y with Lw = L; therefore /? = y, and /? is seen to be aninitial ordinal. So we can state:

366 R. Halin

Lemma 5.2. If G is n-connected with full redundance and n-minimizable, Tn(G) = oc-\- ft, whereoc is the initial ordinal of cardinality \G\ and ft is another initial ordinal ^ oc.

We now show by examples that all these ordinals a + ft also occur as minimization-typesin the case n = 2.

We choose two disjoint copies Kx and K'x of the complete graph of order oc with verticesxv, x'v{y < oc), respectively. We subdivide each edge of Kx and Kx by inserting a new vertexand draw all the edges ev = xvxv. We now add three vertices a, b, c and the edges xoa, ab,be, cx'o; the graph constructed in this way is called X. X is 2-connected and R2(X) consistsof the edges ev. A 2-minimization of A'is obtained if all the er with the exception of exactlyone are deleted. If W is a well-ordering of Rn(X) of type oc, G — Lw will not remain 2-connected; but if W is chosen of order type a + 1 , we get a 2-minimization. Hence

If 1 < ft ^ a, we take ft disjoint copies XA of X and paste them together along the pathwith the vertices a,b,c. We get a 2-connected graph Z/y of order oc with full redundance;Rn(Zfi) = \JA</jRn(XA). If W is a well-ordering of Rn{Z^ of type a + y with y < ft, there isa A such that all elements of Rn(XA) lie in the initial segment of type oc; hence in Z/y = Ln

all these edges are missing. On the other hand, it is easy to find a well-ordering of type oc + /Jsuch that Zti — Lw is 2-minimal. So we have r2(Z^) = a +/?. If Zo arises from X bysubdivision of the edge e^ then clearly T2(Z{)) = a + 0. So we see:

Theorem 5.3. The possible minimization-types for 2-connected graphs of infinite order oc withfull redundance are exactly the ordinals oc-\-ft, where ft is an initial ordinal ^ oc. •

Now we assume that G is not /7-minimizable. Let W = (eA)A < a be a reducing sequence.

Lemma 5.4. L i r /7<2s «<9 greatest element, so its order type is a limit ordinal.

Proof. Assume eA to be the last element of Lw. Then c(G — L(A)) ^ n and eAeRu (G —L(A)),by construction of Lu . But then G — Lw would be /7-connected, hence /7-minimal by Lemma5.1, giving a contradiction. •

So we have:

Lemma 5.5. G — F remains n-connected for every finite Fa L]V.

By Lemma 5.1, c(G — Lw) < n. Hence there exists a smallest 7 c V(G) with \T\ <n—\(possibly empty) such that G — Ln—T has distinct components C^iel) with |/| ^ 2. Forevery partition of/into two non-empty sets V, I" there must be infinitely many edges in L]V

connecting a vertex of [jisI Cf with a vertex of [jieI C, (by Lemma 5.5). It is possible tochoose this partition such that H = G[\J Ct] and J = G[\J Cf] are both connected. Let B be

iel' # iel"

the set of edges e Lw connecting H and / ; B is a bond (or edge-cut) in G— 7\ B U r formsa 'mixed cut' of G (consisting of vertices and edges), which minimally separates H and J.So we have

Theorem 5.6. IfG with c(G) ^ n is no n-minimizable, then there is a mixed cut in G consistingof at most n— 1 vertices and infinitely many edges such that the deletion of finitely many ofthese edges never destroys the n-connectivity.

If G is locally finite, H and J must be infinite and we can find rays U in H and U'mJthat are connected (in G) by infinitely many pairwise disjoint paths (this follows by meansof [3, Satz 5]) and on each of these paths lies at least one edge eLw. We see that in this waythe infinite ladder must be present in G.

A more thorough study of the components C, of G — Lw—T should lead to a deeperinsight into the structure of non ft-minimizable graphs.

6. Tree-decompositions and //-minimization

Before we proceed to our main result some further preparations are necessary.Let G = (V,E) be a graph with \G\ ^ w+ 1 and Ta V with n ^ \T\ < oo. G is called

(n, T)-connected if G U (£) *s ^-connected.The following can be seen by easy applications of Menger's theorem and its well-known

variations.

Lemma 6.1. The following statements are equivalent:(a) G is {n, T)-connected\(b) there is a graph H with V{H) n V(G) = T such that G U H is n-connected;(c) for every ve V — T there is a *¥n(v-> T) ^ G\(d) for every H with c(H) ^ n and V(G) f] V(H ) = T we have G U H is n-connected.

A graph G is called {n, T)-minimal, if it is («, reconnected, but G — e is not (/?, T)-connected for every ee E—(T

2). G is called (n, T)-minimizable if there is an L <= E such thatG — L is (n, r)-minimal.

Lemma 6.2. IfG = (V,E) is n-minimal and T ^ V with n ^ | T\ < oc•, then there exists a finiteL c E-Q such that G-Lis («, T)-minimai

Proof. By Lemma 4.5, Rn(G U Q ) is finite. Let L be a maximal subset of Rn(G U (72))-(0

such that G — L remains («, T)-connected. We claim that G — L is («, r)-minimal.Let eeE—L — ([); e = xy with .Y^ T. Assume that G — L — e remains (/?, T)-connected.

Then there is a ¥n(x,T) and, if v^T, also a ¥„( v, 7) in G-L-e. With// : = (G U (I)) —L —<?, we then conclude /iH(\\y)^n and, by Lemma 2.1, c(H) ^ tucontradicting the maximal choice of L. •

Lemma 6.3. Let G and Gi (iel) be n-minimizable graphs. Let T be a finite subset of V(G) suchthat V(G) fl V{Gi) = Tt. ^ T and \7]\ ^ n for every iel holds. For all i =|=y assume thatV{G( n Gj) c T. Then

G U U G,/ e /

z n-minimizable.

368 R. Halin

Proof. By assumption we have ^-minimizations G' of G and G" of Gt (iel); by Lemma 6.2each G" has an (n, ^-minimization G\. Using Lemma 6.1 (c) we see that H = T[) \JieI G-is (n, 7>minimal. By Lemma 6.1 (d), G' U H is an ^-connected factor of G U \JieI Gv Nowwe put

.J=G'[)H[)

By Lemma 4.5, Rn(G' U Q) is finite. Let eeRn(J), e$Q. The edge e must lie in G' U (£),because otherwise it joins vertices x, y of some G\ not both in T, and therefore G[ — e wouldnot be («, reconnected, which implies c{J—e) < n. Therefore e joins vertices x,y in G', withx<£ 7" (say): There is a &n(x,y) in /—e by choice of e, and this can be formed already in

G'UQ- Hence ee/*n(G'U(D).We find /?„(./) ^ (2) U i?n (G' U (£)), hence it is finite, and therefore / is ft-minimizable.

By Lemma 4.6 G7 U H is ^-minimizable also, and our proof is complete. •

Example 6.4. The graph H of Figure 5 is 2-connected and R2(H) = {ex,e2,...}. ObviouslyH is not 2-minimizable.

0 4

i

(

(

> O v v 0 (

(^ ( r (

> p.^2

e2

Figure 5.

Let G = H— {H'O, W19 H\2, ...} and G, be the circuit through w\ and r, (z = 0,1,2,...). G andthe G? are all 2-minimal. So we see that Lemma 6.3 is no longer true if we drop thehypothesis that the Tf are all contained in a finite subset of V(G).

Let G be a graph, a > 0 an ordinal and J^ = (GA)AeW{(T) be a family of induced subgraphsof G. Put

G|;/ = \J GA and S,, = G|;, D Gft

for each /Y with 0 < ju< a. $F is called a tree-decomposition of G if the following conditionsare satisfied:

2. for every A with 0 < A < a there is a // < A such that SA ^ G/r

If A_ denotes the smallest // with SA c G//9 then T{^) = (W(a), {AA_ 10 < A < a-}) is a tree,called the decomposition tree of J^. We consider the ordinal 0 as the root of T(<F). For eachsubgraph H of T{!F) we denote the subgraph \JveV{H) Gv of G by GH.

$F is of strength n if all the 'attachments' SA have at least « vertices, and it is calledfinitary, if |SA| < oo for all A < a. We speak of a ray less tree-decomposition if itsdecomposition-tree is rayless.

We say that $F satisfies the finite attachment condition [FAC] if for every A < a there isa finite subset FA of V(GA) such that for every fi + A we have V(GA) n V(G/t) £ FA.

It is easily seen that [FAC] is equivalent to each of the following statements:(a) The union of all attachments SM contained in GA is finite (for all A < <r).(b) In each GA there are only finitely many vertices with neighbours in G outside GA.Clearly, if $F satisfies [FAC], it must be finitary. By Lemma 2.4, if <F is of strength n and

all the GA are ^-connected, then G is ^-connected. On the other hand, if all the GA are n-connected and no GA coincides with SA, then from c(G) ^ n we conclude that ^ must beof strength n. For a more thorough treatment of tree-decompositions we refer the readerto [1] and [2].

Now we can prove our main result.

Theorem 6.5. Let $F = (GA)A <(T be a rayless tree-decomposition of a graph G in which themembers GA are n-connected. Assume #" to be of strength n and to satisfy [FAC]. Then G isn-minimizable if and only if each GA is n-minimizable.

Proof1. Assume each GA to be ^-minimizable. We show that G is ft-minimizable, too, by

induction on o(T(^)), the order of T(!F) in the sense of R. Schmidt.If o(r(#")) = 0, then <F is finite, and the assertion follows by a finite series ofapplications of Corollary 4.8.Let o(T(&r))>0. There exists a finite subgraph B of T(^) such that all thecomponents of T(^) — B have order smaller than o(T{?F))\ without loss of generalitywe may suppose that B is a subtree of T(^) containing the root 0. Let Tt(iGl) be thecomponents of T(3F) — B. Let fit be the minimum of the ordinals in K(7^); the tree-order in Tt with respect to the root pti is a restriction of the tree-order in T(JF) (withrespect to the root 0). For each iel, J^ = {GA)AeV{T) is clearly a tree-decomposition ofHi= {JAeV(T)GA, and we have T(^) = T(; by the induction hypothesis, we mayassume that each Hi is fl-minimizable. Furthermore H = [jA€V(B) GA is ft-minimizableby Corollary 4.8. Clearly, H(\Ht = S/If, and by [FAC] there is a finite D c V{H) suchthat V(S/t) <= D for all iel. We see that the conditions of Lemma 6.3 are satisfied;hence G = H [} \<]ielHi is ^-minimizable.

2. Assume now that G has an ^-minimization G\ We select a member GA of 8F. We useG'A to denote the subgraph of G' induced by V(GA). Choose a finite F ^ V(GA) coveringall attachment c GA according to [FAC]. Then GA[)C2) is ^-connected, andKiG' U O)is n n i t e by Lemma 4.5. Clearly Rn(G'A U Q) ^ Rn(G' U Q). Thus G'A U Qis ft-minimizable, hence GA U (0 and GA itself are ^-minimizable also. •

Diestel [2] showed that every rayless ^-connected graph has a rayless tree-decompositionof strength n with finite ^-connected members. By applying Theorem 6.5, we find that eachsuch graph is ^-minimizable. So we may consider Theorem 6.5 to be a generalization ofSchmidt's theorem.

None of the hypotheses in Theorem 6.5 can be omitted. The infinite ladder has a tree-decomposition with 4-circuits as its members, which satisfies all conditions in 6.5 with theexception of raylessness. The graph in Example 6.4 shows that [FAC] cannot be weakenedto the requirement that the given tree-decomposition be finitary.

370 R. Halin

7. The case n = 2 and mx(G) = 1

The question of how possible it is to tackle the minimization problem for simpler cases notcovered by Schmidt's Theorem now suggests itself, i.e. the problem for a graph G with asmall but positive number mx{G) of disjoint rays and small connectivity number.

Let W{G) denote the set of vertices x of G such that there exists a ray U and infinitelymany paths from x to U having pairwise only x in common; we put

\W(G)\ = w{G).

In [5], §2, the following was shown:

Theorem 7.1. If 1 ^ rnx(G) = m < oo, then there exists a finite F^G and a tree-decomposition G?(/ <(o) of G — F with attachments Si = G\{ 0 Gt such that the followingconditions are satisfied:

1. \S\ = mfor all ieN,2. the St are pairwise disjoint,3. 5?. is contained in Gt_x and has no v e r t e x in a G} with j < / — I ,4 . there is an m-tuple of disjoint paths matching S( with Si+1 in each G( (/ ^ 1),5. the G? are rayless.

It is easy to see that F can be chosen as W(G).From Theorem 7.1 we see immediately that for mx{G) ^ 1 we have

Corollary 7.2. c(G) ^ m1(G) + n<G).

If for a rayless G we define w(G) as the number of vertices with infinite degree, thenCorollary 7.2 is also true in the case mx(G) = 0. (For c(G) = 1 it is then Konig's Lemma,and for c(G) ^ 2 it follows recursively by deleting an arbitrary vertex of infinite degree.)

As the rayless graphs can be handled by Schmidt's theorem and the structure of a graphG with m^G) < oo is rather transparent by Theorem 7.1, it seems there is hope for the studyof the ^-minimization problem in these cases.

Let us consider the case n = 2 and mx(G) = w(G) = 1; then W(G) consists of a singlevertex w and the S? reduce to vertices s; that are articulations of the connected graph G — ir[4]. We further see that the block-decomposition of G — w is a refinement of thedecomposition in Theorem 7.1.

Each Gi must be connected (otherwise w would separate G). G, may have infinitely manyblocks, but the decomposition-tree of its block decomposition must be rayless. We denotethe articulation of each endblock B of G, by aB. An endblock B of G? that does not containsi or si+l is called proper; from each proper B we can select a vertex vB in B — aB adjacentto w.

Theorem 7.3. If infinitely many G,. have proper endblocks, G is 2-minimizable.

Proof. We delete all edges from w to vertices different from the vB selected above. Clearly

the resulting graph Gf remains 2-connected and the edges wvB are all necessary in G'. Fora subgraph / of Gf — w we denote by / + w the graph J together with w and all edges fromw to J. Furthermore we find a sequence of integers 0 ^ / 0 < il < i2 < ... such thatc((G0 U . . . U ( / J + i v ) ^ 2 for all k ^ 0. Without restriction, we can assume G = G' andik — k for all k. In particular, there are at least two edges from w into Go and at least oneedge from w into Gi — sf for every / ^ 1.

Let H0 + w be a 2-minimization of Go + w. Then let //2 + w be a 2-minimization of(Ho U Gx) + w, choose a 2-minimization //2 4- w of (//x U G2) + w, and so on. LetG\ = HM[V(GJ\. Then clearly each G\ must be connected and each edge xy of G- ispreserved in all G[+k with A: ^ 2. Namely, if xy e R2(Hi+k + w) for a /: ^ 2, then it would bein R2(Hi+1 + w), since every x, j-path in Hi+k + w either stays in Hi + vv or passes through si+l

into G-+1 and contains w; then we can go from si+1 to a i;B in F(G,'+1) and from there to u\so avoiding the G'i+k with k ^ 2.

Thus we have a sequence of connected graphs G , G'1? G^ ... such that

G* = (G;UG;UG: 2 U . . . )+W

is a connected factor of G. From this representation we see c{G*)consideration for edges xy of GJ, we also see R2(G*) = 0.

2. By the above

If a Gy has no proper endblock, it may be possible to create one by deleting appropriatelychosen edges. We call a Gt (i ^ I) feasible if we can find L{ c: ^(G^ such that

1. Gi — Li is connected,2. there is an endblock of Gi — Li neither containing st nor si+1,3. if B is such an endblock then there is vB in B — aB adjacent to vv.Clearly, if we replace each feasible Gf by its Gi — Li, the resulting graph remains 2-

connected, and if infinitely many Gt are feasible, we know from Theorem 7.3 that then G is2-minimizable. On the other hand, if there is an n0 such that no Gi with / ^ n0 is feasible,we consider an arbitrary 2-connected factor H of G. Let G\ = i/[GJ. Clearly there must beedges from w to infinitely many G\ in H. HwxeE(H) with x in G\, i > n0, then \vxeR2(H),as / / has no proper endblock. Therefore H is not 2-minimal. So we can state:

Theorem 7.4. 4 graph G with c(G) ^ 2 andmx(G) = w(G) = 1 is 2-minimizable if and only ifin any representation of G according to Theorem 7.1 there occur infinitely many feasiblemembers.

Figure 6.

372 R- Halin

If we allow w{G) ^ 2, the condition of Theorem 7.4 in its essence remains necessary forthe 2-minimizability, but examples show that it is not sufficient (Figure 6). There is anenormous jump of difficulty if any parameter in Theorem 7.4 is increased.

8. Other kinds of minimization

If Pn is a property that every ^-minimal graph must have, one may ask whether an n-connected graph that is not ^-minimizable has at least an ^-connected factor with Pn.

Every ^-minimal graph has a vertex of degree n [6]. An ^/-connected graph G is called n-degree minimizable if it has a factor G' with c(G") = n containing a vertex of degree n.

Theorem 8.1. Every (n+ \)-connectedgraph is n-degree minimizable.

(Take an arbitrary vertex v and delete all but n edges incident with v.)

Theorem 8.2. [8, (10)] If G is n-connected and there is a T in G with \T\= n such that G—T

has a finite component, then G is n-degree minimizable.

Every infinite ^-minimal graph G with n ^ 2 has |G| vertices of degree n [7, Satz 2]. Wecall an infinite G with c(G) ^ n fully n-degree minimizable if it has a factor G' with c{G') ^ nand \G'\ vertices of degree n.

The ladder, for instance, is not 2-minimizable, but fully 2-degree minimizable. If Tk is the/c-regular tree, then for integers k ^ 3, Tk x K2 is not 2-degree minimizable. In [6] it is proved(see [3] for the notion of free end) that:

Theorem 8.3. Ifc(G) ^ n ^ 2 is locally finite with at least one free end, then G is fully n-degreeminimizable.

For a cardinal k > n, a graph G with c(G) ^ n is called («, k)-minimal if for all ab e E(G) itis the case that ju,G(a,b) < k. With k = n+ 1, we get the notion '^-minimal'. G is (n,k)-minimizable if it has an (n, A:)-minimal factor. Furthermore, we call G [«, k]-reducible if G hasan ^-connected factor G' with an edge e = ab such that fiG{a, b) < k. Clearly (;?, k)-minimizability implies [«, /r]-reducibility. One could hope that an ^-connected infinite graphG would be at least [n, |G|]-reducible. But we have:

Theorem 8.4. For every integer n ^ 2 and cardinal k ^ w, ///ere zs a graph Gk with \Gk\ = kthat is n-connected and not [n, k]-reducible.

Proof. We construct Gk as follows. Let Ho = Kn+1. If Hr for an integer r ^ 0 is alreadydefined and St (iel) is the collection of Kn contained in Hr, then choose distinct newvertices vu, (v < k) for each / and join each viv to the vertices of Si by edges. Let Hr+1 be thegraph obtained in this way.

Let Gk = H0[jH1[]H2[j .... Clearly Gk is ^-connected and \Gk\ = k. If / is an n-connected factor of Gk and e = abeE(J), then there is a Kn of Gk containing a, b. By


construction Gk — Kn and hence also /— V(Kn) contains k components each of which sendsedges to a and to b. Hence /ij(a, b) = k. •

9. Some open questions

The following question is suggested by Lemma 2.3:

Problem 9.1. What can be said about the existence of an ^-minimal subgraph in an n-connected graph? The answer is trivially positive for n = 2, but the problem seems to behard for all n ^ 3.

In the light of Theorem 8.1, one may ask:

Problem 9.2. Is every (n+ l)-connected graph ^-minimizable? Even the following muchmore modest question seems to be very hard: Is there an n ^ 3 such that every ^-connectedgraph is 2-minimizable?

If G is ^-minimal, the graph arising from G by adding a new vertex x and joining it toevery vertex of G by edges, is (n + l)-minimizable. (Leave only those edges from x that leadto a vertex of degree n.) In this connection we put:

Problem 9.3. If G is (n + l)-connected and there is a finite F such that G — F is ^-minimizable,is G(n+ l)-minimizable?

Let us call a graph G purely ^-connected if it does not contain an (n+ l)-connectedsubgraph.

Clearly the ^-minimal graphs are purely ^-connected.

Problem 9.4. What can be said about the existence of purely ^-connected factors orsubgraphs in ^-connected graphs?

An infinite sequence F15 F2, F3, ... of finite subsets of V{G) is called an infinite chain offinite cuts in G if no Ft contains an Fj for i =|= j and every Fi (i ^ 2) minimally separates eachvertex of Fi_1 — F{ from each vertex of Fi+1 — Fr

Problem 9.5. Does every ^-connected graph that is not ft-minimizable contain an infinitechain of finite cuts? Or at least, can it be reduced by deleting edges to an /7-connected graphwith such a chain?

As such a chain of cuts leads to the existence of a ray, we would get another approachto Schmidt's theorem and a strengthening of it.

Problem 9.6. Does every infinite (n + l)-connected graph contain an infinite set T of vertices(or also of edges) such that G—T remains at least ^-connected?

By Diestel's theorem, the answer is certainly 'yes' for rayless graphs.

374 R. Halin

References

[1] Diestel, R. (1990) Graph decompositions: a study in infinite graph theory. Oxford UniversityPress, Oxford.

[2] Diestel, R. (to appear) On spanning trees and ^-connectedness in infinite graphs. / . Combin.Theory.

[3] Halin, R. (1964) Uber unendliche Wege in Graphen. Math. Ann. 157 125-137.[4] Halin, R. (1965) Charakterisierung der Graphen ohne unendliche Wege. Arch. Math. 16

227-231.[5] Halin, R. (1965) Uber die Maximalzahl fremder unendlicher Wege in Graphen. Math. Nachr.

30 63-85.[6] Halin, R. (1971) Unendliche minimale «-fach zusammenhangende Graphen. Abh. Math. Sem.

Univ. Hamburg 36 75-88.[7] Mader, W. (1972) Uber minimal «-fach zusammenhangende, unendliche Graphen und ein

Extremalproblem. Arch. Math. 23 553-560.[8] Schmidt, R. (1982) Ein Reduktionsverfahren fiir Weg-endliche Graphen, PhD-Thesis Hamburg.[9] Schmidt, R. (1983) Ein Ordnungsbegriff fiir Graphen ohne unendliche Wege mit einer

Anwendung auf «-fach zusammenhangende Graphen. Arch. Math. 40 283-288.

On Universal Threshold Graphs

P. L. HAMMER and A. K. KELMANSf

RUTCOR, Rutgers UniversityNew Brunswick, New Jersey 08903

A graph G is threshold if there exists a 'weight' function w : V(G) -* R such that thetotal weight of any stable set of G is less than the total weight of any non-stable set of G.Let !Tn denote the set of threshold graphs with n vertices. A graph is called ^-universalif it contains every threshold graph with n vertices as an induced subgraph, ^-universalthreshold graphs are of special interest, since they are precisely those STn -universal graphsthat do not contain any non-threshold induced subgraph.In this paper we shall study minimum ^-universal (threshold) graphs, i.e. ^,-universal(threshold) graphs having the minimum number of vertices.It is shown that for any n > 3 there exist minimum 9~n -universal graphs, which arethemselves threshold, and others which are not.Two extremal minimum ^,-universal graphs having respectively the minimum and themaximum number of edges are described, it is proved that they are unique, and that theyare threshold graphs.The set of all minimum ^,-universal threshold graphs is then described constructively; itis shown that it forms a lattice isomorphic to the n — 1 dimensional Boolean cube, andthat the minimum and the maximum elements of this lattice are the two extremal graphsintroduced above.The proofs provide a (polynomial) recursive procedure that determines for any thresholdgraph G with n vertices and for any minimum ^-universal threshold graph T, an inducedsubgraph G' of T isomorphic to G.

1. Introduction

Given a class of graphs # it is natural to find and to study extremal ^-universal graphs, i.e.those graphs that contain every graph from # (e.g. as subgraphs, as induced subgraphs,as a homeomorphic image, etc.), and have some extremal properties (e.g., are of theminimum size).

t The authors gratefully acknowledge the partial support of the National Science Foundation under GrantsNSF-STC88-O9648 and NSF-DMS-8906870, the Air Force Office of Scientific Research under GrantsAFOSR-89-0512 and AFOSR-90-0008 to Rutgers University, the Office of Naval Research under GrantN00014-92-J1375 and the DIMACS Center.

376 P. L. Hammer and A. K. Kelmans

The idea of universal graphs, conceived in 1964 by R. Rado [16], considered todayas fundamental in extremal graph theory, is the subject of numerous investigations (e.g.[3, 6, 7, 8, 14, 15]). Beside its mathematical interest, several recent studies (e.g. [2, 4]) dealwith its applications to various aspects of computer science and engineering.

In this paper we shall concentrate on ^-universal graphs, where # is the set 3Tn ofall threshold graphs with n vertices. The concept of threshold graphs was introducedin [9, 10], and various properties and characterizations of such graphs are known (e.g.[5,9,10,11,13]).

We shall study here ^-universal graphs, i.e. graphs that contain as induced subgraphsall threshold graphs with n vertices. The study of minimum ^-universal graphs, i.e. 9~n-universal graphs having the minimum number of vertices, is one of the main topics ofthis paper. Minimum ^-universal graphs may contain non-threshold graphs as inducedsubgraphs. Irredundant minimum ^-universal graphs could be defined as those minimum^-universal graphs that do not contain as induced subgraphs any non-threshold graph.It is easy to notice that they are precisely those minimum ^,-universal graphs that arethemselves threshold. The second central topic of this paper consists of the study ofminimum ^-universal threshold graphs.

A graph obtained from a ^-universal graph G by deleting or adding an edge, mayor may not be ^-universal. This leads to the question of finding those minimum ?Tn-universal graphs that have the minimum or the maximum number of edges. We constructtwo graphs, Mn and Wn, satisfying, respectively, the above two extremal requirements.These graphs are shown to be themselves threshold, implying that Mn and Wn are alsominimum ^,-universal threshold graphs having the minimum and the maximum numberof edges respectively. We prove that any minimum ^,-universal graph with the minimum(maximum) number of edges is isomorphic to Mn (respectively Wn): in other words Mn

and Wn are the unique minimum ^-universal graphs having, respectively, the minimumand the maximum number of edges.

We give a constructive description of all minimum ^-universal threshold graphs,show that they form a lattice, the minimum and the maximum elements of which arerespectively the extremal graphs Mn and Wn, and show that this lattice is isomorphic tothe n— 1 dimensional Boolean cube. Hence there are 2n~l minimum ^-universal thresholdgraphs. We show in particular that a minimum ^,-universal graph has a surprisingly smallorder equal to 2n — 1 (the lower bound is easy, and so the interesting bit is the upperbound).

A special class of split graphs called n-stair graphs turns out to play an essential role inthe study of minimum ^-universal graphs. It is proved that any minimum ^,-universalgraph is an n-stair graph, and that any n-stair graph is a minimum ^-universal graphif n < 4, or if the graph has an isolated vertex. It is also shown that a threshold graphis a minimum ^,-universal graph if and only if it is an n-stair graph. A family ofgraphs is constructed showing that for any n > 5 there exists an n-stair graph that is not^,-universal.

It is easy to see that every minimum ^,-universal graph with n = 1 or 2 is threshold.We construct a family of graphs that for any n > 3 provides a minimum ^,-universalgraph that is not threshold.

On Universal Threshold Graphs 377

Given an arbitrary threshold graph G with n vertices and an arbitrary minimum 3~n~universal threshold graph T, a simple recursive procedure for finding an induced subgraphG of T isomorphic to G, along with an isomorphism of G and G', can be derived fromthe description of minimum ^,-universal threshold graphs. For the special case when theminimum ^,-universal graph is the extremal graph M", this imbedding can be describeddirectly in terms of the degree sequence of G.

Finally it is shown that the set !Tn of all threshold graphs with n vertices contains aproper subset iVn (of so called uniform threshold graphs), which can be viewed as a kernelof 3Tn because the size of extremal ^-universal graphs is actually determined by iVn. Asa byproduct we get for Wn the main results established for 3~n.

2. Main concepts and notations

We consider undirected graphs without loops or multiple edges [1]. Let V(G) and E(G)denote the set of vertices and edges of G, respectively. Let ^n denote the set of all graphswith n vertices. Two vertices x and y of G are adjacent if (x,y) e E(G). A subset Xof vertices of G is called stable if no two vertices of X are adjacent, and non-stableotherwise.

Let N(x) denote the set of vertices of G adjacent to x:

N(x) = {zeV(G):(x,z)eE(G)},

and let d(x) = \N(x)\ be the degree of x. Two vertices x and y of a graph G are calledequivalent if N(x) \ y = N(y) \ x.

Given X c V(G), the subgraph of G induced by X, denoted by G(X), is the graph whosevertex set is X, and whose edges are those edges of G that have both of their end-verticesinX.

The graph G is said to be the complement of a graph G if there is a one-to-one mappingq> : V(G) —• V(G) such that (w, v) is an edge of G if and only if (<p(u), cp(v)) is not an edgeof G. A self-complementary graph is a graph that is the complement of itself.

Let Kn be the complete graph on n vertices, and let its complement Kn be the empty graphon n vertices. Let Pn and Cn denote the path and the cycle with n vertices, respectively.

Given two graphs A and B, a graph G is called the union of A and B, G = A U B ifV(G) = V(A) U V(B) and E(G) = E(A) U £(£).

If A and B are disjoint graphs, their sum A + B will simply denote the graph A U B,while their product A x B will denote the graph obtained from A + B by adding all theedges (a, b) with a G V(A) and b € V(B). In particular, if B consists of a single vertex b,we write A + b and i x f c instead of A + B and A x B: in these cases b is called an isolated,and, respectively, a universal vertex.

If 7(Kn) = 7(Kn) = {gi,...,gw}, then Kn = g i x g 2 x . . . x g n and KB = g i+g 2 + ... + gn:we shall write simply Kn = g" and Xn = ng.

A graph G is called threshold if there exists a 'weight' function w : F(G) —• K definedon the set of vertices of G such that w(X) < w(Y) for any stable vertex-set X and anynon-stable vertex-set Y of G [9, 10]. Let ?Tn denote the set of threshold graphs with nvertices.


A graph is uniform threshold if it can be obtained from a complete graph either bydeleting the edges of a complete subgraph, or by adding some isolated vertices. Let iVn

denote the set of uniform threshold graphs with n vertices, i.e.

where

a n d

ir{n = {gs(kg):k,se {0,1...,n},k + s = n}.

FutXn = {Kn,Kn}.Given a set # of graphs, a graph G is called ^-universal if it contains every graph from

# as an induced subgraph.A ^-universal graph is called minimum if it has the minimum number of vertices among

all finite ^-universal graphs (if any). A minimum ^-universal graph will be called simplya W-mug.

A ^-mug is called minimum (maximum) if it has the minimum (respectively, the maxi-mum) number of edges among all the ^-mugs.

Let * (#) , fyminW), and ^max(^) denote the sets of all ^-mugs, minimum ^-mugs,and maximum ^-mugs, respectively. Let ^t(^), °Utmin^€\ and °l/tmax(^) denote the setsof all threshold ^-mugs, minimum threshold ^-mugs, and maximum threshold ^-mugs,respectively.

Let v(^) denote the number of vertices of a ^-mug, and let effi) and e(^) denote thenumber of edges of a minimum and of a maximum ^-mug, respectively.

Let vt(^) denote the number of vertices of a threshold ^-mug, and let et(^) andet(^) denote the number of edges of a minimum and of a maximum thresholdrespectively.

3. Preliminaries

The following characterizations of threshold graphs were given in [9, 10].

Theorem 3.1. For every finite graph G the following conditions are equivalent.

(c\) G is threshold,(ci) G is threshold.(CT) G does not contain four vertices a\,a2,b\,bi such that (a\,Gb),(bi,i>i) are edges and

(a\9bi),(a2,b2) are not (or equivalently, G does not contain IKi.P^ and C4 as inducedsubgraphs).

(c4) There exists a partition of the set V(G) into two parts X and Y such that no twovertices in X are adjacent, any two vertices in Y are adjacent, there are orderings

(x i ,X2 , . . . , x / c ) of X and (yi,)>2>-••>}>/) of Y such that

and

where Nx(y) = Xn N(y).

Let 2T\ (respectively, ^\) denote the set of all threshold graphs with n vertices that do(respectively, do not) have an isolated vertex; clearly 9~n = 2T\ U 2T\ and 3~\ n 3~\ = 0.

Corollary 1.

(tl) Each disconnected threshold graph G has an isolated vertex g, hence G = (G\g) + g.(t2) Each connected threshold graph G has a universal vertex g, hence G = (G\g) x g.(t3) If F is a threshold graph and f is an additional vertex, then F + f and F x f are

threshold graphs.Clearly 9\ = {G + g : G <= ^ n _ i } , am* F\ = {G x g : G G #;_!}.

By using Theorem 3.1(cl),(c3) and Corollary 1 one can easily prove the followingcorollary.

Corollary 2.(al) Ifn<(32) ^4 = ^

= {4g, 2g + g2, g + g(2g), g + g3, g(3g), g(g + 2g), g2(2g), g4},#4 = ^4 \ {g + g(2g),g(g + g2)} = {4g, 2g + g\g + g\g(3g),g2(2g),g4},

(a3) ^5 = {5g, 3g + g2,2g + g(2g), 2g + g\g + g(3g), g + g(g + g2), g + g2(2g),g + g\g(4g),g(2g + g2),g(g + g(2g)),g2(3g),g(g + g3),g2(g + g2),g3(2g),g5},#5 = {5g, 3g + g2,2g + g\g + g4, g(4g), g2(3g)g3(2g), g5}.

4. Universal graphs and stair graphs

In this section we shall describe some properties of ^-universal graphs and minimum^-universal graphs for some special classes c€. We introduce the concepts of stair graphsand selfstair graphs, and show that under certain conditions a minimum ^-universal graphis a selfstair graph.

Clearly if <&i c < 2, a ^-universal graph is also a %>\-universal graph. Therefore a^,-universal graph is also a ^-universal graph.

Given a set <€ of graphs, put ^ = {G : G e <€}. Clearly jfB = ^ , and KT$J = ^ .Therefore *iVn = iVn. From Theorem 3.l(cl),(c2) we see that 2Tn=!rn.

Obviously, we have the following propositions.

Proposition 1. Let <6 be a set of graphs, and <& = <£. Then G is a ^-universal (minimum^-universal) graph if and only if G is a ^-universal (respectively, minimum ^-universal)graph.

Corollary 3. Let %> be a set of graphs, and # = c€. G is a minimum ^-mug if and only ifG is a maximum ^-mug.

Lemma 1. Let ^ be a set of graphs containing Kn and Kn, and let G be a %>n-universalgraph, n > 2. Then

(al) G has at least In — 1 vertices, and(a2) if G has exactly 2n — 1 vertices, then V(G) consists of three disjoint subsets An-\,

Bn-\, and c such that An-\ and Bn-\ have n—1 vertices, c is a single vertex, no twovertices of An-\ are adjacent, any two vertices of Bn-\ are adjacent, and the vertexc is connected with no vertex from An-\ and with every vertex from Bn_i (so that

n-i Uc)=Kn and G(Bn_i U c) = Kn).

Proof. Since Kn and Kn are members of #„, the graph G contains Kn and Kn as inducedsubgraphs. Hence G has two subsets Sn and Sn with n vertices such that any two verticesof Sn are adjacent and no two vertices of Sn are adjacent. Clearly Sn and Sn have atmost one vertex in common. Therefore G has at least In— 1 vertices. If G has exactlyIn — 1 vertices, then obviously Sn and Sn have exactly one vertex in common, say c, andAn-x =Sn\c and Bn_{ =Sn\c. •

We shall assume from now on that the vertices of the subsets An-\ and Bn-\ of V(G)desc r ibed in L e m m a 1 a re o r d e r e d : A n - \ = {ai,a2,...,an-i} a n d Bn-\ = {bi,b2,...,bn-i}such t h a t d(a\) > d(a2) > ... > d(an-\) a n d d(b\) < d(b2) < ... < d(bn-i).

Let Qkn be the graph obtained from the complete graph Kn-k on the vertex set Yn-k =

{yuyi^-">yn-k} by adding the set X^ = {xi,X2,.. . ,x^} of fc new vertices, and the set of

edges {(xj,yt) : i = 1,2, . . . ,n — k};j = 1,2,...,/c}. By using the operations + and x on

graphs, the graph Qkn is simply

Qkn = (yxx...x yn-k) x (xi + x2 + . . . + xk).

(see Fig. 1)Let Rk be the graph obtained from the complete graph Kn-k on the vertex set Yn_k =

{yuy2,• • • ?yn-k] by adding a set Xk = {x\,X2,...,Xk} of fe isolated vertices. In other words

R£ = (yi x ... x yn-k) + (xi + x2 + ... + xfe).

Obviously, Rk and Qnn~

k are complementary uniform threshold graphs, and

From Corollary 3 and Lemma 1 we easily obtain the following result.

Proposition 2. Let n = 2,3,.. . . Then R2n-\ an^ Qin-i are tne minimum Jfn~mug and themaximum Jfn-mug, respectively. In particular v(Jfn) = In — 1.

Lemma 2. Let G be a iVn-universal graph with In — 1 vertices, n > 2. Let An-\, Bn-\,and c be the subsets of V(G) described in Lemma I(a2). Then d(an-k-\) > k for any

x, x2

Figure 1 The graph Qkn

Proof. Suppose that d(an-k-\) < k for some k e {1 , . . . , n — 2}. Then in An-\ there are atmost n — k — 2 vertices of degree at least k. Since G is a ^-universal graph and Qn

n~k e #^,

the graph G contains Qnn~

k as an induced subgraph. We can assume that V(Q"~k) <= V(G).Since Xn_/c is a stable set of Qn

n~k and Bn-\ U c induces a clique in G, clearly Xn_/c and

Bn-\ Uc have at most one vertex in common. Therefore Xn-k has at least n — k—1 verticesin common with An-\. Every vertex of Xn-k is of degree k in QJj"*. Therefore every vertexof Xn-k is of degree at least fe in G, and so An-\ should contain at least n — k — l verticesof degree at least fe, a contradiction. D

Lemma 3. Let G be a H^-universal graph with 2n—l vertices, n>2. Let An-\, Bn_i, andc be vertex subsets of G described in Lemma I(a2). Then for any k e {1,2,. ..,n— 1} thereexist at least k vertices in An-\ of degree at most k in G.

Proof. Every vertex in An-\ is of degree at most n — 1. Since An-\ has n — 1 vertices,the statement of the lemma holds for fe = n — 1. Let fe e {l,...,w — 2}. Since G is a^-universal graph and Rk e iTn, the graph G contains Rk as an induced subgraph.We can assume that V(Rk) a V(G). Consider the fe-vertex set Xk of isolated vertices ofRk and the (n — fe)-vertex set Yn-k that induces a clique in Rk. Since Rk is an inducedsubgraph of G, the vertex set Xk is stable in G, and Yn-k induces a clique in G. Sincev4n_i U c is stable in G and 7rt_^ induces a clique in G, clearly Y ^ and An-\ U c haveat most one vertex in common. Therefore \Yn-k n 5w_i| > n — fe — 1. Since fe < n — 2, wehave ft — fe — 1 > 1, so yn_/c and An-\ U c have at least one vertex in common. ThereforeXk c: Xw_i. We know that every vertex adjacent to a vertex from An-\ in G belongs to£„_!. Since Rk is an induced subgraph of G, every vertex a from X/c should be an isolatedvertex in G \ (2?n_i \ yw_fc). Since Bn-\ = n — 1 and | Y ^ n 5n_i| > n — k — 1, it followsthat |J5w_i \ Yn_/d < fe. Therefore every vertex from Xk is of degree at most fe in T. SinceXk <= An-\ and | ^ | = fe, the statement follows. •

Lemma 4. Le^ G be a iVn-universal graph with 2n—\ vertices, n>2. Let An-\, Bn-\, and

c be the subsets of V(G) described in Lemma I(a2). Then d(an-k-\) is either fe or fe + 1 for

any fe G {l, . . . ,n — 2}.

Proof. Follows directly from Lemmas 2 and 3. •

Let ^*[n] denote the set of triples (Gn,An-\,Bn-i) with the following properties,

(pi) Gn is a graph with 2n — 1 vertices,

An-\ = {ai9a2,...9an-i}9

and

Bn_i = {fei,fc2,...A-i}

are disjoint subsets of vertices of G", and

and

(clearly V(Gn) \ (An-\ U Bn-\) consists of a single vertex, say c).(p2) No two vertices of An-\ are adjacent, any two vertices of Bn-\ are adjacent, and

the vertex c is adjacent to no vertex from An-\ and to every vertex from Bn-\ (i.e.G(An-X Uc) = Kn and G(Bn.{ Uc) = Kn\

(p3) J(an_^_i) is either k or fe + 1 for any fc G {l,...,w — 2}.

Given (G,An_i,£n_i), let G be the complement of G, K(G) = K(G), >Jn_i = Bn-U andBn_i = An-\. Let «^[n] denote the set of triples (G,An-\,Bn-i) from £Z[ri\ such that(G,^4w_i,Bn_i) also belongs to SK[ri\.

Let y[n] denote the set of graphs G such that (G,A,B) is isomorphic to a triple from&[n] for some subsets A and B of K(G). The graph set tF[n] is defined similarly.

A graph from 6f[n] is called an n-stair graph, and a graph from J Dz] is called ann-selfstair graph.

From Proposition 1, and Lemmas 1 and 4 we have

Proposition 3. Suppose that G is a ^-universal graph with In— \ vertices, and H^n <^%>.Then G e ^[n] (i.e. G is an n-selfstair graph).

5. Stair graphs with given degree sequences

In this section we are going to classify the stair graphs according to their degree sequences.We shall also describe the set of all threshold stair graphs. This description will be usedin Section 8 to characterize all threshold minimum ^-universal graphs.

Given (G,An-\9Bn-i) € ^ 2 , let the non-increasing sequence

of the degrees of vertices from An-\ in G be called the An^\-sequence of(G,An-i,Bn-i).Let vn~l be the vector (n — 2 , . . . , 1,0) and zn~l be an arbitrary {0, l}-vector of length


n- 1. Let ^(zn~l) denote the set of triples (Gn9An-i9Bn-i) from &[n] with the ^ n _ r

sequence vn~l + zn~l, and let ^(zn~l) denote the set of the corresponding graphs. Clearly

From the definition of the set £f(zn~l) we have

Proposition 4. IfGe 6f(zn~l), then

where \y\ is the sum of coordinates of a vector y.

Let us denote by 0" and 1" the rc-vectors z of length n having all coordinates equal to0 and 1 respectively.

From the above proposition we have the following corollary.

Corollary 4. Among all the graphs in Sf\n\, the graphs from £f(0n~l) and from ^ ( l"" 1 )have the minimum and the maximum number of edges, respectively.

From Theorem 3.1(cl),(c4) it follows easily that if (Gn,An-u Bn-\) G £f" and if Gn is athreshold graph, then Gn is uniquely defined (up to an isomorphism) by its An-\-sequence.Therefore we have

Proposition 5. The set ^ ( z " " 1 ) , z""1 G Bnl, has exactly one triple (Gn,An-UBn_i) (up

to an isomorphism) such that Gn is a threshold graph.

Let us denote this triple by (T(z"~1),^n_i,Bn_i). By using the operations -h and x ongraphs, the graph T(zn~l) can be described as follows:

where a0 = c, zn~x = (zf '^zj""1 , . . .^^}), and for every i = l ,2, . . . ,n - 1

F(0) = d an-i + bn-i x,

and

F\l) = bn-i x (f aw_,- + .

Putting T(0n-{) = Mn and T(V~l) = W\ we notice that

Mn = d an-i + bn-i x (2 (2n_2 4- bn-2 x (3 an_3 + ... G an-,- + bn_i x ... +

V 2 X („_! fl! +fci Xa0 )„_! . . .)l

P^" = bn-i X d an_! + bn-2 X (2 flw-2 + bn-3 X (3 an_3 + . . . +bn-i x G an_i -f ... b\ x (n_i a\ + «o )w-i • • .)i

(see Figs. 2 and 3).Two {0,l}-vectors z""1 and zn - 1 are called complements if z""1 can be obtained from

z"-1 by replacing each 0 by 1 and each 1 by 0, i.e. zn~{ +zn~{ = I""1.Obviously we have the following proposition.


Figure 2 The graph M"

Figure 3 The graph Wn

Proposition 6. T(xn~l) and T(yn~x) are complement graphs if and only if xn~l and yn~l

are complement {0,l}-vectors. In particular, Mn and Wn are complement graphs.

Given a subset @*[n] of triples from «9S[n], put 91.(z) = 9t.[n] n 5S(z); clearly

Let 5^t[n] and J^r[n] denote the sets of all threshold graphs in S?[ri\ andrespectively. Put ^t(zn~l) = &>t[n] n ^(zn~l).

From Propositions 5 and 6 we have the following proposition.

Proposition 7."-1) = {7(2"-')},

«] = ^t[n] = {T(z) : z e

for every n = 1,2,....

y ~

L[O,F] L[hF\

Figure 4 L-operations on graphs

6. Operations on stair graphs and universal graphs

In this section we shall introduce two operations L[0,F] and L[1,F] on graphs. Theseoperations will play an essential role because several classes of graphs we are interested in(classes of stair graphs, selfstair graphs, universal graphs, etc.) turn out to be closed underthese operations. Moreover, we shall see that all threshold stair graphs can be generatedby these operations from a small one.

Given a graph F and two distinct vertices x and y not belonging to F, let us put

L[0,F]=x xF),

and

(see Fig. 4).

Let %i[n] and ^[[n] denote respectively the set of triples (G,An-uBn-i) fromsuch that G is a ^-universal graph, and, respectively, a ^-universal threshold graph.

Similarly let %^[n] and fT[rc] denote, respectively, the set of triples (G,An-i,Bn-\) from£f+\n\ such that G is a ^-universal graph, and, respectively, a ^-universal thresholdgraph.

Obviously

where n > 1 and c is either t or w. Since #^ cz 3Tn, we have

and

Given a subset @t*[n] of triples from Sf*[ri\, let $[n] denote the set of graphs G suchthat (G,A,B) is isomorphic to a triple from 0t+\n\ for some subsets A and B of F(G). Put^*(z) = ®.[n] nSZ(z); clearly «,[«] = \J{@*(z) : z G Bn~1}.

Similar inclusions hold for the graph sets ^[n], &[ri\, ^'[n], <%w[n], ^[n], and i^w[n].


Lemma 5. Let z e {0,1} and n > 1.

(al) L[z,F] is a ^~n+\-universal graph (a Wn+\-universal graph) if and only if F is a 3~n-universal graph (respectively, iVn-universal graph),

(a2) L[z,F] is threshold if and only if F is threshold, and(a3) L[z,F] e 0l\n + 1] if and only if F e 0t\n\, where 0t\n\ stands for any of the graph

sets Sf[ri\, ^[n], Wc[n], Vc\n\, while c is either t or w.

Proof. Let us first prove that if F is ^-universal, then L = L[z,F] is 3~n+\-universal. Todo this, let G be an arbitrary threshold graph with (n+1) vertices. We should prove thatthe graph L has an induced subgraph isomorphic to G. Since G is threshold, by Corollary1 it has a vertex g such that G is either G + g or G' x g, where Gf = G\g. Since G' hasn vertices and F is ^-universal, F has an induced subgraph F' isomorphic to G. By thedefinition of L = L[z,F], the vertices x and y in the graph L are, respectively, adjacent tono vertex of the subgraph L \ {x,y} = F, and to every vertex of it. Therefore F' -f x andFf x y are induced subgraphs of the graph L. By Corollary 1, G is either G + g or G x g.Therefore G is isomorphic to either F + x o r F ' x y.

Let us prove now that if F is not ^-universal, then L = L[z,F] is not 3~n+\-universal.Since F is not ^-universal, there exists a graph G with n vertices such that F has noinduced subgraph isomorphic to G. Obviously G has at least 2 vertices. Suppose thatL = L[0,F], that is, L = x + (y x F). Consider the graph H = G x g with w + 1 vertices.We shall prove that L has no induced subgraph isomorphic to H = G x g. Assume thecontrary, i.e. that L has an induced subgraph H' isomorphic to H. Then H' = G x gr,where g' is a vertex of /T and G is isomorphic to G. Since the vertex g' of H' is adjacentto every other vertex of H\ the graph H' is connected. Since x is an isolated vertex of L,and since H' is an induced subgraph of L, it follows that H1 is an induced subgraph ofL\x = y x F. Clearly G is not a subgraph of F, because F has no induced subgraphisomorphic to G. Therefore y e V(G), that is, N = H' \ y is an induced subgraph of F.Obviously N = gr \ Z, where Z = G \ y. Since y is adjacent to every vertex of F, andsince N a F, we have / / ' = y x N = y x g' x Z. Thus G = Hf \ g' = y x Z, implying thatG' is isomorphic to N. But N is an induced subgraph of F, a contradiction.

Suppose now that L = L[1,F], that is, L = y x (x + F). Then, by using the samearguments as above, one can prove that if F has no induced subgraph isomorphic to agraph G, then L has no induced subgraph isomorphic to a graph G -\- g.

The statements (al) and (a3) follow directly from the corresponding definitions, fromTheorem 3.1 and from (al).

The proof of the lemma for i^n is similar to the above proof for 3~n. •

Lemma 6. Let zn~{ e Bn~l, and let zn = zn~[0 (i.e. the last coordinate znn of zn equals 0).

Then

&(zn) = {L[0,H] : H e &(zn-x)}.

Proof. Let G G &(zn) and znn = 0. Then {G,An-UBn-\) € &(zn) for some vertex

subsets An-\ = {ai,...,aw_i} and 2V-i = {bi,...,fcn-i} of G, and d(an-\) = 0 and d(fcn_i)

is either In — 3 or 2n — 2. Since d(an-\) = 0, we have d(bn-\) = 2n — 3. ThereforeG = an-i + (6n_i x H) = L[0,iJ], where H G J^z""1). D

From Lemmas 5 and 6 we have the following proposition.

Proposition 8. Let zn~l e Bn~\ and let zn = zn~{0 (i.e. the last coordinate znn of zn equals

0). Then

®{zn) = {L[09H] : H e 0t{zn~x%

where St(zn) stands for any of the graph sets i^c(zn), ^c(zn), &(zn) while c is either t or w.

Given a graph G and a {0,l}-vector zn of length n, let us define the graph zn(G), n > 1,recursively:

where the vector zn~x is obtained from zn by deleting the last coordinate z" of zn, andwhere z°(G) = G.

By using Lemma 5, one can easily prove the following proposition.

Proposition 9.

(al) zn(G) is threshold if and only if G is threshold,(a2) zn(G) is a ^n^-universal graph if and only if G is a ^-universal graph, and(a3) zn(G) e 3#[n + k] if and only if G e @[k]; here &[k] stands for any of the graph sets

5?[k], ^[k], q/c[k], rc[k], while c is either t or w.

7. The strict hierarchy of universal, selfstair and stair graphs

In this section we investigate the hierarchy of stair, selfstair, and universal graphs of sometype. We will show that this hierarchy is strict for n > 5. In Section 4 we proved that ifa minimum ^-universal or ^-universal graph has 2n — 1 vertices, then it should belongto ^F[n] (i.e. it should be an rc-selfstair graph). Here we will see that J^fn] does containminimum ^-universal graphs and minimum ^-universal graphs for small n. Later (inSection 8) we will prove that this is true for any n.

By using the descriptions of ^ for n < 5 in Corollary 2, we can find the graph setsJ^jn], £f[n] and i^c[n], %c[n] for n < 5, where c is either t or w. This information enablesus to establish the following proposition.

Proposition 10. For any zn~x e Bnl and n e {1 , . . . , 5}

and

Figure 5 The graph A3

The validity of analogous results for an arbitrary n will be discussed in Section 8 (seeTheorem 8.1).

Put r(zn-x) = r\zn-x\ %{zn-x) = ^ V 1 ) , V[n] = [nl and <%[n] = ^[n].

Proposition 11.

(al) r[2] = <%[2] = &[2] = &[2] = {M2, W2}, and(a2) 1T\S\ c *[3] = 3F\h\ = S?[3]; moreover %[3] \ r[3] = ^(10) \ TT(IO) = {,43}

(implying that A3 is the unique minimum ^-universal graph that is not threshold, seeFig. 5).

Note that according to Proposition I A3 must be self-complementary.

Proposition 12. TT[4] C *[4] = J^[4] c ^[4] . Moreover, \W4 \ rA\ = 8|,= 2 .

From Propositions 8 and 12 we have ^(z30) = «^"(z30). By Lemma 5, {L[z,G] : z G{0,1}, G e *[4]\iT[4]} c *[5] \TT[5] . Since ^[4] \TT[4] ^ 0, it follows that <%[5\\r[5\ ^ 0.

Proposition 13. TT[5] <= *[5] c J^[5] c ^ [5] . Moreover, \<%[5] \ TT[5]| = 34, |J^[5] \= 7 , am* |5^[5] \^[5] | = 31.

From Propositions 9 and 13 we have the following theorem.

Theorem 7.1.

iTs[n] C^5[M] cz^[n] d^[n]

/or any n > 5 and s G {t, w}. Moreover, \Ws[n] \ Vs[n]\ > cu2n~5, \&n \ %n\ > cf2

n~5, and\<fn \ ^n\ > cd2

n~5, where cu > 34, cf > 7, and cd > 31.In particular, the number of minimum ^-universal graphs that are not threshold grows

exponentially as a function of n.

8. Characterizations of threshold universal graphs

Proposition 14. Let zn~l G Bn~x, and n = 2,3,.... Then T(zn~{) is a ^-universal graph.

Proof. Obviously the graph D of one vertex is 2T\-universal, and T(zn~l) = zn~l(D).Therefore the proposition follows from Proposition 9. •

Clearly Xn ^ 1Vn c yn. Since T(zn~l) has In — 1 vertices, Lemma 1 and Propositions 2and 14 imply the following proposition.

Proposition 15. Let n= 1,2,....

(al) / / G is a ^n-mug then G is a Wn-mug, i.e. %{^n) c %{ifn) c %(jQ.(a2) T(zn~l) is a ^n-mug, a O^-mug, and a Jfn-mug.(a3) T(zn~l) is a threshold 3Tn-mug, a threshold 14r

n-mug, and a threshold Jfn-mug.(a4) V(JQ = v(fn) = v{1K) = vt(fn) = vt(1Tn) = 2n-l.

The next proposition follows from Propositions I, 3, 14, and 15, and gives a necessarycondition for a graph to be minimum ^-universal.

Proposition 16. A 3Tn-mug and a it^-mug are selfstair graphs.

We are ready now to give a description of all minimum ^-universal graphs and allminimum ^-universal graphs that are threshold (compare with Proposition 10).

Let us recall that ^t[n] and <Ft[n] denote the sets of all threshold graphs in £f[n] andin &[n], respectively, and ^ f ( ^ ) and ^r(#^) denote the sets of all threshold ^,-mugsand threshold #^-mugs respectively.

Theorem 8.1. Let n= 1,2,.... The following conditions are equivalent:

(cl) G is a minimum threshold ^-universal graph,(c2) G is a minimum threshold ^-universal graph,(c3) G is a threshold n-stair graph,(c4) G is a threshold n-selfstair graph, and(c5) G is a graph T(zn~{) for some {0, l}-vector zn~x of length n—\.

In other words,fn) = &t[ri\ = &t[n] = {T(zn~l) : zn~l e Bn~1}.

Proof. According to Proposition 5, &>t[ri\ = {T{zn~x) : zn~l e Bn~1}.Therefore

^t[n] = £ft[n] (by Proposition 6),6ft[n] c fyt{3rn) (by Proposition 14),

^ ^t(irn) (by Proposition 15), and

c &t[n] (by Proposition 16).

Thus <%t(&n) = °Ut{i^n) = S?t[n]. •

From the above theorem we see that the necessary condition of Proposition 16 for agraph to be minimum ^-universal (minimum ^-universal) is also sufficient if the graphis required to be threshold.

Corollary 5. There are exactly 2n~l minimum ^-universal graphs which are threshold, i.e.

Given two graphs G and H, an injection cp : V(G) —• V(H) is called an induced embed-ding of G into H if <p is an isomorphism of G onto the subgraph of H induced by cp(V(G)).

The proof of Lemma 5 provides a simple recursive procedure for finding an inducedembedding of any given threshold graph with n vertices into the minimum 2Tn-universalthreshold graph T(zn~l\ where zn~x is an arbitrary element of IT"1.

9. Extremal universal graphs

In this section we shall describe all minimum ^-universal graphs and all minimum^,-universal graphs having the minimum or the maximum number of edges.

Lemma 7. Let (G,An-\,Bn-\) be a triple from £%(0n~l) and G be a O^-universal graph.Then G is a threshold graph, implying that (G,An-i,Bn^\) is isomorphic to (Mn,An-i,Bn-\).

Proof. Let (G,An-uBn-i) e ^(O""1). Then from Theorem 3.1(cl),(c4) it follows that Gis threshold if and only if N(an-i) c N(an-2) c • • • c= N(a\) a N(ao), where a0 = c. Let usprove by induction on A: that for any k = 1,..., n — 1

Clearly N(a\) ^ N(OQ). Suppose that N(ak-\) c • • • c N(ao). Since G is a ^-universalgraph and Qk

n is a uniform threshold graph with n vertices, we have: G contains Qkn as an

induced subgraph. We may assume that V(Qkn) c V(G).

Obviously Yn~k ^ Bn-\ and Xk ^ An-\. Since d(aj) = n — j in G, and d{xi) = n — k'mQkn

for any i = l,...,fc, we have Xk = {a^a/c- i , . . . ,^}, and therefore N(ak) = Yn-k <= N(ak-\).

aRemark 1. It is easy to prove that J^O""1) = {Mn} and ^(V~l) = {Wn}. Hence Propo-sition 16 and the above statement also imply Lemma 7.

Theorem 9.1. Let n = 1,2,.... The graphs Mn = T(0n-{) and Wn = T(ln-1) have thefollowing properties:

(pi) Mn and Wn are ?Tn-mugs and iVn-mugs,(p2) Mn is a minimum 3Tn-mug and a minimum iVn-mug,(p3) Wn is a maximum $~n-mug and a maximum Wn-mug,(p4) If G is a minimum ^Tn-mug or a minimum iVn-mug, then G is isomorphic to M", i.e.

Mn is the unique minimum ?Tn-mug and the unique minimum It

(p5) If G is a maximum 3~n-mug or a maximum H^-mug, then G is isomorphic to Wn, i.e.

Wn is the unique maximum 3~n-mug and the unique maximum i^n-mug,

(p6) Mn and Wn are threshold graphs.

In other words,

and

Proof. The properties (pi), (p2) and (p5) follow from Theorem 8.1. The property (p3)

follows from Corollary 4. By Lemma 7, Mn satisfies (p4). Therefore by Corollary 3 and

Proposition 6, Wn satisfies (p4). By Lemma 7, M" is the unique minimum #^-mug. Since

M" is a minimum ^,-mug, every minimum 5^-mug has the same number of edges as

Mn. Suppose that there exist a minimum ^,-mug G non-isomorphic to M". Since every

^,-mug is also a #^-mug, G is also a minimum #^-mug. This contradicts the fact that

Mn is the unique minimum #^-mug. Therefore we have proved property (p4). Now (p5)

follows from (pA) and Corollary 3. Property (p6) follows easily from Theorem 3.1. •

Corollary 6.

effli) = et{1K) = e{3Tn) = et{3Tn) = (n- I)2

and

n) = et(1Tn) = e(<Tn) = et(^n) = n(n - I).

References

[I] Bondy, J. A. and Murty, U. S. R. (1976) Graph Theory with Applications, Macmillan.[2] Bhat, S. N. and Leiserson, C. E. (1984) How to assemble tree machines. In: Preparata, F. (ed.)

Advances in Computing Research 2, JAI Press.[3] Bhat, S. N., Chung, F. R. K., Leighton, T. and Rosenberg, A. L. (1989) Universal graphs for

bounded-degree trees and planar graphs. SI AM J. Discrete Math. 2 145-155.[4] Bhat, S. N., Chung, F. R. K., Leighton, T. and Rosenberg, A. L. (1988). Optimal simulations

by butterfly networks. Proc. 27th Annual ACM Symposium on Theory of Computing, Chicago192-204.

[5] Brandstadt, A. (1991) Special Graph Classes - A Survey, Section Math, Fredrich SchillerUniversitat, Jena, Germany.

[6] Goldberg, M. K. and Lifshitz, E. M. (1968) On minimum universal trees. Mat. Zametki 4371-379.

[7] Chung, F. R. K. (1990) Universal graphs and induced-universal graphs. Journal of Graph Theory14 443-454.

[8] Chung, F. R. K., Graham, R. L. and Shaearer, J. (1981) Universal Caterpillars. J. CombinatorialTheory B 31 348-355.

[9] Chvatal, V. and Hammer, P. L. (1973) Set-packing problem and threshold graphs, University ofWaterloo, CORR 73-21.

[10] Chvatal, V. and Hammer, P. L. (1977) Aggregation of inequalities in integer programming.Annals of Discrete Mathematics 1 145-162.


[11] Erdos, P., Ordan, E. T. and Zalcstein, Y. (1987) Bounds on the threshold dimension and disjointthreshold coverings. SI AM J. of Algebra and Discrete Methods 8 151-154.

[12] Friedman J. and Pippenger, N. (1987) Expanding graphs contain all small trees. Combinatorica1 11-16.

[13] Hammer, P. L., Ibaraki, T. and Peled, U. N. (1981) Threshold numbers and threshold comple-tion. Annals of Discrete Mathematics 11 125-145.

[14] Kannan, S., Naos, M. and Rudich, S. (1988) Implicit representation of graphs. Proceedings ofthe Twentieth Annual ACM Symposium on Theory of Computing 334-343.

[15] Moon, J. W. (1965) On minimal n-universal graphs. Proc. Glasgow Math. Soc. 7 32-33.[16] Rado, R. (1964) Universal graphs and universal functions. Ada Arith. 9 331-340.

Image Partition Regularity of Matrices

NEIL HINDMAN f and IMRE LEADER*+ Department of Mathematics, Howard University, Washington, D.C. 20059, U.S.A.

+ Department of Pure Mathematics and Mathematical Statistics, Cambridge University, England

Many of the classical results of Ramsey Theory, including those of Hilbert, Schur, and vander Waerden, are naturally stated as instances of the following problem: given auxv matrixA with rational entries, is it true, that whenever the set fl of positive integers is finitelycoloured, there must exist some .feN'' such that all entries of Ax are the same colour? Whilethe theorems cited are all consequences of Rado's theorem, the general problem hadremained open. We provide here several solutions for the alternate problem, which asksthat xeZv. Based on this, we solve the general problem, giving various equivalentcharacterizations.

1. Introduction

Consider van der Waerden's Theorem [8]: whenever r\l = {1,2,3,...} is finitely coloured andSeN is given, there exist a and din N such that a, a + d, a + 2d, ..., a + Sd&re all the samecolour (or 'monochrome'). (By a 'finite colouring' we mean, of course, a function definedon l\l with finite range.)

Given t, let

A =

Then van der Waerden's theorem asserts that whenever N is finitely coloured, there is some

x = in f J2 such that the entries of Ax are monochrome. In terminology suggested by

+ This author gratefully acknowledges support received from the National Science Foundation (USA) via grantDMS90-25025.

394 N. Hindman and I. Leader

Walter Deuber, we are talking about the image partition regularity of A, i.e. asking that theimage of x under the map defined by A be monochrome.

By contrast to image partition regularity, the question of which matrices are kernelpartition regular was completely settled by Rado in 1933 [6]. (Here auxv matrix A is kernelpartition regular if and only if, whenever l\l is finitely coloured, there is some J G N I ; withall entries monochrome such that Ax = 0. That is, there is a monochrome member of thekernel of the map defined by A.) For anyone not familiar with it, we will present Rado'sTheorem later in this introduction.

Now van der Waerden's Theorem can be proved as a consequence of Rado's Theoremas follows: given / , one takes x19x2, ...,x,+1 as the terms of an arithmetic progression andcharacterizes the fact that they are in an arithmetic progression by the equationsx2 — x1 = x3 — x2 = x^ — x3= ... = x/+1 — x,. We can rewrite these as

— x1 + 2x2 — x3 = 0

so we are asking for the kernel partition regularity of the matrix

- 1- 1- 1

211

- 110

0- 1

1

00

- 1

... 0

... 0

... 0

000

v—1 1 0 0 0 1 - 1 ,

Alternatively, one can rewrite the equations as

in which case we are asking for the kernel partition regularity of the matrix

- 100

2- 1

0

- 12

- 1

0 ...- 1 ...

2

000

000

000

0 0 0 0 - 1 2 - 1 ,

But there is a problem here! Rado's Theorem does indeed tell us that both of thesematrices are kernel partition regular. But, unfortunately, one monochrome solution hasx1 = x2 = ... = xM = 1; not exactly what we had in mind for our arithmetic progression.(This is not a far-fetched example. The first author made this very error in a talk a fewyears ago until it was brought to his attention by Deuber.)

Image Partition Regularity of Matrices 395

A cure in this case can be obtained by strengthening the conclusion of van der Waerden'sTheorem to require that the increment d also have the same colour as the terms of thearithmetic progression. With this addition, the original matrix for the image partitionregular statement becomes

while one conversion to a kernel partition regular matrix is

100

- 110

0- 1

1

... 0

... 0

... 0

000

0 0 0 1 - 1 ,

But one can surely imagine potential problems. Conceivably the original statement couldhave been valid, while the strengthened one was not. For this reason, as well as for theability to answer a question in the form in which it is naturally stated, we claim our problemis interesting: determine which matrices are image partition regular (in the sense statedearlier).

The theorem of van der Waerden is not the only classical result that is naturally statedin this form. Schur's Theorem [7] says that whenever fJ is finitely coloured there exist xx

and x2 with x19x2, and x± + x2 monochrome. In this case the matrix corresponding to thestatement is

i.e. the first three rows of our strengthened version of van der Waerden's Theorem. Evenolder is the 1892 result of Hilbert [4]: given any / e f\l, whenever fJ is finitely coloured, thereexist aeN and x1,x2,...,x/, in N such that all sums of the form a+YjneFxm where0 =N F^ {1,2,.. . ,/}, are monochrome. Thus, when / = 3, this theorem asserts that thematrix , .

1 1 0 0\0 1 01 1 00 0 11 0 10 f 11 1 1

is image partition regular.Image partition regular matrices have played an important role in Ramsey Theory. In the

terminology of Deuber [2] (modified only slightly), say that a matrix A is an (m,p, c) matrix

(where m,p, and c are in M) if the rows of A consist of all vectors feZm\{0} such

that(1) for each /e{l,2, ...,ra}, \rt\ p, and(2) if t = min{/: ri + 0}, then rt = c.

(Note: two (m,p, c) matrices differ only by the order of their rows.) Deuber showed [2] thatany (m,p,c) matrix is image partition regular. He further showed that if B is any kernelpartition regular matrix, there exist some ra, p and c such that, given an (ra,/?, c) matrix Aand given any xe Nm, one can choose entries for y from among the entries of Ax such thatBy = 0. Since one can also show that (m9p,c) matrices are image partition regular usingRado's Theorem, one might be led to believe that a matrix is image partition regular if andonly if it consists of some of the rows of an (m,p, c) matrix. We shall see, however, that evenweakened versions of this hypothesis are false.

As promised earlier, we now present Rado's Theorem. It depends on a notion called the'columns condition'.

Definition 1.1. Let A be a uxv matrix with entries from Q, and let c1?c2, . . . ,cr be thecolumns of A. Then A satisfies the columns condition if and only if there exist meN and/l5/2, "">Im such that(a) {/1? /2,.. . , IJ partitions {1,2,..., v},

(b) £ , , , / < = 6, and

(c) for each te{2,3, ...,ra} (if any), let Jt = IJJl}^: there exist StieQ for each ieJt such

that Yjiei/t = Ysj^tj-Ci-

Theorem 1.2. (Rado [6].) Let A be a uxv matrix with entries from Q. Then A is kernel

partition regular (i.e. whenever N is finitely coloured, there exists monochrome yeNv such

that Ay = 0) if and only if A satisfies the columns condition.

To describe the results of this paper, we introduce a weaker notion of image partitionregularity (so that what we have been calling 'image partition regular' now becomes'strongly image partition regular').

Definition 1.3. Let A be a uxv matrix with rational entries.(a) A is strongly image partition regular if and only if, whenever 11 is finitely coloured,

there exists xe Nu such that the entries of Ax are monochrome.(b) A is weakly image partition regular if and only if, whenever fl is finitely coloured, there

exists xeZv such that the entries of Ax are monochrome.Since we have allowed the entries of x (in weakly image partition regular matrices) to

range over Z, one could reasonably ask what happens if we talk about colourings of Z.First, of course, one would need to be talking about colourings of Z\{0}. (Otherwise anymatrix would be partition regular, letting x = 0.) If then in (b), one replaces colourings ofl\l with colourings of Z\{0}, one arrives at a statement equivalent to (b). Indeed, oneimplication is trivial. For the other implication, let a colouring of fl be given with say rcolours. Colour the negative members of Z with r new colours, agreeing that a and b getthe same colour if and only if — a and —b had the same colour. If Ax is monochrome, so


is A( — x). (There is a fourth possibility: in (a) one could replace colourings of 11 withcolourings of Z\{0}. This results in a proposition equivalent to the assertion that either Aor — A is strongly image partition regular.)

In Section 2 of this paper we present several characterizations of weak image partitionregularity. Effective solutions are given by statements (II) and (III) of Theorem 2.2. (As faras we know they are new, although the ideas are not: they are in the spirit of the proofsthat Rado's Theorem implies the partition regularity of (ra,/>, c) matrices.) The solution ineither case involves constructing another matrix, and verifying that the new matrix satisfiesthe columns condition. This is a routine, if lengthy, process. (The problem of determiningwhich matrices satisfy the columns condition is NP complete, because it implies the abilityto determine whether a set of numbers has a subset summing to 0.)

In Section 3 we turn our attention to the more difficult problem of characterizing strongpartition regularity. We present several analogues to statements in Theorem 2.2 and somenew conditions, and prove that they are each equivalent to strong partition regularity.

We conclude the introduction with a remark about vector notation. We take f P or Zr

or Qv to consist of column or row vectors as appropriate for the context. Given a row

vectorpe Qv and a u x v matrix A, we denote by the (u+l)xv matrix whose first u rows

\P)are those of A, and whose (w+ l)st row is/7. The meaning of other similar notation shouldbe obvious. We let w = N U {0}.

2. Weak image partition regularity

We begin by introducing a notion based on Deuber's (m,p,c) matrices.

Definition 2.1. Let A be a u x v matrix with rational entries. A satisfies the first entriescondition if and only if each row of A is not 0, and whenever i,je{l,2,...,u} andfe{l,2, ...,v} and t = m\n{k:ai k 4= 0} = m\n{k\aj k 4= 0}, one has at t = aj t> 0.

It is a fact (Theorem 2.11) that if A satisfies the first entries condition, then A is stronglyimage partition regular. One also easily sees that rearranging the columns of a matrix doesnot affect its partition regularity. Thus one would be tempted to conjecture that a matrixA is strongly or weakly partition regular if and only if the columns of A could be rearrangedso that the resulting matrix satisfied the first entries condition. This is easily seen to be false,

however. Consider -4 = 1, ? I • Then neither A nor I satisfy the first entries

condition, while A is in fact strongly partition regular. (Simply let xx = x2.)We now state the main result of this section. Its proof will be pieced together as we

proceed through the section.

Theorem 2.2. Let A be a uxv matrix with rational entries. Then the following statements areequivalent:

(I) A is weakly image partition regular.(II) Let ^ = rank(^4). Rearrange the rows of A so that the first / rows are linearly

independent. Let f1,f2,...,fu denote the rows of A. For each te{i?+ l , / + 2 , . . . , u} {if

any), let ytl, yt2, ...,ytJ be the members ofQ determined by rt = YIi=\7t,Ji- Ifu>£,

let D be the {u-/)xu matrix such that, for te{\,2, ...,u-/} and / e {1,2, . . . ,«},

Then either £ = u or the matrix D satisfies the columns condition.

(III) Let cx,c2, ...,cv be the columns of A. Then there exist tx, t2,..., tv in {xeQ\ x =}= 0} suchthat the matrix

- 1 0 ... 00 - 1 0

t1c\ t2c2 ... tvcv .

0 0 ... -}

is kernel partition regular.( A\

(IV) For each peZv\{0}, there exists beQ\{0} such that \ is weakly image partition\bp)

regular.(V) There exist bx,b2, ...,bvin Q\{0} such that

Ab, 0 0 ... 00 b2 0 ... 00 0 b3 ... 0

, 0 0 0 ...

is weakly image partition regular.(VI) There exist an m ^ u and a uxm matrix B that satisfies the first entries condition such

that for each yeZm there exists xeZm such that Ax = By.

Before beginning the proof of Theorem 2.2, a few remarks about the special features ofeach of the equivalent conditions are in order. As we observed in the introduction,statements (II) and (III) allow us to determine in finite time whether a matrix is weaklyimage partition regular. The added information conveyed by statement (IV) is clear, but wefeel remarkable: a weakly image partition regular matrix can be expanded almost at will.Statement (V) tells us, for example, that given any weakly image partition regular u x vmatrix A, there is a subset P of {1,2, ...,v} such that whenever M is finitely coloured, thereis an xeJ.v such that the entries of Ax are monochrome and if ieP, xt > 0, and ifze{l,2, ...,v}\P,xt < 0. (In particular we may insist that the entries of x are not 0.) Finally,statement (VI) tells us that the first entries condition, which one might have hoped wasnecessary for weak image partition regularity, does provide a characterization.

The argument in the proof of the following lemma is standard. At various stages insubsequent arguments we shall need to consider common multiples in order to make some

variables integers. We remark to the interested reader that an alternative approach is toreplace Z by Q and N by Q+ = {xe Q: x > 0} and at the end use a compactness argument.

Lemma 2.3. Let A be a uxv matrix with rational entries. Then statements (I) and (II) of

Theorem 2.2 are equivalent.

Proof. Assume (I) holds, assume / < u, and let D be as defined in statement (II). We showthat D is kernel partition regular, so that, by Rado's Theorem (Theorem 1.2), D satisfiesthe columns condition. Let l\l be finitely coloured, and pick xeZv such that the entries ofAx are monochrome. Let w = Ax. We claim that Dw = 0. To see this, let t e {1,2,..., u — £}be given. Then u u v

= 0.

Now assume (II) holds and at first that / = u. Then we may assume that the first fcolumns of A are linearly independent. Let A* consist of the first / columns of A and choosex1? x2,..., xf in Q such that / 1

(i

Let d be a common multiple of the denominators in x. For ie {1,2,..., {}, let yt = dx(, andfor ie{f+l,f + 2,...,v} (if any), let yi = 0. Then

Now assume that u > / and that the matrix D of statement (II) satisfies the columnscondition. We may assume that the upper left / x / corner A* of A has rank /, byrearranging rows and columns if necessary. Let c be the absolute value of the determinantof A*. It is immediate (or see Theorem 2.11) that Nc is 'large', that is, whenever Nc isfinitely coloured Nc contains monochrome solutions to any kernel partition regular matrix.So let l\l be finitely coloured and pick monochrome xl5x2, ...,xu in Nc such that

r\0


For ze{l,2,...,«}, let zt = xjc, and choose w15 w2,..., w in Q solving

A*w =

For ye{l ,2 , . . . , /} let j ^ = v^c, and observe that since c = |deM*|, each yêZ. Forje{£+\,£ + 2,...,v} (if any), let j>; = 0. We show that Ay = x, which will complete theproof. If / e{ l , 2 , . . . , / } , one has immediately that YjVj=iaijyj = YjUiaijwjc = zic = xfNow let f e { ^ + i y + 2,...,M} be given. Then given y we have atj = YjLi7t.i-aij'Dx = 0, so M ,

1=1

so f

•^t La

Thus we haveV /

i .W;.C

= xt. U

We have already observed that statement (II) of Theorem 2.2 is one that is effectivelydecidable. It is also easy to work with, and as a consequence it will be heavily utilizedthroughout the rest of the paper, beginning with the next lemma.

Lemma 2.4. Let A be a uxv matrix with rational entries that satisfies statement (II) of

(A \Theorem 2.2, and letpeZv\{0}. There exists beQ\{0} such that satisfies statement (II)

of Theorem 2.2.

Proof. Let / = rank ( 4). We may presume the first £ rows of A are linearly independent. Letthe rows of A be f19 f2, ...,fu.

( A\ ( A\Case 1. (*f = u) If p$span{f^r,, ...,/>}, then rank = / + l = w+1, so satisfies

\PJ \PJstatement (II) of Theorem 2.2. Thus we assume pespan{f^fg, . . . ,f /}. Let aâ., aêQsuch that p = X/=iai^-- Since ^ =|= 6, we may pickye{l ,2, . . . , /} such that a; 4= 0 and let

( A\b = I/a,. Then bp = y,/-=1baifi, so the matrix D determined by statement II for is

\°P)(bocvboc2, ...,botf, - 1 ) . Let 71 = { y , / + l } , 72 = {1,2, . . . , /}\{y}, 82j = 0, and let^2y+i = ~SLieiJDCLi- Then we have shown that D satisfies the columns condition.

Case 2. (/ < u) Let D be the matrix determined by statement (II) for D. Let c1 ?c2, . . . ,cu bethe columns of D. Then D satisfies the columns condition, so pick me{1 ,2 , . . . , u)and 71?72, . . . ,7m such that {71572, ...,7m} is a partition of {1,2, ...,w} and X*e/,^- = -For re{2 ,3 , . . . , m}, if any, let / , = (J ' :}7; and pick (,8t t}ieJ in Q such that

Assume first that p$span{f15f2, ..., j>}. Then let Z? = 1. Then rearrange the rows of

by adding p as f0. Let / ) ' be the matrix determined by statement (II) for I I. Then D' is

Z) with a new column 0 added in front as c0. Then letting 7 = 7X U {0} and letting 7 = 7 andS'tJ = StJ for te{2,...,m} and ze/^, one sees that D' satisfies the columns condition.

Thus we assumepespan{f^f2, . . . , f / } , and pick a19a2, . . . ,a^in Q such t h a t ^ = X d a / - ^ -For z e { / + l , ^ + 2, . . . , M } , let a?: = 0. If X ^ / ^ + O, let Z? = l /%-6 / ia? : and let fc= 1. If^ i 6 / ocf = 0 and there is somefce{2,3, .. . ,w} such that £ \ e / at. + ^ i e J Sk t.at, let ^ be thefirst such, and let b = 1 / ( I ^ 6 / ^ - L , e J ^ . a ? ; ) . If L 6 / l ^ = 0 and for all re{2,3, . . . ,m},E<e/f

a< = Z ^ J ^ M - ^ ' letfc = w + l and let b= 1.Define the matrix 7)' as follows: for te{l,2,...,u — £} and / e{ l , 2 , . . . , «} , let d'ti = dti

and let d't M+1 = 0; for / e { l , 2 , . . . , M}, let d'u_/+lA = ba( and let d'u_/+l w+1 = — 1. T h e n i ) ' is the

matrix determined by statement (II) for . Let c[, c'2,..., c'u+1 be the columns of D\ We

need to show that Df satisfies the columns condition. To do so, we consider the possibilitiesk = m+ 1 and k ^ m separately.

Assume first that k = rn+\. For te{\,2,...,m} let 7 = 7, and Jt=Jt, and let4 + 1 = {M+l}. For /e{2,3 , . . . ,m} and ieJt, let ^^; = Sti. Then ^ i e / ; q = 0, and forfe{2,3, . . . ,m}, Xiie/'^/ = Z i i e j ' ^ , / - ^ ' s o w e o n l y n e e ^ to define 8'm+lJ for / e{ l , 2 , . . . , «} .Since ^#=0 , pick je{\,2,...,/} such that a;. 4= 0, let ^ + l j = —l/aj 5 and for / e{ l , 2 ,. . . , / } \ { A let (Tm+li< = 0. For / e { l , 2 , . . . , ^ - / } , let 8'm+1J+t =-d'tJ*y Then ^ + 1 =E?=i*m+i, i-^ as required.

Now assume k^m. Let Tk = 7 U {w+ 1}, and for fe{ l ,2 , ...,w}\{fc} let Tt=lv For/E{2,3, . . . ,m}, let / ; = U^1 /* - F o r ^ { 2 , 3 , ...,m} and z e ^ , let Vui = Sti. For te{k+l,k + 2,...,m}, let ^ u+1 = Z i G j # ' .£•# , . — £ i e / 6.a?:. We see then that 7X does satisfy thecolumns condition. •

Lemma 2.5. Statements (III) and (V) o/ Theorem 2.2 are equivalent.

Proof. We show first that statement (III) implies statement (V). Let tl,t2,...,tv be as instatement (III), and for ye{ l ,2 , ...,v}, let bi = \/tj. Let d be a common multiple of thedenominators of the ^s. To see that b19b2, ...,bv are as required by statement (V), let f l be

402 TV. Hindman and I. Leader

finitely coloured. As we remarked in the proof of Lemma 2.3, Nd\s large, so we may choose

monochrome zl 9z2, . . . ,zv , w15 vv2, ..., wu in Pyj such that

- 1 0 ...

tl c t2 c2 ... trcr 0 — 1 ...

0 0 ... - 1w1

vP9

= 0 .

Then given any /e {1,2,..., u), one has vv? = ^ J = 1 1 } at j zy For je {1,2, . . . , v}, let x} = t} z- and

observe that each XJEZ, since ZjENd. Then

wi\

A

bx 0 ... 0

0 b9 ... 0

0 0

\ ^

The proof that statement (V) implies statement (III) is similar, though somewhat easier,since we do not need to worry about 'large' sets. Given a finite colouring of f\l, one picksx19x2, ...,xv in Z such that if

0 ...

0 b2

,0 0 . . . bj \xv

then y is monochrome. Letting t} = \/b} for J'E{1,2, ...,r}, one sees that

t,c\,

- 1 00 - 1

0 0 - 1= 0.

•We can now establish most of Theorem 2.2.

Lemma 2.6. Statements (I), (II), (III), (IV) and (V) of Theorem 2.2 are equivalent.

Proof. By Lemma 2.3, statements (I) and (II) are equivalent. Consequently, Lemma 2.4 tellsus that statement (I) implies statement (IV) (which trivially implies statement (I)). Applying

Lemma 2.4 v times in succession to the vectors (1,0,..., 0), (0,1, . . . , 0),..., (0,0,..., 1) showsus that statement (I) implies statement (V), which in turn implies statement (I). By Lemma2.5, statements (III) and (V) are equivalent. •

We now set out to establish the equivalence of statement (VI). We prove in Lemma 2.7a statement stronger than needed here, but which will be used in the next section. As aconsequence, our proof that statement (II) implies statement (VI) may seem morecomplicated than it really is.

Lemma 2.7. Let A be auxv matrix with rational entries such that rank A = / < u and assume

that A satisfies statement (II) of Theorem 2.2. Let 7l9 72, . . . , Im and for te {2 ,3 , . . . , m] let Jt and

(8t tyiej be as given in the columns condition for the matrix D of statement (II). Then there

is a uxm matrix B satisfying the first entries condition such that for each yeZm there exists

xeZv such that Ax = By. If for each re{2 ,3 , ...,m} andeach ieJt n { 1 , 2 , . . . , / } , 8tJ < 0, then

for each ZG{ 1,2,. . . , /} and each f e{ l , 2 , . . . ,w} , bLt ^ 0, where bLt is the entry in row i and

column t of B.

Proof. Assume, as in statement (II), that the first / rows of A are linearly independent. Nowthe matrix

has rank / , so we may rearrange the columns of A so that the upper £ x / corner, ^4*, hasnonzero determinant.

Let d be a common multiple of the denominators in A, and let E = Ad. Then D is alsothe matrix determined for E by statement (II). Let E* be the upper left £ x / corner of E,and let w = |det(£*)|. Let cl9c2, ...,cu be the columns of D. Now D satisfies the columnscondition, so pick me{l,2,. . . ,u} and 7l572, ...,7m such that {71972, ...,Im} is a partition of{1,2,..., u} and X? e / ic ? = 6. For f e{2,3, ...,m}, if any, let Jt = ( J j : ^ and pick (Sti}ieJ( inQ such that ^ ? 6 / ct = Yaiej ^tj^i- Let Jx = 0 and let n be a common positive multiple ofthe denominators in 8tJ for te{2,3, ...,m} and ieJt. Define the u x m matrix B by

Then B satisfies the first entries condition. Further, if 8tJ < 0, then bit > 0, as claimed.Now let yeZm be given and let z = By. Since each St t:.neZ, we have that w divides each

entry off. Let PeQ/ be such that


and note that (for example by Cramer's rule) each vrweZ. Define xeZv by

(w.v, if ye{l,2,...,/}Xj [0 if . / e K + i y + 2,...,1;}.

We claim that Ex = z (so A(dx) = Ex = z = By as required). For z e { l , 2 , . . . , / } , we have

as required. So let te{£+ l , / + 2 , . . . , w} be given. Now we have ft = YJi=i7t ?•>% s o for each

1,2,. . . ,^}, etj= YuUjtj-eu- Thus

so it suffices to show that Yfi=\7t i-zi = zo ^ a t is? w e want to show that YJ=\^t-/ \-z\ = 0.

Now, given any se{l,2,...,m}, we have X? 6 . / S ^u-^ = X/e/,^- (where, if s= 1, we treat

Z i ^ 0 ^ s , i - ^ a s 5). Thus, for each s e { l , 2 , . . . ,m},

SO

(s=\ \ieJ

fs=l \i=l

1=1 S=l

as required. D

Lemma 2.8. Let A be auxv matrix with rational entries. Then statement (II) of Theorem 2.2

implies statement (VI) of Theorem 2.2.

Proof. Let / = rank ,4. If / > u, this follows from Lemma 2.7, so we assume that £ = u. Let

vDeQK be such that A*w = T, where

Myand let d be a common positive multiple of the denominators in w. Let

Then B satisfies the first entries condition. Let yeZbe given, and define xeZv by

ydw\ if / e{ l ,2 , . . . , /}0 if U

T h e n lya

Ax = A*(ydw) = I yd \= By. D

\y'd,The following lemma completes the proof of Theorem 2.2.

Lemma 2.9. Let A be a uxv matrix with rational entries. Then statement (VI) of Theorem

2.2 implies statement (I).

Proof. Let I\J be finitely coloured. For each y e { l , 2 , ...,m} pick d^N such that for any

ie{1,2,..., M}, ify = min{t:bitt 4= 0}, t h e n ^ ; = ^ (which we can do, since B satisfies the first

entries condition). Let c be a common multiple of d1,d2, ...,dm. Define a new matrix E as

follows: for / e { l , 2 , . . . , « } and y e { l , 2 , . . . ,m}, et j = bt r(c/dj). Let p = max{|^.<;.|:ze{l,

2, ...,w} and ye {1,2, . . . ,w}}. Then ^ c o n s i s t s of some of the rows of an (m,/?, c) matrix,

so pick, by Deuber's (m,/?, c)-sets theorem [3], vi>e^m such that the entries of Ew are

monochrome. Define jef^Jm by y. = wr{c/d^) for y e { l , 2 , . . . ,m}. Then By = Ew. Pick

?' such that Ax = By. Then the entries of Ax are monochrome. •

In the course of proving Lemma 2.9 we showed, using Deuber's (m,/?,c)-sets theorem,that any matrix satisfying the first entries condition is strongly image partition regular. Wedigress now to prove a much stronger assertion in Theorem 2.11, which we believe isinteresting in its own right. Our proof is similar to the proof of Rado's Theorem in [3,Theorem 8.22].

We shall have need of a result from [3]. The result refers to the notion of a 'central' subsetof l\l. The definition of central in either of two equivalent forms involves the introductionof considerable terminology, and we will not do this here. For our purposes two facts aboutcentral sets are all we need to know. First, if N is finitely coloured, there is some colour suchthat the set of n receiving that colour is central. (See [3] or [1].) Second, if C is a central setand de N, then C n Nd is central. (See [5, Theorem 2.7].)

Theorem 2.10. Let C be a central subset ofN, let SeN, and for /e{l, 2, . . . , /} let (yLnYn=x

be a sequence in N. There exists a sequence <^n)J=1 in N and a sequence <i/n>^=i ofpairwisedisjoint finite nonempty subsets of N such that, whenever F is a finite nonempty subset of N

n G F n e F teHn

Proof. [3, Proposition 8.21] or see [1, Theorem 4.12]. •Theorem 2.11. Let A be a uxv matrix with rational entries that satisfies the first entriescondition, and let C be a central set in M. There exist sequences (xUn}*=1 in N forie{\,2, ...,v} so that, whenever F is a finite nonempty subset of N and

X —

ly x \neF

y x92.71

one has AxeC". In particular, if N is finitely coloured, there is a colour-class C as above.

Proof. We proceed by induction on v. Assume first that v = 1. Then there is some positiverational d such that A = (d). (We may presume A has no repeated rows.) Write d = p/q,wherep, qeN. Then, as we have observed, C(1 Np is central, so choose, by Theorem 2.7,some sequence <6n>n=i w i t n YjneFbn eC f]Np whenever Fis a finite nonempty subset of l\l.Let xln = (bjp).q.

Now let veN and assume the statement is valid for v. Let A be a w x ( r + l ) matrixsatisfying the positive first entries condition. We may assume we have somete{\,2, ...,M— 1}, and some positive rational dsuch that if ie{1,2, ...,t}, thenaul = 0, whileif ie{t+ 1, t + 2,..., M}, then aiX = d. (Additional rows may be added if need be to ensurethat such a t exists.) Let B be the t x v matrix defined by bui = aiJ+1. Let a central set C begiven, and let, for /e{l,2, ...,v}, <v?<n>J=i be a sequence in fl as guaranteed by the inductionhypothesis for B and C. For each ie{t+l,t + 2,...,u} and each neN, letzUn = YJ]=2

ai j-yj-i n- Write d = p/q, wherep,qeN. Then C n Np is central, so choose, byTheorem 2.10, <6n>J=1 and (Hn}*=1 such that, for each finite nonempty F^M, one has

1, t + 2, ..., u},

For each n let xx n = (bn/p).q, and forye{2,3, ...,v+ 1}, let x;<n = X«e//w>j-i.s-To see that the sequences <x; n>J=1 are as required, let finite nonempty F ^ N be given,


and let /e{l, 2,...,«}. We need to show that Y^lflu- L e F x j . » e ^ Assume first that/e{l,2, . . . , t} . Then

X>«- E *;.» = E «,-.,•• E E yt-i..

where G=[JneFHn. This sum is in C by the induction hypothesis. Now assumeie{t+\,t + 2, . . . ,M}. Then

XXr L *,.„ = * £ (&-A0 + XX,- Z S JVi,sj = 2 neF seHn

v+1

seHnj=2

seHn I

The reader may wonder why we speak of matrices with rational entries rather thaninteger entries. Indeed, if A is a matrix with rational entries and dis a positive multiple ofthe denominators in A, it is easy to see that A is weakly (respectively strongly) imagepartition regular if and only if dA is weakly (respectively strongly) image partition regular.Certainly, if (dA)x is monochrome, then A(dx) is monochrome, so the sufficiency isimmediate. To see the necessity, assume that A is weakly (respectively strongly) partitionregular, and let cp: N ->{1,2, ...,r} be a finite colouring of l\l. Define r: N ->{1,2, ...,r} byT(H) = cp(drc), and pick xeZv (respectively Nv) such that Ax is monochrome with respect tor. Then (dA)x is monochrome with respect to cp.

The reason for choosing to use matrices with rational entries is reflected in statements(IV) and (V) of Theorem 2.2. As we shall see in the final result of this section, even if onestarts with an integer matrix, one may not end up with one. The proof illustrates theapplication of statement (II) of Theorem 2.2.

Theorem 2.12. Let A = I . Then A is strongly image partition regular, but there do not

exist integers bx and b2 such that

is weakly image partition regular.

Proof. The matrix D given by statement (II) of Theorem 2.2 for the matrix


is/ 8^ _ibi _ ! 0 \

\ - * b 2 l b 2 o - l ) '

Now if (b19b2) = (7/2,7), we see that c1 + c2 + c3 + c4 = 0. Consequently by Theorem 2.2,

is weakly image partition regular. Then given any xl9x2 with (l/2)x1eN and lx2eN, onemust have xt > 0 and x2 > 0. Consequently the matrix is strongly partition regular.

Now assume we have (b19 b2), making

weakly image partition regular, and observe (since 0 + 0^ l\l) that b± + 0 and b2 =}= 0. NowD must satisfy the columns condition (by Theorem 2.2). The only possible choices for /x

(which do not obviously force b± = 0 or b2 = 0) are I± = {1,3,4}, ^ = {2,3,4}, andIx = {1,2,3,4}. These choices force (b19b2) to be (7/3, - 7 / 2 ) , ( - 7 , 7/3), and (7/2, 7)respectively. •

3. Strong image partition regularity

In this section we turn our attention to strong image partition regularity. Just as in Section2, our aim is to give several equivalent characterizations. These are analogues of theconditions in Theorem 2.2. Howevei, the proofs are not just analogues of the proofs inTheorem 2.2, because we are now dealing not only with linear algebra: the ordering on Nis important. In fact, when we come to prove that strong image partition regularity impliesvarious properties, we shall need to construct some explicit colourings of f J, rather thanrelying on the columns property.

Theorem 3.1. Let A be auxv matrix with rational entries. Then the following statements areequivalent.(A) The matrix A is strongly image partition regular.(B) Let c15 c2, ..., cv be the columns of A. Then there exist t1912,..., tv in {xeQ: x > 0} such

that the matrix- 1 0 ... 0

0 - 1 ... 0t1c1 t2c2 ... tvcv . .

0 0 - 1is kernel partition regular.


(C) There exist b±,b2, ...,bv in {xeQ:x> 0} such that

00

0b2

0

A00

... 0

... 0

... 0

0 0 0 tv

is weakly image partition regular.

(A \(D) For each peo/\{0\ there exists beQ with b > 0 such that is strongly image partition

J \bpregular. v 7

(E) There exist meN and auxm matrix B that satisfies the first entries condition such thatfor each ye Nm there exists xeHv such that Ax = By.

We remark that both statements (B) and (C) of Theorem 3.2 provide us with effectivemeans of determining whether a given matrix is strongly image partition regular. In the caseof statement (B), one simply determines whether one can find tx,t2,...,tv such that thespecified matrix satisfies the columns condition. Since statement (C) refers to weak imagepartition regularity, one may utilize statement (II) of Theorem 2.2 to see if there existbx,b2, ...,bv making the resulting matrix partition regular.

We now record some trivial implications.

Lemma 3.2. Statements (D) and (E) of Theorem 3.1 each imply statement (A).

Proof. The only assertion that is not completely obvious is that (E) implies (A). To see this,one simply applies Theorem 2.11. (If By is monochrome and Ax = By, then Ax ismonochrome.) •

We would not characterize the following as 'trivial', but it does follow quickly from (thehardest part of) Theorem 2.2.

Lemma 3.3. Let A beaux v matrix with rational entries. Then statement (C) of Theorem 3.1implies statement (D).

Proof. Applying Theorem 2.2 to the weakly image partition regular matrix

bx 0 0 ... 0 *0 b2 0 ... 00 0 6, ... 0

\ 0 0 0 bJ

we obtain some de Q\{0} such that the matrix

00

0b2

0

A0 ..0 ..b3 ..

. 0

. 0

. 0

0 0 0 ... £dp

is weakly image partition regular. Since given any /, bt > 0 and bi xt > 0 implies xi > 0, we

( A\see that this latter matrix is strongly image partition regular and hence so is . Finally,

\dp)

given any xe Nl\ we have p.x > 0, since peajv\{0}. We know there exists xe NV such that(dp) .xe N (by the strong partition regularity) so we conclude that d > 0. •

We have one more routine implication.

Lemma 3.4. Let A be a uxv matrix with rational entries. Then statement (B) of Theorem 3.1implies statement (C).

Proof. This may be taken verbatim from the first half of the proof of Lemma 2.5, notingthat bj > 0 since t} > 0 (and further that each .V;G f J). •

We now set out to show, in Lemma 3.6, that statement (C) of Theorem 3.1 impliesstatement (E).

Lemma 3.5. Let A be a ux(u + v) matrix such that for i,je{\,2,..., w},

That isa12

A = ao, a9 a2r

0- 1

if iif i

- 10

= 10 ...

- 1 ...00

iaul a,,2 ... aur 0 0 ... - 1

If A satisfies the columns condition and l x , L 2 , -Jm and, for t = {2, . . . , m } , Jt and ( S t i } i e J

are as given by the columns c o n d i t i o n , then one may assume that for t e { 2 , . . . , w } and{l,2, ...,i;}, St f < 0.

Proof. For each fe{2,3, ...,w}, let J* = Jt n {1,2, . . . , r}, and for each re{l,2, ...,m}, letI* = It Pi {1,2, ...,i;}. We proceed by induction on t, producing </ .,->,-eJ such thatL«6/,^ = L ? e ^ ^ . ? . ^ and for ieJ*, fitJ < 0.

Since the columns ct with / > v have no positive entries, we can assume /? + 0 . Pick/*. Then for each / G { 2 , 3 , ...,m}, A:eJ*.


Let [i2 k = min{ — \+82 k — 82 f.jeJ*}, and for jeJ*\{k}, let ji2; = /i2 k + (82 j — 8.2 k).

Then for e a c h y e / f ,/i2j ^ — 1. F o r y e / 2 \ { 1 , 2 , ...,v}, let

/*2J= L^2,i'^J-v,i- L aj-v,i'

Now we show thatLa Lj La r2,j- Lj-

We show this line by line, so let / e { l , 2 , . . . , u } be given. Assume first that S + veJ2. Then

2^ jti2j.a/j= 2 J Jbt2j-a/j—JuYj+vJeJ2 jeJ*

= L /*2,r^j- L /*2.r^j- L «/jjeJ* Ve-/* je/2

v^

j e l .

Next assume £ + v$J2. Now / * = /* and f + v^I^ so 0 = ^ ; e / an = E;e/*^/;- Thus«/.* = Le./?v*(-«/.;)-Then

jeJ2 jeJ*

= X P>2J-a/J+P>2.k-af.kjeJ*\{k\

= E P>2J'a/J~ E V>2.k'a/JjeJ*\{k\ jeJ*\\k\

ieJ*\{k\

= E

= E *2.*.«M*jeJ*

= E S2J-a'JjeJ,

Now let t > 2 and assume the induction has proceeded through t—l. For jelfL, let

lit_x j = — 1 and observe that for a l lye7*, / V u < 0. Also observe that i f / e { 1,2,..., w} and

f + v$Jt, thenE « M = E ^ j = E Pt-u-tyj,

jelf_, jel(^ jeJt_,

-M>t-i,k-<*/.*:= l i (-«/ , ; )+ E Pt-i,ra/.i= X P-t-ij-t/j-

Consequently ^ fc = ^ ^ ^ ^ ( ( - ^ - u V ^ . ^ J . ^ ^ .Now let

For jeJ*\{k}, let

ForjeJt\{l,2,...,vl let

We now letye/f and show that / 7j < 0. Note that ^t-u/^t_1A. > 0. Now

SO

^ j = (Pt-u/M-t-i.kXj

Now let ^e{l ,2 , . . . , w}, and assume first that f + veJt. Then

Z M>tj-<*/j= X Ptj-<*,j-M-t.v+,je./t jeJ?

= E i « / - ( E

= E «/.;•

Finally assume £ + v$Jt. Then

*

Eje./*\{k\

= E Ptj-a/j+V>t.k E ( ( -je.J?\{k\ je./*\{k\

= E Qitj-p>t.k-p>t-ij/P't-i,k)-JeJ?\{k\

= E ((Vt-ij/Pt-i.k)(Mt.k-S

*

YJ (St,i-dt,k-M>t-lj/Pt-l.lc)-<l/Jjejf\\k}

jejf\\k\

JeJt

•

Lemma 3.6. Let A be a u x v matrix with rational entries. Then statement (C) of Theorem 3.1

implies statement (E).

Proof. Pick b1,b2, ...,bv in {xeQ:x > 0} so that the matrix

lbx 0 0 ...

0 b, 0 ...

A* 0 0 Z?3 ...A * = . .

, 0 0 0 ...

\ A

is weakly partition regular. Note that rank.4* = v. Then A* satisfies statement (II) of

Theorem 2.2, so the matrix D given by statement (II) satisfies the columns condition. By

Lemma 3.5, we may assume that for /e{2,3 , . . . ,m} and jeJt 0 {1,2, . . . , r } , Stj<0.

Consequently, by Lemma 2.7, we may pick a.(u + v)xm matrix B* satisfying the first entries

condition, so that for every ye~Lm there exists xeZv such that A*x = By, and such that

bi t ^ 0 whenever / e { l , 2 , ...,v} and / e { l , 2 , . . . , m ) .

Let ^cons is t of the bottom u rows of B*. Then if A*x = B*y, we also have that Ax — By.

It thus suffices to show that if ye Nrn,xeZv, and A*x = B*y\ then all of the entries of .v are

positive. To this end, let ye Nm and / e{ l , 2 , . . . , v} be given. Then the /th entry of ^*.fis &,...v,.,

while the /th entry of B*y is X!?=i^.^>V- Since each bit 0 and at least one is positive,

one then has bi.xi > 0, so xi > 0. •

We have now established the following pattern of implications:

(B)=>(C)=>(D)

To complete the proof of Theorem 3.1, we now set out to show that statement (A) of

Theorem 3.1 implies statement (B).

Definition 3.7. Let cx,c2,...,cv be in Uu and let I^ {1,2, ...,v}. The I-restricted span of

(q,e2, ...,cr) is {^'=1a,.<* :each a?.elR and if is I, then af ^ 0}.We shall need two very easy facts about linear spans, which we present below. We give

proofs for the sake of completeness.

Lemma 3.8. Let c1?c2, ...,cv be in Qu and let I ^ {1,2, ...,v}. Let S be the I-restricted span

of(cl9c2,...,ct).

(a) S is closed in M".

(b) IfyeSf)Qu, then there exist S1,S2,...,8V in Q with 8i ^ 0 whenever is I, such that

Proof.(a) We proceed by induction on |/| (for all v). If / = 0 , this is simply the assertion thatany vector subspace of Uu is closed. So we assume / =1= 0 and assume, without loss ofgenerality, that 1 el. Let 7 be the (7\{l})-restricted span of (c2,c2, ...,cr). By the inductionhypothesis, T is closed.

To see that S is closed, let EsS1 the closure of S. We show BsS. For each nsN, pick<a/(«)»L1 such that a,(«) > 0 when / e / a n d | |5-][Xia*(w)^ll < l/w-

1.that 5Then

Then

^w): ne N} is bounded) Pick S a limit point of the sequenceThen B-Sc.eT. (Given e > 0, pick « > 2/e such that la

c 51, and we are done.

1(«)>J=1, and note) - ^ < e/(2 HcJ).

Case 2. ({oc^n): nsN} is unbounded) We claim then that —c1eT. To see this, let e > 0 begiven and pick n such that oc^ri) > (1 + ||5||)/e. For ze{2, 3, ...,y}, let ^ = oi^/oi^n), andnote that for z'e/\{lK ^ ^ 0. Then

< \/(nax(n)) + \\b\\/

b/ax{n)

< e.

Since T is closed, it follows that cx e T. Thus cl and — cx are in 51, from which it followsimmediately that S is in fact the (7\{ 1 })-restricted span of (q, c2,..., cr). Thus 5 is closed byinduction.(b) Again we proceed by induction on |/|. The case / = 0 is immediate, being merely theassertion that a rational vector in the linear span of some other rational vectors is actuallyin their rational linear span (which is true because we are solving linear equations withrational coefficients).

So assume 1+0. Let X = {xs Uv: £]j'=1 xi ct = f}. Thus A'is an affine subspace of W\ andwe are told there is some xsX with xj > 0 for all is I. Also (by the case / = 0 ) , there is

some zeXwith z(eQ for all i. If z{ ^ 0 for all iel, then we are done, so suppose that zt < 0for some iel. Choose te[0,1] maximal such that the vector w = (\—t)x+tz satisfieswt ^ 0 for all iel we have wt = 0. Say 1 el and wx = 0. Then y is in the (7\{l})-restrictedspan of (c19 c3, ...,cv), and we are done, by induction. •

To prove that statement (A) implies statement (B), we shall need a special class ofcolourings, which we introduce now.

Definition 3.9. Let /?ef^l\{l}. The start base p colouring is the function crp :N^{l ,2 ,...,/?— 1} x {0,1,...,/?— 1} x {0,1} defined as follows: given_ye f\l, writey = Yjt=oatP*> w n e r eeach ate{0,\,...,p—\} and an 4= 0; if n > 0, ap(y) = (a^a^J), where « = /mod2; ifn = 0, <rp(y) = (ao,0,0).

For example, given p > 8, if x = 8320100, y = 503011, z = 834, and w = 834012(all written in base /?, of course), then crp(x) = crp(z) = (8,3,0), orp(w) = (8,3,1), and(7P(>;) = (5,0,0).

Lemma 3.10. Let A be a uxv matrix with rational entries. Then statement (A) of Theorem

3.1 implies statement (B).

Proof. As usual, let c19c2, . . . , c r denote the columns of A, and let rf^rfg, ...,du denote the

columns of the uxu identity matrix. Let B be the matrix

(tlc1 t2c2 ... trcr —dl —d2 ... — du),

where t1,t2,...,tv are as yet unspecified positive rationals. Denote the columns of B by

519 62 , . . . , bu+v. Then ^ j- ^ ^ i f ,- ^

— 5,_,. if i>v.

Given any/7ef^J\{l} and any JCGI^I, let y(^,x) = mzx{n:pn ^ .v}. Now let /?el\l\{l} begiven. We obtain m = m(/?) and an ordered partition {Dx{p), D2(p), ...,Dm(p)) of {1,2,..., u) as follows. Pick xe Nv such that ^.f = y is monochrome with respect to the start basep colouring. Now divide up {1,2,...,u} according to which of the y(s start furthest to theleft in their base p representation. That is, we get Dx(p), D2(p),...,Dm(p) so that(1) if ke{\,2, ...,ra} and iJeDk(p), then y(p, y,) = y(/',yJ), and(2) if ke{2,3, ...,m} and ieDk(p) and jeDK_x(p), then y(/?, y;) > y(/?, y,.). (We also

observe that since 0 (3 .) = crp(yf), we have y(/>,})) = (/?,>\)mod2, and hence

There are only finitely many ordered partitions of {1,2,..., u}. Therefore we may pick aninfinite subset P of 11 \{1}, me N, and an ordered partition (D^D.,, ...,Dm) of {1,2,..., u\ sothat for all peP, m(p) = m and (D^p), D2(p),...,Dm(p)) = (D1,D2,...,Dm). We shall

utilize (D19D2,...,Dm) to find a partition of {1,2,...,u + v}, as required for the columnscondition.

We proceed by induction. First we shall find E1 c {1,2, . . . , r}, specify tjeQ+={teQ:t> 0} for ieE^ let Ix = E1[) (v + DJ, and show that Yjteifi = 6. That is, we willshow that YsieEjiCi+YaieD^-di) = 0. In order to do this, we show that X/ez)^ ls m

the {1,2,...,^-restricted span of (cx,c2, ...,cr). (For then, by Lemma 3.8(6), one has

YjieDî = YjVi=iaiCi> w n e r e e a c h a ^ e Q a n d e a c h at 0 . L e t Ex — { / e { l , 2 , . . . , v } : a i > 0 } ,and for ieEx, let tt = oct.)

Let S be the {1,2,...,^-restricted span of (c1?c2, ...,cv). In order to show that YIED 4is in S, it suffices, by Lemma 3.8(a), to show that YteD 4 ls m the closure of S. To thisend, let e > 0 be given, and pickpeP withp > u/e. Pick the xeNv and yeNu that we usedto get (D^p), D2(p), ...,Dm(p)). That is Ax = y and y is monochrome with respectto the start base p colouring, and {DX,D2, ...,Dm) is the ordered partition of {1,2,..., w}induced by the starting positions of the yts. Pick y so that for all ieD^y(p,y{) = y. Pick (a,6, c)e{1,2,...,/?-1} x {0,1,...,/?-1} x {0,1} such that <rp(>>,) = (a,b,c)for all /e{l,2, . . . ,M}. Let £ = a + b/p and observe that 1 < / < / ? . For / e / ) ^>>f = a.py + b.pyl + z4./?

r"2, where 0 < z4 < /?, and henceyjpy = / + z,.//?2; let A?. = r,.//?2 andnote that 0 < A?: < 1/^. For ie {JjL2DP we have y(p,yt) < y - 2 ; let Af = yjpy and note thatOÂ^l/^.

Now, Ax = y so v u m

i = l i=l ieDl }=2 iEDj

Thus v ^

i=l ieD1 i=l

and consequently

ieD, i=l

, / / \<u/p< e.i=l

: is in S, we have that Z?ez)4 is in t n e closure of S, as required.Now let /ce{2,3, ...,m}, and assume we have chosen EÊ2, ...,Ek_l ^ {1,2, . . . , r}, and

^ e Q + for ie^Jf'iEj, and /; = £ . U (y +/)_;), as required for the columns condition. LetLk = [j^zlEj and let Mk = [jjli Dj, and enumerate Mk in order as (#(1),#(2), ...,q(s)).We claim that it suffices to show that YjieD d{ is in the {1,2, ...,i?}-restricted span of(q,c2, ...,cv,dg(1)dQ{2), ...,dq(s)), which we will again denote by S. Indeed, assume we havedone this and pick by Lemma 3.8(b), a1?a2, . . . ,ar in Q with each oct 0, and^( 1 ) ,^ ( 2 ) , . . . ,^ ( s ) in Q such that

Let Ek = {/e{l,2, ...,v}\Lk:oct > 0} and for ieEk, let ti = a,.. Let 4 = £A. U (r + /)A). Then

ieLk

k-1

where jit = —oci/ti if ieLk and /??. = if ieMk.

Image Partition Regularity of Matrices All

To see that YjieD 4 *s m ^ ^ suffices, by Lemma 3.8(a), to show it is in the closure ofS, so let e > 0 be given and pickpeP withp > u/e. Pick the xe Nv and yeNu that we usedto get (D^p), D2(p\ ..., Dm(p)). Pick y so that for all ieDk9 y(p,yf) = y. Pick(a,b,c)e{l,2,...,/?-l}x{0,1,...,/?-l}x{0,1} such that <rp(j?-) = (fl,6,c) for all/e{l,2, ...,w}. Le t / = a + b/p. For ieDk,yi = a.py + b.pr~1-\-zi.p

7~2, where 0 ^ z, </7, andhence j?://>

r = / + zt/p2; let A = z?://?

2 and note that 0 ^ A? < l//>. For ie [JjLk+1 Dp we haveyiP^yd ^ y —2; let A = yjp7 and note that 0 ^ Af < 1//?. (Of course we have no controlon the size of yjp7 for ieMk.)

Now Ax = v so

JeMfc * G ^ t j=A:+l ?e£>;-

Thus

Consequently,

< e,as required.

Having chosen IxJ^...J)ir if {\,2, ...,u + v} = \Jl"=lI}, we are done. Otherwise, let/m+1 = {l,2,...,K + i ; } \ 0 ^and we can write Yuieim+X^i a s a linear combination of {br.ie \Jf=1Ij\. •

Observe that we have in fact shown that a matrix is strongly partition regular if and onlyif it is partition regular with respect to the start base p colourings for all (or even forinfinitely many) p in N\{1}.

Let us close with an illustration of the use of the effectively computable conditions,namely conditions (II) and (III) of Theorem 2.2 and condition (B) of Theorem 3.1. Weutilize them to determine whether or not

- 1

is strongly image partition regular (and at the same time whether or not it is weakly imagepartition regular). In the process, we will be developing anecdotal evidence that the solutionmethod based on statement (II) of Theorem 2.2 (in conjunction with statement (V) ofTheorem 2.2 and statement (C) of Theorem 3.1) is the more efficient of the two methods.


To utilize statement (II) of Theorem 2.2, we let bx and b2 be as yet undetermined non-

zero rationals such that

1 - 1

3 2

4 6

bx 0

>0 b2

is weakly partition regular. Then

f3 = - 2 ^

Thus the matrix D of statement (II) of Theorem 2.2 is/ - 2 2 - 1 0 0\I 2bJ5 bJ5 0 - 1 0\-3b2/5 bJ5 0 0 - 1 / .

We consider possibilities for I1. Clearly /2 $ {3,4,5}. Thus, looking at row 1 we see we musthave Ix as some one of {1,2}, {1,2,4}, {1,2,5}, or {1,2,4,5}. Any of the first three alternativesrequires either that bx = 0 or b2 = 0. Consequently, one has Ix must be {1,2,4,5}. This canhappen if and only if b1 = 5/3 and b2 = —5/2. Thus, in one stroke we see that

is weakly but not strongly image partition regular.For comparison, let us examine the same matrix using statements (III) of Theorem 2.2

and (B) of Theorem 3.1 instead. To this end we let tx and /., be as yet undetermined non-zero rationals such that

/ tx -t2 - 1 0 0 \I 3fx 2t2 0 - 1 0 j\4 / 1 6t2 0 0 - 1 /

is kernel partition regular. As before, one quickly sees {1,2} ^ Iv Further, if 3 ^/1? one hastx = t2, so 3tl + 2t2 = 5/x and 4t1-\-6t2 = 10^-conditions that are incompatible with anypossibilities for Iv This leaves the possibilities for /x as {1,2,3}, {1,2,3,4}, {1,2,3,5}, or{1,2,3,4,5}. Each choice leads to an inconsistent set of equations except / = {1,2,3,4},which forces tx = 3/5 and t2 = —2/5. Thus again we see, with somewhat more effort, thatthe system is weakly but not strongly image partition regular.

Acknowledgements

The authors would like to thank Walter Deuber, Ronald Graham, and Hanno Lefmannfor some helpful correspondence.


References

[1] Bergelson, V., Deuber, W. and Hindman, N. (1990) Nonmetrizable topological dynamics andRamsey Theory. Trans. Amer. Math. Soc. 320 293-320.

[2] Deuber, W. (1973) Partitionen und lineare Gleichungssysteme. Math. Zeit. 133 109-123.[3] Furstenberg, H. (1981) Recurrence in ergodic theory and combinatorial number theory, Princeton

Univ. Press, Princeton.[4] Hilbert, D. (1982) Uber die Irreducibilitat ganzer Rationaler Funktionen mit ganzzahligen

Koeffizienten. J. Reine Angew Math. 110 104-129.[5] Hindman, N. and Woan, W. (to appear) Central sets in semigroups and partition regularity of

systems of linear equations. Mathematika.[6] Rado, R. (1933) Studien zur Kombinatorik. Math. Zeit. 36 424-480.[7] Schur, I. (1916) Uber die Kongruenz xm+ym = zm (modp). Jahresbericht der Deutschen

Math.- Verein. 25 114-117.[8] van der Waerden, B. (1927) Beweis einer Baudetschen Vermutung. Nieuw Arch. Wisk. 15

212-216.

Extremal Graph Problems for Graphs with aColor-Critical Vertex

CHRISTOPH HUNDACK, HANS JURGEN PROMELand ANGELIKA STEGER

Institut fur Diskrete Mathematik, Universitat Bonn, Nassestr. 2, 53113 Bonn, Germany

In this paper we consider the following problem, given a graph //, what is the structureof a typical, i.e. random, //-free graph? We completely solve this problem for all graphsH containing a critical vertex. While this result subsumes a sequence of known results, itsshort proof is self contained.

1. Introduction

What does a typical triangle-free graph look like? This question was answered by Erdos,Kleitman and Rothschild [3] proving that almost every triangle-free graph is bipartite,i.e., is two-colorable.

From the point of view of extremal graph theory, this result resembles an old result ofMantel [6] stating that the complete bipartite graph is the extremal, i.e., edge-maximum,triangle-free graph. Mantel's solution was a kind of forerunner of extremal graph theory.Its starting point is usually considered to be Turan's celebrated generalization [14] ofMantel's result, characterizing the extremal graphs T/(w) on n vertices which do notcontain a complete graph K\+\ on / +1 vertices as a subgraph. Turan's result stimulated avariety of deep results in graph theory, the reader is referred to [1] and [12], two excellentsources on these problems. For our purposes we will just mention two strengthenings ofTuran's theorem.

Let H be a graph of chromatic number / + 1. By ^orbn(H) we denote the class of allgraphs on n vertices that do not contain H as a weak subgraph, i.e., the class of all H-freegraphs. Basic problems in extremal graph theory are the following: if a graph H is given,what is the maximum number of edges a graph in ^Forbn(H) can have, and, provided G issuch an extremal graph, what can be said about the structure of G? If H — K\+\, Turan'stheorem gives the answer to both questions.

Research supported by DFG-project Pr 296/2-1.

422 C. Hundack, H.J. Promel and A. Steger

An edge e of H is called critical if the deletion of e from H results in a graphwith chromatic number /. H is called edge-critical if H contains a critical edge. Thefirst strengthening of Turan's theorem we are going to mention is Simonovits's inverseextremal theorem [11] which gives a complete characterization of those graphs H sharingthe property that Tn(l) is the extremal graph in 3Forbn(H).

Theorem 1.1. [11] Let I > 1 and let H be a graph of chromatic number I + 1. Then theTurdn graph Tn(l) is an extremal graph in ^Forbn(H) if and only if H is edge-critical.

If Gi,.. . ,G/ is a family of graphs, we denote by X{=1G,- the product of the graphs G,,that is, the graph obtained by taking the disjoint union of G\,... , G/ and joining every twovertices belonging to different G/'s. Let K/+i(ro,... ,/*/) denote the complete (/ + l)-partitegraph with r, vertices in the /th class. In particular, K/+i = K/+i(l,... ,1). A vertex v ofH is called critical if the deletion of v from H results in a graph with chromatic numberless than H. If H contains a critical vertex, H is called vertex-critical. Obviously, if Hcontains a critical edge, it is vertex-critical. Moreover, observe that each K/+i(l,ri,... , r/)is vertex-critical.

The second extension of Turan's theorem we mention, also due to Simonovits [10], andlater generalized by Erdos and Simonovits [4] gives a characterization of the extremalgraphs in classes of graphs that are defined by forbidding certain vertex-critical graphs,viz. K/ + i( l , r i , . . . ,r/)-graphs.

Theorem 1.2. [10] Let I > 1,1 < r\ < ... < r/ and n be sufficiently large. If G is an extremali

graph in 3Forbn(Ki+\(\,r\,... , r/)), then G admits a representation as X G/» wherei=\

( 1 ) \V(Gi)\ = « / / + o ( n ) , for each i = 1 , . . . ,/, and( 2 ) Gi is an extremal graph in ^ o r b n ( K 2 ( \ , r \ ) ) for each i — 1 , . . . ,/.

The general random graph question we are considering in this paper is: given a graphH, what is the number of graphs in ^orbn(H) and, moreover, what does a typical /f-freegraph look like? In answer to a question of Paul Erdos, a counterpart to Simonovits's[11] inverse extremal theorem was proved in [8]. For convenience, we denote by #„(/) theclass of /-colorable graphs on n vertices.

Theorem 1.3. [8] Let I > 1 and let H be a graph of chromatic number / + 1. Then

\&orbn(H)\ = ( l+o(l)H#n(/) |

if and only if H is edge-critical.

In particular, choosing H = /C3, gives as a corollary the result of Erdos, Kleitman andRothschild that almost every triangle-free graph is two-colorable.

Let H be a vertex-critical graph and let v be a critical vertex in H. We say that vhas criticallity d, if there exist d edges e\,...,et\ incident to v such that H \ {<?i,...,<?j}

Extremal Graph Problems for Graphs with a Color-Critical Vertex 423

has chromatic number /, and d is minimal with respect to this property. Furthermore wedenote by 0>n{l,d) the class of graphs on n vertices that are subgraphs of some productof / graphs, each of which has maximal degree at most d, i.e.,

{ 1@>n(l,d) : = < G | G • £ X G,-, w h e r e ^ | K ( G , ) | = n , A(G, ) <d for a l l i = 1 , . . . , / \ .( /=. ,=, J

The main result of this paper is the following theorem.

Theorem 1.4. If H has chromatic number I + 1 and contains a color-critical vertex VQ withcriticallity d, then

\<Forbn(H)\ = (14- 0(2-™)) • \&orbn(H) n &n&d - 1)|,

/or som^ constant c > 0.

Note that this result on random graphs resembles Theorem 1.2 on extremal graphs in asimiliar way to that in which the Erdos, Kleitman, Rothschild theorem resembles Mantel'sresult. For general vertex-critical graphs //, different from a K/+i(l,ri,...,r/), Theorem1.4, together with some additional counting, also implies results on the structure of theextremal graph in ^Forbn(H) that are slightly stronger than the ones following from theAsymptotic Structure Theorem of Erdos and Simonovits (cf [1]).

2. The Proof

We prove Theorem 1.4 by partitioning the class ôrbn(H) into a finite number ofsubclasses and showing that all but one of these subclasses are asymptotically negligible.This then proves that almost all members 3?orbn(H) have the properties of this oneremaining class. In addition, the rate of convergence can be expressed in terms of twofunctions that bound the growth rate of the negligible classes. For a proof of the followingtheorem, compare [13] or [7], where proofs of similar theorems can be found. For thesake of completeness, we also include a sketch of the proof.

Theorem 2.1. [7], [13] Let k be a positive integer, let S(n) ^ ^F{n) and î(n), i = l , . . . , /c,

be k + 2 families of graphs on n vertices, and let <x,p>Obe reals such that for all sufficiently

large n we have

(1) l o g . J f ( ^ ' . < - x - n + P ^ and\S(n 4- 1)|

A:

(2) &(n) c S(n)u{jî(n).;=1

Moreover, let z, = z,(n) be positive integral functions such that z, = o(n), and let e{ > 0 be

constants such that for all sufficiently large n we have

(3) log ' ^ ' W ' < ( a -e / ) - z , -n for every i = ! , . . . , * .

424 C. Hundack, HJ. Promel and A. Steger

Then there exist c > 1 and y > 1 such that the following inequality holds for every n e IN:

Proof (Sketch). Let e = mini</<&£•/ and y = 26^4, and choose no sufficiently large suchthat (1) - (3) of the Theorem and all inequalities below are fulfilled for all n > no. Finally,choose c > 1 sufficiently large such that the claim of the Theorem holds for all n < no.We proceed by induction on n. Using (2), we conclude that it suffices to show that

Using (3), the induction hypothesis, and (1), we obtain

-Zi)\ fr \S(n-j)\\S(n)\ - \&(n-Zi)\ \£{n-Zi)\ v}x\S(r

< 2 ( a - f ) - n - ( l + c y - ( w - ' ' ) ) - 2 ^ ' ( " a " (

From this the claim follows by straightforward calculations. •In the application of Theorem 2.1 we will have S(n) = ôrbn(H) C\P?n{Ud — 1) and

!F{ri) = ôrbn(H). We first determine the constants a and /? such that (1) of Theorem 2.1is satisfied.

Lemma 2.1. There exists a constant D > 0 such that for all sufficiently large n:

\ôrbn{H)C\2Pn{lJ-X)\ / - I

Proof. Let S(n) = ôrbn(H) n0>n(l,d - 1), let <£oln(l) denote the set of all /-colorablegraphs on n vertices, and let !Fn(d) denote the set of all graphs of order n with maximaldegree at most d. Then, obviously,

It is well-known, cf e.g. [9], that the number of /-colorable graphs satisfies

l«ww(/)l = e

Furthermore, we easily obtain the following crude bound for 'Tn{d):

\3Tn(d)\ <2dtllog".

We need some more notation. Let I be a property that an /-colorable graph may have.(For example, the existence of a color-critical vertex is such a property.) Let

<gn(£) = {G e <£oln(l) | G has property 1}

and

£n($) = {G e ôrbn(H) | G = G{ U G2,G{ e ,Tn(d - 1),G2 e %oln(l),G2 has property i ) .

Extremal Graph Problems for Graphs with a Color-Critical Vertex

Obviously, for all such properties 1:

That is, if

425

01, (1)

then \£nm=o{\£(n)\).We will consider the following two properties:

J i = there exists a coloring with color classes P i , . . . , P / such that mini</</ |P,| < ^, andJ 2 = for all colorings with color classes P i , . . . , P / such that \P\\ = mini<(</ |P,-| > ^

there exist sets A\j, 1 < i < y/n, 2 < j < I such that— Atj^Pj, \Au\ = \V(H)l— Aij D Ahj = 0 for all i\ =fc i2, and

{veP\\ U2<,</ Ay c n

One easily checks thatgraphs contained in #W

< \V(H)\ for all 1 < i < ^ .

as well as 12 satisfy inequality (1). (Overestimate the number of,-) by first choosing appropriate color classes (less than /" ways),y

the sets Ajj (less than (\V"H)\) ways), and then choosing the edges between the colorclasses.) More precisely,

dnlo&n) ' \Voln(l)\,

and (here we let h = \V(H)\)

. /2(/-n/i _ niPiiv/^

The cardinality of S(n + 1) can now be bounded as follows. Fix an arbitrary vertexo £ Vn+\> We claim that there are at least 2Ti("~/|K(//)l^") possibilities to connect v0 to

any graph G e $n(M\ A J2) induced on Vn+\ \ vo. Indeed, by the definition of 1\ and J2

every graph G e Sn(I\ A 12) may be written as G = G\ U G2 with G2 e ^ n (J i A J2). Let|Pi| < • • • < |P/| be the color classes of an appropriate coloring of G. Then |Pi| > n/2l,and for / = 2, . . . , / there exist sets St ^ P, of size |5,-| = y/n • \V{H)\ such that any tuple(X2,...,Xi) with Xt c p, \ s / and | ^ | = \V(H)\ has the property that U2<i<ixi h a s a t

least \V(H)\ many common neighbors in Pi. In particular, this implies that vo may beconnected to UL2W \ ") m a n arbitrary way without generating a copy of the forbiddensubgraph H. Therefore,

<f(n+ 1) > \Sn(lx A i 2 )

> (\£{n)\ - \Sn

> (1 -

from which the desired inequality follows easily. •For the rest of this section, fix integers / > 2 and 1 < d < k and let H be an arbitrary

but fixed (/ + l)-partite graph containing a color-critical vertex of criticality d. The mainpart of the proof is devoted to defining appropriate sets J^,(n).

We proceed with some definitions. Throughout the rest of this section we fix ei :=1/(23J+8 • /3) and an e0 > 0 such that

e0)) eh

Observe that this implies n2 • (f"J < 2^€in for n sufficiently large. All logarithms inthis paper are to base 2. The kth iteration of logn is denoted by log{k) n, i.e. \og(k) n =log(log(A"u n) and log(1) n = logn.

For 0 < k < /, let fk denote the integral function

= \log{k2-k+2)n].

A pk-set is a subset of the vertex set Vn of size kfk(n) with a partition into k subsetsPi = {va-\)fk(n)+\,... ,Vifk(n)}, i = 1,... ,/c, of size fk(n) such that the following conditionsare satisfied:

(1) Pi DPj = 0 for every \ < i < j < k,(2) IP,-1 = fk(n) for every \ <i <k and(3) every pair of sets P, and Pj is completely connected, that is {x, y] G E

for all x e P{ and y e Pj, and all 1 < / < j < k.

For a pfc-set P = U t i p» w e denote by Tfc(P) = Tk{Pu...,Pk) the set

\T(v) n P/| > eo\Pi\ for every / = 1,... , ki=\

Observe that, by definition, a po-set is empty and To(0) = Vn. For simplicity, a p/_i -set Qis also called a q-set. The corresponding set T/_i(Q) is also denoted by R(Q) and calledthe r-set of Q. In addition, we let q denote the size of a g-set, i.e. q = (I — l)//_j. A g-set

i; is a g-set contained in the neighborhood T(v) of i\

Lemma 2.2. Lef 1 < k < I — 1 be an integer and let <o/k(n) denote the set of all graphsG e 3?orbn{H) that contain a pk-set P such that the set

\Tk{P)\<[ LJL-e,)n.

Then

/or all sufficiently large n.

Proof. Construct all graphs in ,tfk(n) as follows. First choose the pk-sct P = \Jk=l Pi and

an //-free graph on Vn \ P (at most nkfk • \^orbn-kfk{H)\ ways). Then choose edges insideP (less than 2~-(kfk)~ ways), choose the set Tk(P) (less than 2" ways) and connect P to

Tk(P) (at most 2kfklTk{P)l ways). Finally, connect P to Vn \ (P U Tk(P)). Do this by firstchoosing for each v E Vn \ (P U Tk(P)) an index /, 1 < i < fe, so that v has less than eofkneighbors in P,. In this way the total number of ways to connect P to Vn \ (P U Tk(P))can be bounded, for sufficiently large n, by

n-|7|t(P)|

Together this gives

log f m i < fe/felogn+\^orbn-kfk{h)\ I

(k-l)fkY(n-\Tk(P)\)

J -fkn<-

for n sufficiently large. •

Lemma 2.3. Let 0 < k ( ] - *i) n and \Y(v) n Tk(P)\ < fk(n).

Then

\&k(n)\

for all sufficiently large n.

Proof. Construct all graphs in &k(ri) as follows. First choose the vertex v and an /f-freegraph on Vn \ v (at most n - \^Forbn-\(H)\ ways). Then choose the pk-set P (less than nk^k

ways) — observe that this implicitly defines the set Tk(P). Next connect v with Tk(P) (lessthan X£o (/) ^ 2fkl°g" w ays) a n d w i t h ^ \ (P U Tk(P)) (less than 2n- |T^P) l ways).

Together this gives

/ - I 1—"a)"

for n sufficiently large. D

Lemma 2.4. For n sufficiently large, every graph G = (Vn,E) in ^Forbn(H)\{{JkL\^k(n)U U/c=o^(n)) has the following property: the neighbourhood T(v) of everyvertex v G Vn contains a q-set Q.

The proof of this lemma is analogous to the proof of a similar result in [5]. In particularwe need the following lemma from [5], which generalizes a result of [2].

Lemma 2.5. [5] Let 0 < e < 1 and h G IN be given. Then there exists NQ = No(e,k) suchthat the following is true. Let G be a graph with vertex set AQ U • • • U Ak, where the A-x arepairwise disjoint sets each of size N > No, and suppose that \F(v) C\Ai\ > eN for all v G Aoand all 1 < i < k. Then there exist sets A\ c Ait 0 < i < k, such that \Af

t\ = [log12^1 N] foreach i = 0,...,kand such that ( J t i A\ - r(v) for al1 V e Ao- D

Proof of Lemma 2.4. Let v be an arbitrary vertex of G. By definition of the set Jwe observe that V(v) contains a set P\\ of size f\(n). We conclude the proof by inductionon k. Suppose for some /c, 1 < k < I — 2, there exist sets P^i,..., Pkk satisfying the followingtwo conditions:(i) the Pki, 1 fk(n).

Let P be an arbitrary subset of F(v) Pi Tk(Pk\ U • • • U Pkk) of size fk(n). Apply Lemma 2.5to Ao = P and Ax = Pki, 1 < / < k to obtain sets Pk+\,\,...Pk+\,k+\ satisfying conditions(i) and (ii) for fc+ 1. Inductively, we thus obtain sets P/_u,...,P/_i,/_i that form a q-seicontained in T(v). •

Observe also that another immediate consequence of Lemma 2.5 is that a vertex maynot have many neighbours in the r-set of a g-set contained in its neighbourhood.

Corollary 2.1. For all sufficiently large n, every graph G = (Vn,F) in orbn(H) has thefollowing property: if v G Vn is a vertex with q-set Q,

\r(v)nR(Q)\<q. •

Lemma 2.6. Let (€{n) denote the set of all graphs G G J^orbn(H) that contain a vertex vwith q-set Q such that

\R(Q)\ > I 7 +

Then\<#(n)\ < (Ij2± X


Proof. Construct all graphs in (€(n) as follows. First choose the vertex v and an //-freegraph on Vn \ v (at most n • \^Forbn-\{H)\ ways). Then choose the g-set Q in Vn \ v (less

than nq ways) and connect v to Vn\ R(Q) (less than 2"~^(e)l ways). Finally, connect v toR(Q). Observe that by Corollary 2.1 there are at most

ways to do this.Together this gives

log

/=0

< logn + qlogn + n — \R(Q)\ + qlogn

1-1 11 21

for n sufficiently large. •Corollary 2.2. For n sufficiently large, every graph G = (Vn,E) in ôrbn(H) \

has the following property: if v is a vertex with q-set Q,

I _ e\ n < |R(<2)| < î +

D

Lemma 2.7. Let 0 < k < I — 2 and m G {2,rf + l} be integers, and let ^^,m(n) denote the set

of all graphs G G JForbn(H) that contain distinct vertices v\,...,vm with q-sets Q\,.,Qm*

respectively, and a p^-set P contained in f)™=i F(i;;) such that the following properties are

satisfied simultaneously:

\Tk(P)\> ( y - f]T(Vi)nTk(P)i=\

< fk(n),

> n,

and

Then

j-e,jn<\R(Qi)\<(j+e,)n for i=\,...,m.


Proof. Similarly, as in Lemma 2.3, we construct all graphs in @k,m(n) as follows. Wefirst choose the vertices v\,...,vm, an //-free graph on Vn \ {v\,...,vm}, choose the ^-setsg i , . . . , Qm, and the p^-set P (less than nm • \ôrbn-m(H)\ - nmq • nkik ways). Then we choosethe edges among the v{ (less than 2m~ ways), connect v\ to R\ := R(Q\) and the viy i > 2,to R := H^2 ^(6/) (less t h a n 2nvqlogn ways, cf. Corollary 2.1), connect vh i > 2, io R{ \ R

430 C. Hundack, HJ. Promel and A. Steger

and v\ to R \ R\ (less than 2(m W*1' \RnRi\) • 2'^' \RnRi\ ways), and connect t?,-, i > 1, toVn$RuRiU Tk(P)) (less than 2m^-lr^p^^u/?^l-l^l-lRil+^n/?il) ways). Finally, we connectthe i?,-, i > 1, to Tk(P) \(RUR$. This can be done by first choosing the at most fk{n) manyvertices of Tk(P) \ (R U R\) connected to all the vi and then connecting the remainingvertices to at most m — 1 of the i?,-. That is, there are at most

-2-'"-1)|TJt(P)$RUR1;§0)'(2""ways to connect the vt to ^ ( P ) \ (R U R$.

Together this gives

l o g i^r ^ (mi ^ m\ogn + mq logn + fe/fc logn + m2 + mq logn + (m - \)\R{\\r oroti)\

+ \R\+m-{n- \Tk(P) $RU Ri)\ - \R\ - \R{ |)

+ / , logn + (m - ^j—\ \Tk(P) \ (R U Rx)\

/ 1 / O3J+6 i i

m n e / ^ 2 e / e /

for n sufficiently large. •The following corollaries can be proved in exactly the same way as Lemma 2.4 if one

uses the ^m(rc)-sets instead of the

Corollary 2.3. For n sufficiently large, every graph G = {Vn,E) in ^Forbn(H)\( U ^ i ^k(n)U Lfco^/cM U#(n) U(Jfc=o^u(w)) ^ s the following property: for every twovertices vuv2 6 Kn r/iat /za^ f-s ts Qi and Q2, such that \R(Q$ n R(Q2)\ > 2M+b€\n, thereexists a q-set Q contained in T(v\) n F(i?2).

Corollary 2.4. For n sufficiently large, every graph G = (Vn,E) in ^Forbn(H)\

(Ut 1 ! */k(n) U I j t o &k{n) U ^(w) U UL=o M+i (")) ^«5 r/l^ following property: for every

d + \ vertices v\,...9Vd+\ that have q-sets Q\,..,Qd+\, respectively, such that \ f]^1 R{Qi)\ >\, e(]- — 2d+4ei)n, there exists a q-set Q contained in C\i=

Lemma 2.8. Let ${n) denote the set of all graphs G G orbn(H) that contain a vertex vwith two q-sets Q\ and Q2 such that the following two properties are satisfied simultaneously:

\R(Q\)\ > ( \ ~ *i ) n and \R(Q2) \ R(Q{)\ >

Then

Proof. Construct all graphs in $(ri) as follows. First choose the vertex v and an H-freegraph on Vn\v (at most n • \^Forbn-$H)\ ways). Then choose the g-sets Q\ and Q2 (lessthan n2* ways) and connect v with Vn \ (R(Q{) u R(Q2)) (less than 2"-|/?(<2I)HK(<22)\K(<2I)I

ways). Finally connect v to R(Q$UR(Q2). Observe that by Corollary 2.1 there are at most

Together this gives

log l ^ ( " ) ' m , <

-2e,\n

for n sufficiently large. •

Corollary 2.5. For n sufficiently large, every graph G = (Vn,E) in ôrbn(H)\

(Ut 1 ! sfk(n) U U t o ^k(n) U <g(n) U IJô ®wW U <f (n)) /z«5 the following property: ifvu v2

are two vertices with q-sets Q\ and Q2, resp., then the corresponding r-sets either 'coincide'

or are 'disjoint', that is, either

\R(Qi) n ^(62)1 > (] ~ 9ei\ n or \R(Q{) n R(Q2)\ <

Proof. Let v\ and v2 be two vertices with f-sets Q\ and Qi, respectively. Assume that\R(Q\) n R(Q2)\ > 23d+6ein. According to Corollary 2.3, there exists a g-set Qi2, which isa f-set of v\ as well as of v2. As G is not in S(n), this implies that

\ ^(Qi2)l < 4e,n and \R(Ql2) \ R(Q2)\ <

Therefore,

> \R(Q{)\ - |/J(QO \ R(Ql2)\ - \R(Ql2) \ R(Q2))\

. •

Corollary 2.6. For n sufficiently large,

(/-1 i-i 1-2

(J s/k(n) U |J «fc(n) U V(n) U | J (û n(l,d- 1).

Proof. Let G = (Vn,£) be a graph contained in the set on the left-hand side and assumen is large enough for all of the above lemmas and corollaries hold. We now construct apartition Vn = IJ,. î a n d show that it has the desired properties.

By Lemma 2.4, the neighbourhood of every vertex v e Vn contains at least one g-set.For every vertex v, fix one such g-set, let us call it Qv, and denote the corresponding r-set

by Rv = R(QV)> Choose a maximum number of vertices v\,...,vs G Vn such that

\RVi n RVj\ < 23d+6ein for all pairs \ ~ \ .

Claim 1. The sets X( partition the set Vn.

Choose x G Vn \ {v\,...,vs} arbitrarily. By the maximality of the set {t;i,...,i?s} there

has to exist at least one Vi such that \RX n RVi\ > 2M+6e\n. By Corollary 2.5 this implies

\RX nRVj\ > \y, that is x e X(. Assume now there also exists an 1 < j < s, j ^ i, such that

xeXj. Then

^ y < \RX n RVi\ + l^x n Kr, n

which is a contradiction.

Claim 2. G[X,] has maximal degree at most d — 1, for all f = l , . . . , s .

Assume there exist 1 < io < s and vertices x o , x i , . . . , x j G ^/0 such that xo is connected to

all vertices x i , . . . , x j . Then the definition of X^ and Corollaries 2.2 and 2.5 imply

\RXi n Rx.| > ( y - 9f/ j n for all 0 < i < j < d.

Therefore

|^.X/ \ RXj\ < lOf/n for all 0 < i < j < d

and

i=0

This means that by Corollary 2.4 we know that there exists a g-set Q = (J / = 1 Q, con-

tained in p|f=o T(x/). Recall now that the definition of the r-set together with Lemma

2.5 implies that there exist sets Hi c R(Q) and Hi+\ c Q/9 / = 1 , . . . , / — 1 so that

{xo},//i U {xi, . . . ,Xf/},/ /2, . . . ,H/ form the parts of the forbidden subgraph H, which is a

contradiction.

Claim 3. s = I.

By Claim 2, at most d — 1 vertices of QV] can belong to Xj. Similarly, we observe that

the / — 1 parts of QVl must belong to different sets X{. This shows s > I. As, on the other

hand,

n >i=\

n -

we immediately observe that also s < /, that is s = I.

References

[1] Bollobas, B. (1978) Extremal Graph Theory, Academic Press, New York, London.

•


[2] Bollobas, B. and Erdos, P. (1973) On the structure of edge graphs. Bull. London Math. Soc. 5317-321.

[3] Erdos, P., Kleitman, D.J. and Rothschild, B.L. (1976) Asymptotic enumeration of K,,-freegraphs. In: International Colloquium on Combinatorial Theory. Atti dei Convegni Lincei 17(2) Rome 19-27.

[4] Erdos, P. and Simonovits, M. (1971) An extremal graph problem. Acta Mathematica AcademiaeScientiarum Hungaricae 22 275-282.

[5] Kolaitis, Ph. G., Promel, H.J. and Rothschild, B.L. (1987) X/+1-free graphs: asymptotic structureand a 0 - 1 law. Trans. Amer. Math. Soc. 303 637-671.

[6] Mantel, W. (1907) Problem 28, soln. by H. Gouwentak, W. Mantel, J. Teixeira de Mattes, F.Schuh and W.A. Wythoff. Wiskundige Opgaven 10 60-61.

[7] Promel, H.J. and Steger, A. (1992) Coloring clique-free graphs in linear expected time. RandomStructures and Algorithms 3 375-402.

[8] Promel, H.J. and Steger, A. (1992) The asymptotic number of graphs not containing a fixedcolor-critical subgraph. Combinatorica 12 463-473.

[9] Promel, H.J. and Steger, A. (1993) Random l-colorable graphs, Forschungsinstitut fiir DiskreteMathematik, Universitat Bonn.

[10] Simonovits M. (1966) A method for solving extremal problems in graph theory, stabilityproblems. In: Theory of Graphs, Proc. Coll. held at Tihany, Hungary.

[11] Simonovits, M. (1974) Extremal graph problems with symmetrical extremal graphs. Additionalchromatic conditions. Discrete Math. 7 349-376.

[12] Simonovits, M. (1983) Extremal graph theory. In: Beineke, L.W. and Wilson R.J. (eds) SelectedTopics in Graph Theory 2, Academic Press, London, 161-200.

[13] Steger, A. (1990) Die Kleitman-Rothschild Methode, Dissertation Universitat Bonn.[14] Turan, P. (1941) Egy grafelmeleti szelsoertekfeladatrol. Mat. Fiz. Lapok 48 436-452.

A Note on coi —• co\ Functions

PETER KOMJATHf

Dept. Comp. Sci. Eotvos University, Budapest, Muzeum krt 6-8, 1088, Hungarye-mail: [email protected]

A well known and widely investigated statement in model theory is Chang's conjecture.It asserts that whenever M is some structure on a>2 of countable length then there isan elementary substructure N of cardinal Ki such that N D co\ is a countable ordinal.See [1]. This statement fails under the axiom of constructibility, e.g., a Kurepa-tree givesa counter-example. Namely, Chang's conjecture implies that there are no K2 functionsu>\ —> to such that any two differ on a co-countable set. It is even true that there are noX2 functions co\ —> co such that any two differ on a closed unbounded set as the followingargument shows. Assume that {/a : a < co2} is such a family. Let the model M contain thefollowing functions F and G. F(a, £) = /a(£), and for a, ft < C02, a jS, £ < co\ G(a,/?, £) isthe <f th element of a closed unbounded set on which / a , fp differ. If N is an uncountableelementary substructure with S = N n co\ < co\ then for a =/= /3 in N / a , fp differ on theclosed unbounded subset {G(a,/J,£) : £ < 3} of (5 so necessarily fu(S) i= fp(d), that is, theKi functions {/a : a G N} get different values at S which is impossible.

The consistency of Chang's conjecture was first established (from the consistency of theexistence of an coi-Erdos cardinal) by J. Silver (unpublished, but see e.g., in [2]). See alsoin [5], pp. 395-400.

In the seventies the investigation of the generalized continuum hypothesis led to theresearch of co\ —> a>\ functions under eventual and club domination (see [3]). The "first"co2 of them can easily be constructed, in fact they are quite determined (see the Statement).If the axiom of constructibility holds there is a function that eventually dominates all ofthem and our above argument shows that Chang's conjecture implies that no co\ —• co\function can dominate all of them on a closed unbounded set. Here we show that it isconsistent that there is a function dominating in the weak sense, but there is no one whichdominates in the strong sense. I suspect that this result is known but have not been ableto trace an explicit reference.

t Research partially supported by Hungarian National Science Research Fund OTKA No. 1908

436 P. Komjdth

Definition 1. If f,g : co\ -> co\ then f <* g, / < g denote that {£ : /(£) <res/?. {£, : /(£) ^ g(£)} is co-countable, f < + g, / < + g denote that these hold for a closedunbounded set.

Assume that for every limit a < co2 a cofinal sequence {aT /" a : T < cf(a)} is fixed.

Definition 2. For a < C02, /ia : coi —• coi is the following function: /zo(£) = 0. /ia+i(O =/za(£) + 1. If cf(a) = co, and ocn / a, f/zerc /za(£) = sup{/zan(£) : n < co}. If cf(a) = coi andaT >* a, put /za(O = sup{/iar(£) : T < £}.

Notice that these functions {/ia : a < C02} depend on the particular choice of theconvergent sequences aT /* a : T < cf(a).

Statement

(a) For OL < ft < a>2, ha <* hp holds.(b) If {/a : a < C02} is a <+-increasing sequence of functions, then ha ^ + / a holds for

every a < (02.

Proof, (a) By induction on /?. (b) By induction on a. The only non-trivial case is whencf(a) = CDI. Assume that a = sup{aT : T < cwi}. hat, / a t < / a on some club CT. On a clubset of £ < c»i, /za(O = sup{^aT(O : T < <J}, and also £ e CT holds (T < ^). But then,

KM) ^ Ud) ^ fa(O (T < 0, SO M£) < /a(^). •

Theorem, /r is consistent relative to the existence of an coi-Erdos cardinal that there isa function f : co\ —> a>\ such that ha <+ f for every a < a>2, but there is no functiong : CD\ —• a>\ with ha <* g for every a < coi-

We in fact prove that if CH holds and {ha : oc < C02} is an arbitrary family of a>\ —• a>\functions that is not <* bounded by an co\ —• CD\ function then there is a countably closed,K2-C.C. notion of forcing which adds a function dominating each ha on a closed unboundedset but no such function in the extension can dominate each ha on a co-countable set.

The result is connected to the paper of Kanai [4] where it is claimed that if there is a<+-chain of a>\ —> co\ functions of length C02 + 1 then there is even a <*-chain of lengtha)2 + l. The proof in [4], however, seems not to be complete ( ^ in the proof of Claim 2really depends on fi as well).

Proof. Let V be a model of GCH in which the second statement of the Theorem holds,e.g., if Chang's Conjecture is true.

Let P be the following poset. p = (a , / ,5,C) e P, if a < co\, f : a -> co\, S e [co2]*\ C isa function on S, C(£) is a closed subset of a for { G S , and, if 0 e C(£), then f(0) > h*{p).(ar, / ' , S', C) ^ (a,/, S, C) iff ar ^ a, / ' 3 / , S' 3 S, and for ]8 G 5, Cr(j?) n a = C(jB) holds.

Claim 1. (i5,^) is oj\-closed.

A Note on co\ —• co\ Functions 437

Proof. If po ^ pi ^ • • •, pn = ((xn,fn,Sn,Cn) then p = (a,/ ,S,C) extends all pn wherea = sup{an : n < co}, / = |J{/« : n < ^ l ' ^ = U{^« '. n < co] and, if £ G Sn, then

i > n}. D

Claim 2. Ify<<x>\, then the set Dy = {(a,/,S,C) : a ^ 7} is dense m P.

Proof. By induction on 7. Using Claim 1, it suffices to show that p = (a , / ,S, C) has anextension of the form (a + 1,/',S, C"). Take /'(a) bigger than every h^(cc) (£ G S), and

{a}. D

Claim 3. If £ < co2 then {(a,/,S,C) G P : ^ G 5} is dense in (P,<).

Proof. If (a,/ ,S,C) e P, £ (£ S then (a,/ ,S U {^},C) extends it where C'(£) = 0 andC(C) = C(C) for C G S. •

Claim 4. /n (P, ^ ) among any X2 conditions some K2 «re pairwise compatible, so (P,<)/zas t/ie a>2-chain condition.

Proof. By CH and the A-system lemma, among K2 conditions there are K2 such thatany two are of the form (a,/ ,S U S',Cr) and (a,/ ,S U 5r/,C/r) with C'\S = C"\S. Then(a, f,SuS'U S", C U C") is a common extension. •

Claim 5. IfG^P is generic, £ < co2, then the set E^ = \J{C(£) : (a,/ ,S,C) G G,^ G S}is a closed unbounded subset of a>\.

Proof. If p = (a , / ,S, C) forces that some T < coi is a limit point of E% and a > T then bythe definition of P, T G C(^) = £^ n a and so E^ is closed. That £^ is unbounded can beproved by an argument as in Claim 2. •

Claim 6. IfG^P is generic, then F = | J { / : (a, / ,S,C) G G} is an coi —• co\ functionsuch that h^ <+ F for every a < co2.

Proof. By Claims 3 and 5. •

Claim 7. In V[G], there is no g : coi —• a>\ function such that h^ <* g for every (£ < 02).

Proof. Assume that 1 \\—g : co\ —> o\ is such a function. For every I; < C02 there is aPt £ P which determines the point from which g(a) > /i^(a) holds. For a set X c co2,|X| = X2, this point is the same, call it /?. We may as well assume that p^ = (a,/,SuS<*, Q )where the sets {5,5^ : ( G l } are disjoint and Q | 5 = C are the same.

Assume first that for every 7 > ft, there are gr(7) < co\, Z(y) G [X]Kl such that if^ G X — Z(y)9 then ^(7) < gr(7). Then the function g' : <x>\ -> coi dominates every /z(^ G X — |J{Z(7) : 7 < coi}) on a co-bounded subset, so, it dominates every h% (£, < 02), acontradiction to the assumption on V. There is, therefore, a 7 > /? such that no such g'(7)

438 P. Komjdth

exists. Select p' = (OL'J\S\C) ^ (a,/ ,S,C) such that p' ||—g{y) = S and then choose a

T G X such that SznSf = S , and My) > <5. Put r = (a' ,/ ' ,S'uST,C") where C » = CT(v)

if v G ST, C"(v) = Cr(v) for v G iSr. r is clearly a condition, as CT(v) has no new element

beyond a. r extends px and // , so it forces that hx(y) > g(y) = S and also that hT(y) < g(y)

(as 7 > P), a contradiction. D

Acknowledgment. The author is grateful to the referee for a very careful job.

References

[1] Chang, C.C. and Keisler, J. (1973) Model Theory, North-Holland.[2] Devlin, K. (1975) A note on a problem of Erdos and Hajnal, Discrete Math. 11 9-22.[3] Galvin, F. and Hajnal, A. (1975) Inequalities for cardinal powers, Annals of Mathematics 101

491-498.[4] Kanai, Y. (1991) On a variant of Chang's conjecture, Zeitschr.fur math. Logic und Grundlagen

d. Math. 37 289-292.[5] Shelah, S. Proper Forcing, Lecture Notes 940, Springer-Verlag.

Topological Cliques in Graphs

JANOS KOMLOS and ENDRE SZEMEREDI

Rutgers University and Hungarian Academy of Sciences

Let f(t) be the largest integer such that every graph with average degree t has a topologicalclique with f(t) vertices. It is widely believed that f(t) > Cyjt. Here we prove the weakerestimate f(t) > cjt/{\ogtf.

1. Introduction

A subdivision of a graph G is obtained by replacing some of the edges by independent(vertex disjoint) paths. We say that the graph H is a topological subgraph of the graph G(and write H < G) if there is a subgraph H' of G that is isomorphic to a subdivision ofH.

If Kr < G, where Kr is the complete graph on r vertices, we say that G has a topologicalr-clique. We also define the topological clique number

tcl(G) = max {r : Kr < G},

and write f(t) for the largest integer such that every graph with average degree at least thas a topological clique with f(t) vertices:

/(f) = min{tc/(G) : t(G) > t},

where t(G) is the average degree of the graph G.It is an easy exercise to show that tcl(G) « c^Jn for most graphs G on n vertices. (This

is the Erdos-Fajtlowicz theorem [4], see also the papers of Bollobas and Catlin [3], andalso [1].)

The following lower bound was conjectured by Mader [8], and Erdos and Hajna [5].

Conjecture 1. fit) > ctl/2.

The weaker bound f(t) > c(logt)1/2 was proved by Mader [9].

440 J- Komlos and E. Szemeredi

Theorem 1.1. For any n > 1 there is a positive cn such that

fi for t>2.

1.1. Sketch of the proof We use the following greedy-type algorithm. We pick r points farfrom each other, and pairwise connect them with the shortest possible paths that avoid alarge neighbourhood of the other r — 2 points.

To guarantee that these paths do not use up too many vertices, we need to control thediameter. We do this by controlling degrees: we start with a subgraph G of minimumdegree ct and maximum degree at most t2 (Section 2.1).

To avoid the main obstruction - small bottlenecks in the graph - we select a furthersubgraph G" with the following expanding property: any m vertices in G" have at leastm/(\ogm)K neighbours (Section 2.3).

In fact, we show that if the number of vertices in G" is sufficiently large (in terms of f)then G" even has a topological clique of order ct. The bottleneck of the problem is whenthis number of vertices is small (e.g. a power of t) but not as small as O(t).

We shall also point out in Section 2.2 a very simple proof of Theorem 1.1 for graphswith positive density. The proof is entirely self-contained; moreover, the general proof ofTheorem 1.1 does not make use of any material from Section 2.2.

1.2. Definitions and Notation All our graphs will be simple and have no isolated vertices.The vertex-set and the edge-set of the graph G will be denoted by V(G) and £(G), and welet n(G) = \V(G)\ and e(G) = \E(G)\. By an n-graph, we mean a graph of order n (that is,a graph with n vertices). We write (A, B,E) for the bipartite graph with bipartition (A, B)and edge set E a A x B.

The set of neighbours of v e V including v itself will be denoted by N(v). Hence\N(v)\ = deg(v) + 1, where deg(v) is the degree of v. More generally, for A a V(G) wewrite N(A) = \JveA N(v). For v e V, U, V c V, U n U' = 0, we write deg(v, U) for thenumber of edges from v to U, and e(U, V) for the number of edges between U and V.For non-empty A and B, we write

for the density of the graph between A and B.The graph G restricted to the vertex-set S is denoted G|s, and G — S is shorthand for

G\v(G)-s, the graph obtained by deleting the vertex set S. We also use the abbreviatione(U) = e(G\u). An important definition is that of the boundary of the vertex-set S, whichis denoted by dS:

dS = N(S)-S = {v e V-S : {s,v} e E(G) for some s e S}.

The notation H a G means that H is a subgraph of G, and H < G means that H is atopological subgraph of G. When a graph H is subdivided into a graph H\ the originalvertices of H will be called the principal vertices in H'.

The average degree, minimum degree and maximum degree of the graph G will be

Topological Cliques in Graphs 441

denoted by t(G), 3(G) and A(G), respectively. A graph G is called t-maximal if t(G) =max//cG t(H) (which implies 6{G) > V2 t(G)\ and it is T-maximal if t(G) = max//<G *(#)•

Finally, Co, ci, C2,... will stand for positive absolute constants. The notation f(x) f (/(x) | )will signify that the function f(x) is monotone increasing (decreasing).

2. Preliminaries

2.7. Almost regular subgraphs Given a constant c, an almost regular graph is one withA(G) < cd(G). We would like to be able to select, for every graph G, an almost regularsubgraph with only a constant loss in average degree. Unfortunately, this is not alwayspossible. (It is not too hard to construct counterexamples.) The following lemma sufficesfor our present purpose.

Lemma 2.1. Every graph G has a topological subgraph H such that

t(H) > \ t(G) and \ t(H) < S(H) < A(H) < 72 t2(H). (1)6 2

In fact, we have the following more detailed picture.

L e m m a 2.2 . Let G be a T-maximal graph. Then either G has a subgraph H for which (I)holds, or tcl(G) > ( l / 4 ) t ( G ) .

Remarks. The bound 12t2(H) in (1) can be replaced by cpt^(H) for any j? > 1. Also, withsome loss in the average degree, we can get an almost regular topological subgraph. Thefollowing lemma may help improve Theorem 1.1 for not too sparse graphs, but we willneither use it nor prove it in this paper.

Lemma 2.3. Every graph G with average degree t > 2 has an almost regular topologicalsubgraph H with

t(H) > c{t(G)/log t(G) and U(H) < 3(H) < A(H) < c2t(H).

Proofs of Lemmas 2.1 and 2.2. These are trivial if t(G) < 12 (if G is a forest, pick anedge, otherwise pick any cycle as H). Thus, we will assume that t(G) > 12, in which caseLemma 2.2 implies Lemma 2.1.

Now, write t = t(G), and set S = {v e V : deg(v) < 2f2}, L=V-S.Since \L\ < n/(2t) < n/24 (by Markov's inequality), and all degrees are at least f/2, thereare two possibilities:(A) #{seS : deg(s,S)>t/4}> \n

(B) Sf = {s £ S : deg(s,L) > t/4} is of size at least \n

442 J. Komlos and E. Szemeredi

In case (A), we choose H = G\s. We have t(H) > (2/3)(t/4) = t/6 and A(H) < It2 <12t\H\ as needed.

In case (B), we construct a topological subgraph of G with principal points in L asfollows. We pick vertices of Sf, one-by-one, and for each s' G S' we find two verticesx,y e L adjacent to s'. The path xsfy (of length 2) will represent an edge between x andy. We do the selections in an arbitrary fashion; the only thing to keep in mind is not touse a vertex s! e S' twice, or a pair (x,y), x,y e L twice (no multiple edges). We only stopwhen there are no more vertices sf e Sf to use.

Since the resulting topological subgraph is of average degree at most t (by the T-maximality of G), we could pick at most t\L\/2 < n/4 such vertices sr e S". But \S'\ > n/4.Hence there is at least one sf £ Sf that is unusable. Why couldn't we use it ? Because anytwo of its t/4 (or more) neighbours have already been connected via previously chosenvertices from Sf. In other words, there is a topological clique of size at least t/4 in G.

•

2.2. Dense graphs While the content of this section is not necessary for proving Theorem1.1, it may illuminate the general proof, and it is also interesting in its own right.

It is a standard exercise in graph theory classes to show that tcl(G) = &(y/n) for mostn-graphs G. It is not mentioned, however, that in fact tcl(G) > c^Jn for all dense n-graphs.

Theorem 2.1. For every c > 0 there is a d > 0 such that, for all n,

tcl(G) > c\Jn for all n-graphs with at least en2 edges.

A trivial proof is through the use of the Regularity Lemma [10, 11]. Indeed, every densen-graph has a subgraph that is a regular pair with minimum degree c'n, and such graphsare easily seen to contain topological cliques of size c"\Jn. (Use the greedy algorithm andthe fact that such a regular pair has a diameter at most 4.) Here are more details. Let0 < 6 < 1/2.

Definition 1. The bipartite graph (A,B,E) is called an e-regular pair if

I c i , Y a 5, \X\> e\A\, \Y\ > e\B\

imply

\d(X,Y)-d(A,B)\ <e.

The Regularity Lemma guarantees, among other things, that for any c,e > 0 there isa c" > 0 such that every n-graph with more than en2 edges contains as a subgraph an8-regular pair H = (A,B,E), \A\ = \B\ > c"n, with density greater than c.

Let e < 3 < 1/2 be positive numbers.

Definition 2. The bipartite graph G = (A,B,E), \A\ = \B\ = n, is called an (e,(5)-expander

if S(G)>Sn, and

e(X, Y) > 0 for all X a A, Y c £, |X|, |71 > an.

It easily follows from the above definitions that any e-regular pair (A,B,E), \A\ = \B\ = n,of density d > 2e has an (e'^^-expander subgraph (A',B'9E

r), where

\A'\ = \B'\ > (1 - e)n9 sf = 8/(1 - 8), (5' = d - 2e.

Now the proof of Theorem 2.1 goes as follows. Let e = c/5. Apply the RegularityLemma to find an a-regular subgraph H of G of order c"n with density d > c = 5s. Thenselect a (2s, 3a)-expander subgraph H' of H of order at least c"n/2. Apply the followinglemma to conclude the proof.

Lemma 2.4. Let H = (A,B,£), \A\ = |B| = n, fo? an (e,S)-expander. Then tcl(H) >

Proof of the lemma. Pick r arbitrary vertices, where r = [((S — s)n)1/2 J. Connect thempairwise with vertex disjoint shortest paths using the greedy algorithm, going throughthe pairs one-by-one. To show that the deletion of previous paths does not increase thediameter, use the following trivial observation: let H = (A,B,E), \A\ = \B\ = n, be an(a, (5)-expander. Then H has a diameter at most 4, and this remains true even after deletingarbitrary (S — s)n arbitrary vertices.

This proves the lemma, and hence Theorem 2.1. •

How efficient is this method? The Regularity Lemma says much more than we haveused (see [11]), and thus it is not surprising that the constant d obtained in this way isquite small; for s = c/2 (say), \/c' is a tower function with about 1/c levels, that is, the\/c times iterated logarithm of 1/c' is about 1.

Since we only use a small part of the Regularity Lemma, it is more prudent, and fitsour present purpose better, to find a more direct approach. Using graph functional ofthe form ip(G) = ipi(S(G))\p2(n(G))9 we can directly select expanding subgraphs of G, andthese subgraphs are much bigger than those guaranteed by the Regularity Lemma (see

[6]).We will generalize this method in the next section.

2.3. s-expanders We will generalize the expander method used for the case of densegraphs by extending the definition to expanding by a varying degree.

Note that s = e(l),e(2)9... will always denote a non-negative sequence.

Definition 3. A graph G = (V,E) is an e-expander if

\dX\\X\

Theorem 2.2. Let

> E(\X\) for all subsets X aV, \X\ < (1/2)|7|. (2)

e(k) | , ke(k) | and ^e(/c)//c < 1/6. (3)

Then every graph G has an s-expander subgraph H with

t{H)>l-t{G) and S(H) > ^t(H). (4)

If s(k) I, ks(k) | are only assumed for k > /co, then some subgraph H (satisfying (4)) willstill be expanding in the sense that (2) holds for 3/co/2 < \X\ < 2n(H)/3.

Let us define the graph functional

where cp(k) is a monotone decreasing positive sequence.

Definition 4. G is called ^-maximal if xp(G) = max//cc w(H).

Note that y;-maximality implies f-maximality.

Lemma 2.5. Let G be a \p-maximal graph on n vertices. Then, for all Y cz V,

\dY\ > cp(\X\) - cp(n) > (p(\X\)-cp(n)

\X\ ~ l + <p(\X\) ~ l + <p(l) '

where X = Y U dY.

Proof. By the f-maximality of G, t(G\y) < f(G), which implies

2e(G) = 2e(Y) + 2e(Y, Y) + 2e(Y)< 2e(Y) + 2e(Y,T) + t(G)\Y\ < 2e(X) + f(G)|Y|,

whence

t(G)\Y\ < 2e(X) = t(G\x)\X\ < t(G)\X$l + <p(n))/(\ + q>(\X$)9

and the rest is highschool algebra. •

Lemma 2.6. Let s satisfy (3), and let

cp(k) = 6 \ J s(i)/U V>(G) = t(G)(l + cp(n(G))).i>2k/3

Then every \p-maximal graph is an e-expander.If s(k) | , ks(k) | are only assumed for k > /co, every xp-maximal graph G is still expanding

in the weaker sense that (2) holds for 3/co/2 < \X\ < 2n(G)/3.

Proof. It is easy to see that we have cp(l) < 1, and

(1 + (p(l))~l [q>(k) - q>(3k/2)] > 3 V e(i)/i > s(k) (5)2k/3<i<k

for k > 3/co/2. Now let Y cz V, \Y| < |F| /2, be arbitrary, and let X = Y UdY. In the case\X\ < 2n/3, Lemma 2.5 and (5) imply

If \X\ > 2n/3 then \dY\ > n/6 > \Y\/3 > s(\Y\)\Y|. •

Proof of Theorem 2.2. We simply choose a y;-maximal subgraph H of G. By the previous

lemma, H is an ^-expander. We still have to estimate t(H) and S(H). We have

xp(H) =

whence t(H) > t(G)/(l + (p(l)) > t(G)/2. Also, a tp-maximal graph is obviously r-maximal,too. Hence S(H) > l/2 t(H). D

Fix a constant K > 1. From now on, we will always use the same sequence

0 i f i<r /4 ,c3log-K(8i7r) i f i>r /4 ,

and cp(k) = 6 V^ s(i)/i9 ip(G) = t(G)(l + c/)(n(G))). (6)i>2k/3

3. Proof of Theorem 1.1

According to Lemma 2.1 and Theorem 2.2, we can restrict our attention to graphs Gsatisfying t(G) = t > to and t/2 < S(G) < A(G) < 12t2. Moreover, we may assume G istp-maximal, and so is e-expanding with cp and £ defined in (??).

At this point we introduce some more notation. The distance p(u,v) between verticesu,v is the length of the shortest path between them. We also define balls and spheresaround a vertex v, in the natural way, as

B(v, r) = {u e V : p(u, v) < r), S(v, r) = {u € V : p(u, v) = r}.

Note that dB(v,r) a S(v,r+1). The bottleneck for a vertex v is defined as min{|S(i;, r + l)| :r > 0, \B(v,r)\ < (l/2)n(G)}, and the bottleneck of G is then the minimum of those of itsvertices.

Here are some of the properties of G that we shall use. First, by the choice of ip, thetp-maximality of G implies that

if \B(v,r)\ < l/2n(G) then \S(v,r + l)\ >e(\B(v,r)\)\B(v,r)\.

Given this expansion inequality for balls, it can then be shown (say by induction on r)that

min{exp{c4r1/(1+K)}, l/2 n) < \B(v, r)\ < (9t)2r for all v, r.

It now follows fairly straightforwardly that the diameter of G is less than C5(logn)1+K, andalso that G has a bottleneck greater than c^t.

It is important that the properties of G just mentioned are robust, in the sense thatthe removal of a few vertices from G will not affect the properties (apart from slightlychanged values of the constants). In particular, the properties remain if at most (l/2)cetvertices are removed from G (the bottleneck will still be at least (l/2)c6f). The propertieswill also remain if a number of vertices are deleted from G in such a way that the numberremoved from any sphere S(v,r) is very small in comparison to the size of the sphereitself.

446 J. Komlos and E. Szemeredi

In the next two subsections, we will take K > 1 fixed, and use an arbitrary parametera > K2 + 3K + 3 > 7.

3.1. Not too sparse graphs Here we prove Theorem 1.1 for n-graphs with logn < (log£)a.In fact, in this case we simply use the greedy algorithm.

Choose arbitrary vertices v\,V2 to start with, and connect them with a shortest path.Select any vertex 173 outside this path. In general, choose an arbitrary non-used vertex,and connect it, one by one, with shortest paths to the old vertices, each time making sureto use only vertices that have not been used before. As remarked earlier, provided weuse fewer than (l/2)cet in the total construction, we can always find, between any twovertices, a path of length at most cy(logn)1+K. For a topological clique of order r, we needless than r2 such paths, so the greedy algorithm provides a topological clique of order rwith r > c8V^(logn)"(1+/c)/2 > csy/i{logt)~rl, where n = a(l + K)/2 > 7. •

3.2. Sparse graphs

Theorem 3.1. Let G be an e-expander graph on n vertices, with s defined in (2.3), withaverage degree t > to such that logn > (\ogt)a. Then tcl(G) > cgt.

Miki Simonovits noted that what we really prove here is that any graph G as in Theo-rem 3.1 is highly connected in the following sense:

Let k satisfy log/c < c(\og t)^, for some ft < (a—2 — K)/(1 + K)2. Then, given any k verticesin G pairwise far from each other, and given any graph H on k vertices and maximum degreeA < c\ot, there is a topological H in G with the given k vertices as principal vertices.

We will only use this for the very small value k = ct (that is, /? = 1).

Here the algorithm is somewhat different. Let

R = 0.1 log n/ log t and r = cn(\ogk + loglogn)1+K.

We start by selecting k = cnt vertices v\,...,Vk such that any two are at a distance at least2R. These will be our principal points. Then we connect these points pairwise as follows.We surround each of our k vertices with balls of radius r, and whenever we are to connectvi by Vj, we first connect the spheres S(vt,R) and S(VJ,R) in such a way that the path goesentirely outside all the balls JB(tv,r), ^ 7 Uj- We choose a shortest such connection. Thenwe connect the contact points on these two /^-spheres to their respective centres withshortest paths. We continue the algorithm in the graph obtained by deleting all interiorvertices of the path constructed.

(At this point, we mentally rebuild the spheres, in that we recompute diameter, bottle-necks, etc., in the new graph obtained by deleting the vertices used. But this only concernsthe analysis of the algorithm, not the algorithm itself.)

Lemma 3.1. The conditions of Theorem 3.1 imply that G has k vertices such that any twoare at a distance at least 2R. (Here k is as defined in the remark after the theorem.)

Furthermore, for any positive integer w < n/4, and for any set S a V, \S\ < vve(vv), the

union of all components of G — S of size at most n/2 is of size less than w.

In particular, if the pairwise disjoint sets T\,Ti,S a V satisfy |Ti | , IT2I > w and \S\ <

ws(w), then S cannot separate T\ and Ti.

Proof. First, we can find k such vertices, since the size of a 2K-ball is at most (9t)4R, andk(9t)4R < n. The second claim is also true. Indeed, otherwise the union of some of thesmaller components would form a set W c V, w < \W\ < 2w < n/2 such that dW c S,whence \dW\ < \S\ < ws(w) < \W\e(\W\) (since xe(x) | ) , so contradicting expansion. Thefinal claim follows directly, for there can be at most one component of size greater thann/2. - •

Proof of Theorem 3.1. The crucial observation is that, for any vt and any p < r, wedeleted at most A < ct vertices from any one sphere S(y,,p), which is less than thebottleneck in the graph. Also, we deleted at most Qd vertices from the whole graphaltogether, where d is an upper bound on the maximum distance between two of ourprincipal points vt at the end of the construction after we made all the deletions.

Now, as we said earlier, if the number of deleted vertices turns out to be muchless than the number of vertices on any sphere S(vt,p), p > r, then the lower bounds\B(vi,p)\ > c exp{cr1^1+K^} still hold for all /, so we can take d = c(logn)1+K for someconstant c.

We use Lemma 3.1 with T\, Ti being ^-spheres around two distant points, and S beingthe union of r-spheres around all the other k — 2 points, and the interior points of allpaths constructed before.

Each Tt has at least exp{c#1/(1+K)} points, while S has size less than k(9t)2r + (k2)d. It is

easy to check that the conditions of the lemma hold.The lengths of the connections are at most the diameters in the current graph, thus it

is easy to check that on each sphere S(vi,p), p > r, there are more points than the totallengths of all the paths combined.

It remains to choose the parameters such that k > ct. This holds i f a > 2 + /c + (l + K)2.•

Addendum. Noga Alon has remarked that an unpublished theorem of Robertson andSeymour about graph minors together with a theorem of Mader and a bound of Thomasonand Kostochka imply Theorem 1.1 with a better constant (n = 1/4). The approach seemsto be completely different from ours.

References

[1] Ajtai, M , Komlos, J. and Szemeredi, E. (1979) Topological complete subgraphs in randomgraphs. Studia Sci. Math. Hung. 14, 293-297.

[2] Bollobas, B. (1978) Extremal Graph Theory, Academic Press, London.[3] Bollobas, B. and Catlin, P. (1981) Topological cliques of random graphs. J. Combinatorial

Theory 30B, 224-227.


[4] Erdos, P. and Fajtlowicz, S. (1981) On the conjecture of Hajos. Combinatorica 1, 141-143.[5] Erdos, P. and Hajnal, A. (1969) On complete topological subgraphs of certain graphs. Annales

Univ. Sci. Budapest 7, 193-199.[6] Komlos, J. and Sos, V. (manuscript) Regular subgraphs of graphs.[7] Lovasz, L. (1979) Combinatorial Problems and Exercises, Akademiai Kiado, Budapest.[8] Mader, W. (1967) Homomorphieeigenschaften und mittlere Kantendichte von Graphen. Math.

Annalen 174, 265-268.[9] Mader, W. (1972) Hinreichende Bedingungen fiir die Existenz von Teilgraphen die zu einem

vollstandigen Graphen homoomorph sind. Math. Nachr. 53, 145-150.[10] Szemeredi, E. (1976) Regular partitions of graphs. Colloques Internationaux C.N.R.S. N- 260 -

Problemes Combinatoires et Theorie des Graphes, Orsay, 399-401.[11] Szemeredi, E. (1975) On a set containing no k elements in arithmetic progression. Ada

Arithmetica XXVII, 199-245.

Local-Global Phenomena in Graphs

NATHAN LINIAL

Institute of Computer Science, Hebrew University, Jerusalem, Israel

This is a survey of a number of recent papers dealing with graphs from a geometricperspective. The main theme of these studies is the relationship between graph propertiesthat are local in nature, and global graph parameters. Connections with the theory ofdistributed computing are pointed out and many open problems are presented.

1. Introduction

How well can global properties of a graph be inferred from observations that are purelylocal? This general question gives rise to numerous interesting problems that we wantto discuss here. Such a local-global approach is often taken in geometry, where it has along and successful history, but a systematic study of graphs from this perspective hasnot begun until recently. Nevertheless, a number of older results in graph theory do fitvery nicely into this framework, as we later point out. Most of the specific problems fallin two categories. In the first, local structural information on the graph is collected andthen used to derive certain consequences for the graph as a whole. The other class ofproblems concerns consistency of local data. Namely, one asks to characterize those setsof local data that may come from some graphs.

As the reader will soon see, the local-global paradigm leads to many questions inwhich graphs are viewed as geometric objects, a point of view that we believe can greatlybenefit graph theory. Besides the geometric connection, ties also exist with the theoryof combinatorial algorithms. We suggest a specific test case for the heuristic notion thatpolynomial-time algorithms are capable of examining only local phenomena. In distributedcomputing, locality of computation is an already recognized and studied notion, and someconnections with this discipline are pointed out as well.

2. Packing and covering with spheres and local-global averaging

Let W c: V(G) be a set of vertices in a graph G. If the vertices in W form a majority inevery ball of radius between 1 and r in G, does this imply that W has a large cardinality?

450 N. Linial

As an illustration, consider the following example with r = 1. In this graph, W is aclique of yjn vertices. Each vertex in W has a set of y/n — 1 neighbors not in W, each ofwhich has degree 1. It is a routine matter to check that this graph satisfies the assumptionfor r = 1. It is also not hard to modify the construction for any fixed a < 1 so that Woccupies a fraction > a of any 1-neighborhood, while \W\ = O(y/n) (here a was 1/2).

Let us introduce some notation. The ball' of radius k centered at x, denoted Bk(x)consists of all vertices y whose distance from x does not exceed k, and its cardinality\Bk(x)\ is denoted fik(x). Our question is how small \W\ may be in terms of r and n, theorder of G.

If we represent W by its characteristic function, we are led to consider a more generalproblem. Namely, let / be a nonnegative function defined on the vertices of an n-vertexgraph G. Suppose that we have a lower bound on the average of / on every ball in G ofradius between 1 and r. What can we conclude for the overall average o f / ?

This subject has been recently taken up by Linial, Peleg, Rabinovich and Saks [23] whoshow the following.

Theorem 2.1. (Local Averages) Let f be a nonnegative function defined on the vertices ofan n-vertex graph G. Suppose that the average of f over every ball of radius r > t > 1 inG is at least \i. Then, the average of' f over all of V is at least \i • n-0(i/i°sr)t j n e \)oun^i ;s

tight.

Consequently, if we let r be nc for some positive constant c, local averages do reflectthe true global behavior of / . Examples are given in [23] showing that smaller r's willnot do. It is natural to ask at this point what happens if we only know a lower boundfor the average of / over balls of radius r (and not for every r > t > 1). Examples aregiven showing that only very weak conclusions can be drawn about the overall averageof/, however big r may be. Namely, it may be that the average of/ is only O(n~1/3). Itis also worthwhile noting that the conclusions of the theorems remain unchanged even ifwe make the assumption only for balls whose radius r > t > 1 is a power of 2.

The result for local averages is proved as a consequence of tight theorems about spherepacking and about covering by spheres in general graphs. Either 0-1 or fractional packingand covering results will do for this purpose.

Theorem 2.2. (Covering by Spheres) For integers n > r, the vertices of an n-vertex graph

can be covered by a collection of balls with radii in the range [ l , . . . , r ] , that cover no vertex

more than n0{X/Xogr) times. The bound is tight.

Theorem 2.3. (Sphere Packing) In any n-vertex graph, there is a collection of disjoint balls

whose radii are in the range [ l , . . . , r ] , which together cover at least ^ -W/iogr) vertices^ j n e

bound is tight.

The words ball and sphere are used interchangeably here.

Local-Global Phenomena in Graphs 451

It would be very interesting to understand how various properties of a graph affect theefficiency of sphere packing and of covering by spheres. Also, it is not hard to extendthese results to general finite metric spaces. We still do not know, for example, whathappens if the metric space is embedded in a d -dimensional Euclidean space or otherlow-dimensional normed space. These questions lead us to our next subject.

2.1. Connections with the theory of maximal functions

There is an appealing connection between this class of problems and the theory ofmaximal functions in analysis (e.g. [32]). This observation came up in discussions withMetanya Ben-Artzi.

Briefly, the connection is this: again let Br(x) denote the ball of radius r centered atx G R , the d -dimensional Euclidean space. Let / be a real function on R , and let ar(x)be the average of / over Br(x). Define f*(x) as the supremum of ar(x) over all r > 0.The function /* is called the maximal function of / . Numerous results have been derivedover the years concerning maximal functions. Informally speaking, among the most basicfindings is that '/* is not much larger than / ' .

Our proof for Theorem 2.1 shows a significant similarity with the methods used inanalysis to compare the p-norms of /* and / . Specifically, the most traditional prooftechnique involves some geometric covering arguments (Vitali's Lemma), and a similarargument underlies some of our proofs as well. In analysis, such arguments lead to resultsof the form

Wfh < cdjf\\p

where Q,p grows exponentially with the dimension d. This bad dependency is unavoidablein this method, since the bounds in Vitali's lemma do grow this way. More modern resultsconcerning maximal functions (e.g. [33]) manage to bypass this difficulty. It is conceivablethat these methods may help settle our questions on low-dimensional finite metric spaces.It would also be interesting to see if similar ideas can be developed for other classes ofgraphs.

3. Locality in distributed systems

The theory of distributed computing concerns a set of processors connected through acommunication network. The network is depicted as a graph in whose vertices computersor processors reside. Communication takes place as messages are exchanged betweenneighboring vertices. The processors' goal is to perform some computational task together.Let us restrict our attention to determiriistic and synchronized networks - the simplestamong this class of computational models In such an environment it is easy to see thatin t time units a processor can only learn about the situation at processors that arewithin distance at most t from itself in the graph. This observation gives rise to numerousquestions of the local-global type. In studying such questions, some care has to be givento symmetry breaking. If processors 'have no identity' and cannot be told apart by otherprocessors, then almost nothing interesting can be done. We do not elaborate on this,

452 N. Linial

but rather say that the common practice in this area is to assume that processors areequipped with individual (distinct) ID-numbers, and so symmetry causes no problems.

3.1. Low-diameter decompositions of graphs

Perhaps the most fundamental difficulty in distributed processing, as compared with moretraditional computational models, is the absence of central control. It is very difficult tohave many processors perform in concert when there is no conductor around. Indeed,much research effort in distributed processing concerns efficient and reliable methods forelecting a leader. We will not pursue this fascinating subject, and only point out someof the shortcomings of this approach. It creates a communication bottleneck around theelected leader. It is also very sensitive to failures, or latency of the leader and its neighbors.Moreover, if the graph underlying the communication network has a large diameter, thismethod is also very wasteful in terms of communication.

In view of the difficulties involved with such a 'central government' the next thing totry is a set of cooperative local authorities'. Namely, in the previous section we werecovering vertices by balls; now we consider decomposing the vertices, subject to a certainupper bound on the diameter of each part. Let us introduce some notation: if IT is adecomposition of the vertices of graph G into subsets, V(G) = (JS,-. The diameter of thisdecomposition is defined as the maximum over all diam(Si).

Remark 3.1. In defining the diameter, we may consider the graphs induced by the parts,and compute distances within these graphs. Alternatively, we may consider distancesas inherited from the whole graph. Our statements, slightly modified, hold for eitherdefinition.

The graph induced by IT has one vertex per S/, with vertices ij adjacent iff there is avertex in St and one in Sj that are adjacent in G. The goal is to find partitions Yl withsmall diameter and favorable properties for the induced graph. Linial and Saks [25] show(see also [6, 7]):

Theorem 3.2. An n-vertex graph has a decomposition of diameter r, where the inducedgraph has chromatic number < %, if both

flogn\ (logn\= Q( i and r = Q ——

Vlogry Vlog*/

hold. Examples exist showing these bounds are tight. A randomized distributed algorithm oflogO(1) n run time is provided to obtain such decompositions.

We briefly discuss some extreme examples for Theorem 3.2. It is easily seen that there

are two interesting ranges to this theorem:

log/tr > > y.

log log n

In this range, the tradeoff between r and x is given by:

x = i

The known extreme examples in this range are graphs corresponding to triangulations ofEuclidean spaces. For example, the graph whose vertices are all lattice points in log n/ log rdimensions, with adjacency between x,y iff ||x — y||x = 1.

X> l 0 g W > r ,log log ft

where the condition is

Here trees and expander graphs provide extreme examples.

Remark 3.3. Notice that radius log ft along with / = 0 (log ft) are possible. Consequently,if every ball or radius log ft in G is /c-colorable, x{G) = O(k\ogn). So, up to a logarithmicfactor, the coloring number can be inferred from radius log n views of G.

More on coloring from the local-global perspective will be said later.So far we have considered only the chromatic number of the graph induced by a

decomposition. Other properties of this graph are of interest as well. Let us point outthe analogy between these questions and notions from dimension theory in topology [19].The following question is inspired by the notion of covering dimension of metric spaces.Let n : V(G) = \JSt be, again, a decomposition of the vertices of graph G. For a vertexx, let y(x) be the number of S,- in which x has a neighbor. A(FI) is defined as maxx(y(x)).

Problem 3.4. What is the least D = D(r, ft), such that any ft-vertex graph has a decompo-sition n of diameter < r with A(I1) < D?

Possibly, the tradeoff between D,r and n is the same as the one for /, r and n inTheorem 3.2.

3.2. Applications of low-diameter decompositions

Low diameter graph decompositions have found numerous applications in distributedcomputing. We briefly sketch some of these. We begin with the Maximal Independent Set(MIS) problem. (We mean inclusion-maximal. This problem is not to be confused withthe search for an independent set of largest cardinality, which is NP-complete.) There is,of course, a most simple sequential algorithm, which at each step adds a new vertex tothe MIS and eliminates all its neighbors from the graph. While such a naive sequentialalgorithm solves the problem in optimal time, finding efficient parallel algorithms forthis question is not nearly as obvious. An efficient parallel algorithm was first found byKarp and Wigderson [20] with numerous improvements and ramifications by others {e.g.[1, 27]). In fact, Luby's algorithm [27] works also in the distributed model, but it does userandomization, however. One of the tantalizing questions that remain in this area is:

454 N. Linial

Problem 3.5. Is there a deterministic, distributed, polylog-time algorithm to find amaximal independent set (MIS) in a graph?

It has been observed in [4] that low-diameter decompositions with a low-chromaticdecomposition graph may help provide such an algorithm. Assuming such (an alreadycolored) decomposition is available, we construct, in parallel, an MIS within each partcolored 1. Since each part has diameter at most r, an MIS for it can be constructed byan elected leader, where both election and construction take time O(r). Also, there areno edges between different parts of color 1, so these activities in different parts can beperformed in parallel without affecting each other. Vertices selected so far for the MISeliminate their neighbors (also in other parts), and we move on to parts colored 2 etc.Using the terminology of Theorem 3.2, a time bound of O(rx) can be achieved, which byproper choice of parameters may be made O(\og2n). The difficulty is, of course, that thisargument assumes a partition to be already available. Currently, however, only randomizeddistributed algorithms are known that find such decompositions in polylogarithmic time(Theorem 3.2 and [3]). Problem 3.5 thus remains open.

Another problem for which low-diameter decompositions help is distributed job schedul-ing [5]. In this problem, processors try to efficiently share their workloads. Initially, eachprocessor is assigned a number of (unit-cost) jobs to perform. In each step, a processor canperform one of its assigned jobs, as well as send some of its assigned jobs to neighbors. Itcan also communicate messages to its neighbors. A processor knows only its own historyand the contents of the messages it receives. An algorithm is sought where the completiontime is early as possible. Moreover, the following strong ('competitive') criterion is applied:the time for completion should compare favorably with the best that can be achieved byan optimal central controller having a complete view of the situation at all times (and notjust local views at a certain processor). Using low-diameter decompositions, [5] managesto guarantee a completion time that is only O (log ft) longer than can be attained by aknowledgable central controller. This result is shown to be almost optimal for certainfamilies of graphs in [2]. For further applications see [6, 7].

4. Distributed coloring and related problems

The systematic study of locality in distributed processing was begun in [22]. Our firstresult concerned the time required to 3-color an rc-cycle of processors. A clever algorithmby Cole and Vishkin [11] does this in time O(log* n). (Recall that log* n is the number oftimes one has to iterate the log function to come down from n to 1).

The first result in [22] says that this algorithm is optimal.

Theorem 4.1. A distributed algorithm that properly colors the n-cycle with only 3 colorsrequires time Q(log* n). The bound is tight.

It was later shown by Naor [30] that the same statement also holds for randomizedalgorithms.

Assuming n is even, how long does it take to 2-color an n-cycle? A huge differenceshows up, in comparison with the time complexity of 3-coloring.

Theorem 4.2. A distributed algorithm that properly colors the n-cycle (n even) with only 2colors requires time £l(n). The bound is tight.

This example captures a big difference in locality of 2 and 3-coloring of cycles. Ofcourse, a 2-coloring requires perfect global coordination, which results in an excessivetime complexity.

Other results from [22] are:

Theorem 4.3. Let T be the d-regular tree of radius r. Any algorithm that properly colorsT and runs for time < 2r/3 requires at least Q{\fd) colors.

This result can probably be improved to Q(d/\ogd). An intriguing open question in thisarea is:

Problem 4.4. Consider distributed algorithms that properly color n-vertex graphs andtake time logO(1) n. What can be said about the least number of colors required by suchan algorithm? Specifically, is it possible that A + 1 colors suffice, where A is the largestvertex degree of the graph?

This question is closely related to Problem 3.5. Some partial results have been providedin [22].

Theorem 4.5. An 0(log* n)-time algorithm exists to color any n-vertex graph G with O(A2)colors, where A = A(G) is the largest vertex degree in G.

See also [34] for some recent progress in this area. Naor and Stockmeyer [31] haverecently investigated the limits of what can be computed with a constant diameter oflocality.

Another related problem is that of finding happy partitions. A partition of the vertexset of a graph V = A\JB is called happy if every x G A has most of its neighbors in B andvice versa. That such partitions always exist is easy to show, and a sequential algorithmto construct such partitions is easy to find. The distributed time complexity of this is stillunknown: Linial and Saks conjecture (unpublished) as follows.

Conjecture 4.6. There are n-vertex graphs where a distributed algorithm to find a happypartition requires time Q(v/n).

5. Coloring

The chromatic number of a graph is a good example of a global parameter where thebehavior of small induced subgraphs seems to be a weak indicator of global properties.

456 N. Linial

(But notice Remark 3.3.) Up to this point local' has always been taken in the sense ofdistance. It is also interesting to examine assumptions about the behavior of (cardinality)small sets of vertices. In this section we consider sets that are small in either diameter orcardinality. One easy consequence of Theorem 3.2 is the following.

Theorem 5.1. / / the subgraph spanned by every k vertices in G is 2-colorable, /(G) =O(nO(1/^). This bound is tight. Moreover, it is possible to find a proper coloring with thisnumber of colors in polynomial time.

Dealing with 2-colorabilty is usually much easier than with any larger coloring number.Is there, perhaps, a similar result for graphs that are, say, locally 3-colorable? To simplifyour notation, we will only consider 3-colorability, leaving out the obvious extension tomore colors.

Problem 5.2. Let #(n, k) be the largest chromatic number of an n-vertex graph G if thesubgraph spanned by every k vertices in G is 3-colorable. Determine the behavior ofX(n,k).

It is not hard to see that

where the o(l) terms tend to zero as k grows. The upper bound follows, e.g., fromWigderson's argument [35] mentioned below. The lower bound combines an argumentfrom [24] with a lower bound due to Gallai [15] on the least number of edges inminimally non-3-colorable graphs. Note the difference compared with locally 2-colorablegraphs (Theorem 5.1).

Besides the interest in the problem per se, it is related to approximating chromaticnumbers in polynomial time. That it is NP-hard to determine the chromatic numberhas been known for a long time [16]. How well this quantity may be approximated isstill unknown, although considerable progress has been made. An early positive resulton approximating chromatic numbers is a polynomial-time algorithm by Wigderson [35],which colors any n-vertex 3-colorable graph with O(^/n)-colors. Here is the argument: aslong as you can find a vertex x of degree > -y/w, allot two fresh colors for the neighbors ofx and discard them (they are two-colorable, since /(G) = 3). When the remaining graphhas all degrees < y/n, it can be y^-colored by a greedy algorithm. Altogether, only O(y/n)colors are utilized.

Observe that the algorithm actually applies not only to 3-colorable graphs, but, in fact,to every graph in which the neighborhood of every vertex is 2-colorable. Now, bounds onRamsey numbers naturally fit into the local-global framework. For example, the fact that

(see [17] for the sharpest known bounds) answers the following question: given thatthe neighbors of any vertex in G form an anti-clique, what is the best lower bound onthe largest anti-clique in G (answer: n1/2~o(1)). In particular, triangle-free graphs exist of

chromatic number Q(jt1//2~0^). But in a triangle-free graph, the neighborhood of everyvertex is, in fact an independent set, so under the more general assumptions, Wigderson'salgorithm is in fact optimal.

These arguments were further improved by A. Blum [9], who showed how to colora 3-colorable graph with n3/8 colors in polynomial time. Interestingly, Blum's algorithm(which is much more involved than that of [35]) also exploits only local (neighborhoodsof radius 2) properties of 3-colorable graphs. This leads us to ask some questions tocapture the heuristic claim that polynomial-time graph-coloring algorithms can only checklocal properties. We first observe that the answer to Problem 5.2 yields a completely trivialalgorithm to tell 3-chromatic graphs from those not colorable in nc colors. We expect thisalgorithm to be better than Blum's in this respect.

Conjecture 5.3. Let G be an n-vertex graph in which every induced subgraph of order k is3-colorable. Then, #(G) < ny+0^ for some 3/8 > y > 7/20, and where the o(\) term tendsto zero with k —> oo.

An exhaustive algorithm running in time O(nk) can obviously test this condition. It isan interesting possibility that this procedure may be transformed into an algorithm thatactually provides a ny+o(1) coloring.

If this conjecture fails, it may be possible to save it by adding an assumption such as thatthe neighborhood of every vertex is 2-colorable, a condition that is again polynomial-timeverifiable.

A more daring conjecture is:

Conjecture 5.4. If P ^ NP, then no polynomial time algorithm can color every 3-chromaticn-vertex graph with fewer than ne colors for some 9 > 0.

There have been many new and exciting results on the difficulty of approximatingNP-hard problems. The first step in establishing such a result for coloring has been takenby Lund and Yanakakis [28], who establish a separation between coloring numbers ri'[

and nc'2 for some fixed 1 > c\ > c^ > 0. A simpler proof has been provided recently byKhanna, Linial and Safra [21], who also show that it is NP-hard to 5-color 3-colorablegraphs. All this is, obviously, still a far cry from Conjecture 5.4, but some progress in thisdirection is likely to occur in the foreseeable future.

6. Sizes of neighborhoods

Perhaps the most obvious local' information about a graph is the degree sequence,classically characterized by Erdos and Gallai [12]. Briefly, d\ > ...dn > 0 is such asequence iff (i) £d,- is even and (ii) for all 1 < k < n it is the case that YA^J —k(k — 1) + ^7->fcmin{fc,d7-}. The necessity of these conditions is easy to establish and thethrust of the theorem is that they are also sufficient. Pursuing our local-global approachwe ask: what else can be said about the possible rate of growth of (balls in) a graph?

Recall that ft(x) is the number of vertices y whose distance from x does not exceed

458 N. Linial

k. In a connected n-vertex graph, one obtains n integer sequences, one for each vertex,1 =i /?o(x) < /3\(x)... < pn(x) = n. Following Erdos-Gallai's result, it is appealing to ask:

Problem 6.1. Characterize those sets of n integer sequences of the type

that are obtained from connected graphs.

This question, in full generality, is presently too difficult, and at this time one shouldsettle for less. Here are some illustrative special cases of this problem:

— Is it possible to characterize sets of n pairs (3\(x) ^ Pi(x) that come from graphs?One possible approach would be to get sufficient information on squares of graphsand then resort to Erdos-Gallai. Note, however, that Motwani and Sudan [29] haveshown that it is NP-complete to decide whether a given graph is a square.

— In the context of the previous question, it is not hard to derive some necessaryconditions, e.g., that

with equality iff girth(G) > 5. This inequality suggests that there might exist somecomparison theorems between norms of the various vectors /?,• = (/?;(x)|x e V).Obviously, for any fixed x, the sequence 1 = /?o(x) < fi\(x)... < pn(x) — n is unre-stricted. It may be possible to characterize pairs of such sequences, one for vertex xand one for y. Such an analysis could start by considering for any i,j the number ofvertices z that are at distance i from x and j from y.For which parameters is it possible that all x satisfy /3\(x) = d + I, while for everyi < clogn it is the case that /?,+i(x) > (1 +<5)/J,-(x)? This question is clearly relatedto the existence of constant-degree expanders. Methods developed in that area mayprove helpful in studying growth rates of graphs in general.What is the largest girth of a d-regular n-vertex graph? Specifically, is it

This is also an instance of the general problem. We conjecture the answer to benegative. The best current lower bound [26, 8] gives 4/3 instead of the 2.

A problem related to the last item in this list concerns the ratio between girth anddiameter. Consider the distance between two vertices that are antipodal in a shortest cycle.This consideration shows that 2 • diameter(G) > girth(G). Equality holds for even cycles,but what if all degrees are > 3? Examples are known with girth(G) = 2 • diameter(G),where the numbers are small, e.g., the points-lines graph of a projective plane. We are notaware of similar constructions with large girth, so we ask:

Problem 6.2. Consider graphs G with all degrees > 3. What are possible values for thepair (girth(G),diameter(G))l In particular, can their ratio be kept as close to 2 as we wish?

See [14] for a related classical work.

Together with S. Hoori [18], we have recently obtained some results concerning theexistence of a 'center of mass' in both graphs and sets in Euclidean spaces. Namely, weare looking for a vertex x where we can establish a tight lower bound on the numbers

7. Cliques

A number of people have investigated how well the clique number of a graph can beinferred from local behavior. The earliest work we are aware of is by Erdos and Rogers[13]. Recent work on the subject can be found in [10, 24] and the references therein.The main question studied here can be stated in terms of computing, or estimating, thequantities related to the following arrow relation. Say that a graph G has property (p, q)if every set of p vertices in G contains a ^-clique. We say that (/?, q) —• (/, n), if everyG of order > p having the (p, q)-property must satisfy (/, n) as well. The question thenis, for given p,q,n estimate the least / = f(p,q,n) for which (p,q) —• (f,n). The exactdetermination of / includes, as a special case, the exact evaluation of Ramsey numbers,so it is more realistic to ask for estimates, or to settle for special cases. We use both thearrow notation and the function / to describe the results.

Bollobas and Hind [10] concentrate on the case of large p and small q,n. Among theirresults are:

Theorem 7.1. For s > r > 3,

(n,r)^(cns"r+1,s)

for some constant 1 > c > 0. Also, for r > 3 and n large enough

Linial and Rabinovich [24] consider fixed p, q and n tending to infinity. Their main resultbreaks down into three cases, roughly according to whether p/q is smaller than, equal toor bigger than 2.

Theorem 7.2.

— For p <2q — 2 and all n,

f(p,q,n) = n + p-q.

— For p = 2q — 1 and all n> p ,

— For all n> p > q,

(p9q) -> (R(r,n)+p- l,n),

where r = [-^yl, R(r9n) is the Ramsey number and c is an absolute constant. On theother hand,

460 N. Linial

where T is a Turan number: the least number of edges in a p-vertex graph without a

q-anticlique.

All o(l) terms are for fixed p,q and growing n.

References

[I] Alon, N., Babai, L. and Itai, A. (1986) A fast and simple randomized algorithm for the maximalindependent set problem. J. of Algorithms 7 567-583.

[2] Alon, N., Kalai, G., Ricklin, M. and Stockmeyer, L. (1992) Lower bounds on the competitiveratio for mobile user tracking and distributed job scheduling. FOCS 33 334-343.

[3] Awerbuch, B., Berger, B., Cowen, L. and Peleg, D. (1992) Fast distributed network decomposi-tion. PODC 11 169-178.

[4] Awerbuch, B., Goldberg, A. V., Luby, M. and Plotkin, S. A. (1989) Network decompositionand locality in distributed computation. FOCS 30 364-369.

[5] Awerbuch, B., Kutten, S. and Peleg, D. (1992) Competitive distributed job scheduling. STOC24 571-580.

[6] Awerbuch, B. and Peleg, D. (1990) Sparse partitions. FOCS 31 503-513.[7] Awerbuch, B. and Peleg, D. (1990) Network synchronization with polylogarithmic overhead.

FOCS 31 514-522.[8] Biggs, N. L. and Boshier, A. G. (1990) Note on the girth of Ramanujan graphs. J. Combin. Th.

ser. B 49 190-194.[9] Blum, A. (1990) Some tools for approximate 3-coloring. FOCS 31 554-562.[10] Bollobas, B. and Hind, H. R. (1991) Graphs without large triangle free subgraphs. Discrete

Math. 87 119-131.[II] Cole, R. and Vishkin, U. (1986) Deterministic coin tossing and accelerating cascades: micro

and macro techniques for designing parallel algorithms. STOC 18 206-219.[12] Erdos, P. and Gallai, T. (1960) Graphen mit Punkten vorgeschriebenen Grades. Mat. Lapok 11

264-274.[13] Erdos, P. and Rogers, C. A. (1962) The construction of certain graphs. Canad. J. Math. 14

702-707.[14] Feit, W. and Higman, G. (1964) The nonexistence of certain generalized polygons. J. Algebra

1 114-131.[15] Gallai, T. (1963) Kritische Graphen I. Publ. Math. Inst. Hungar. Acad. Sci. 8 165-192.[16] Garey, M. R. and Johnson, D. S. (1979) Computers and Intractability: A Guide to NP-

completeness, W. H. Freeman.[17] Graham, R. L., Rothschild, B. L. and Spencer, J. (1980) Ramsey Theory, Wiley, New York.[18] Hoori, S. and Linial, N. (March 1993) Work in progress.[19] Hurewicz, W. and Wallman, H. (1948) Dimension Theory, Princeton University Press.[20] Karp, R. and Wigderson, A. (1985) A fast parallel algorithm for the maximal independent set

problem. J. ACM 32 762-773.[21] Khanna, S., Linial, N. and Safra, S. (to appear) On the hardness of approximating the chromatic

number. ISTCS.[22] Linial, N. (1992) Locality in distributed graph algorithms. SIAM J. Comp. 21 193-201. (Pre-

liminary version: Linial, N, (1987) Distributive graph algorithms - global solutions from localdata. FOCS 331-335.

[23] Linial, N., Peleg, D., Rabinovich, Yu. and Saks, M. (to appear) Sphere packing and localmajorities in graphs. ISTCS.

[24] Linial, N. and Rabinovich, Yu. (in press) Local and global clique numbers. J. Combin. Th. ser.B.

[25] Linial, N. and Saks, M. (to appear) Low diameter graph decompositions. Combinatorial.(Preliminary version (1991) published in SODA 2 320-330.)


[26] Lubotsky, A., Phillips, R. and Sarnak, P. (1988) Ramanujan graphs. Combinatorica 8 261-278.[27] Luby, M. (1986) A simple parallel algorithm for the maximal independent set problem. SI AM

J. Comp. 15 1036-1053.[28] Lund, C. and Yanakakis, M. (1992) On the Hardness of Approximating Minimization Problems,

manuscript.[29] Motwani, R. and Sudan, M. (1991) Computing roots of graphs is hard, manuscript, Stanford.[30] Naor, M. (1991) A lower bound on probabilistic algorithms for distributive ring coloring.

SI AM J. Disc. Math. 4 409-412.[31] Naor, M. and Stockmeyer, L. (1993, to appear) What can be Computed Locally? STOC 25.[32] Stein, E. M. (1970) Topics in Harmonic Analysis. Annals of Math. Study 63, Princeton University

Press.[33] Stein, E. M. (1985) Three variations on the theme of maximal functions. In: Peral, I. and

Rubio de Francia, J.-L. (eds.) Recent Progress in Fourier Analysis, North-Holland MathematicsStudies 111, North-Holland, 229-244.

[34] Szegedy, M. and Vishwanatan, S. (1993, to appear) Locality-based graph coloring. STOC 25.[35] Wigderson, A. (1983) Improving the performance for approximate graph coloring. J. ACM 30

325-329.

On Random Generation of the Symmetric Group

TOMASZ LUCZAKf* and LASZLO PYBER***+ Mathematical Institute of the Polish Academy of Sciences, Poznari, Poland

* Mathematical Institute of the Hungarian Academy of Sciences, Budapest, Hungary

We prove that the probability /(«, k) that a random permutation of an n element set has aninvariant subset of precisely k elements decreases as a power of k, for k ^ n/2. Using thisfact, we prove that the fraction of elements of Sn belong to transitive subgroups other thanSn or An tends to 0 when n -> oo, as conjectured by Cameron. Finally, we show that for everye > 0 there exists a constant C such that C elements of the symmetric group Sn, chosenrandomly and independently, generate invariably Sn with probability at least 1 — e. Thisconfirms a conjecture of McKay.

1. Introduction

Let nn be a permutation picked randomly from Sn in such a way that each element of Sn

is equally likely to be chosen as nn, and let i(n, k) denote the probability that some set kelements remains invariant under nn. We show that there exists an absolute constant A suchthat i(n,k) ^ Ak~0'01, whenever k^n/2. This fact has been known to have importantimplications for the statistical theory of the symmetric group. In particular, we confirm aconjecture of Cameron [3], showing that only a small part of the symmetric group Sn canbe covered by non-trivial transitive subgroups.

Theorem 1. Let tn denote the number of elements of the symmetric group Sn that belong totransitive subgroups different from Sn and An. Then

lim tjn\ = 0.n ->• oo

This theorem has various applications. A classical result of Dixon [4] states that arandom pair of permutations generates either An or Sn with large probability. As it is very

* On leave from Adam Mickiewicz University, Poznari, Poland. Research partially supported by KBN grant2 1087 91 01.** Research partially supported by the Hungarian National Foundation for Scientific Research, Grant No4267.

464 T. Luczak and L. Pyber

easy to see that such a random pair with probability 1 — 0(1) generates a transitive group,Theorem 1 can be viewed as a natural extension of Dixon's result.

Another consequence of Theorem 1 is mentioned by Cameron. He observed that it wouldimmediately imply his result from [3] that for almost all Latin squares L the groupgenerated by the rows of L is the symmetric or alternating group.

Dixon [5] noticed another application of non-trivial upper bounds for i(n,k). Let us saythat a group G is generated invariably by the elements xx,x2, ...,xm, if the elementsyvy2, ->,ym generate G whenever yi is conjugate to xt for / = 1,2, ...,ra (this definition ismotivated by problems emerging in computational Galois theory (see (5])). Dixon [5]showed the existence of a constant b such that, with probability 1—0(1), b\/\nn randomlychosen permutations generate Sn invariably. Furthermore, he noted that a good upperbound for i(n,k) would imply that O(\nlnn) random permutations suffice. Using ourprevious result, we can prove an even stronger statement, confirming a conjecture ofMcKay (see [5]).

Theorem 2. For every e > 0 there exists a constant C = C(e) such that Cpermutations, chosenfrom Sn uniformly and independently, generate invariably Sn with probability larger than1-e.

Finally, let us mention that our argument does not require the classification of finitesimple groups.

2. Properties of a random permutation nn

In this part of the note we study the cyclic structure of nn, i.e. the asymptotic behaviour ofthe random variables L = L(n) and Ci = Ct(ri), i = 1,2, ...,L, where Cx^ C2 ^ ... ^ CL

denote the lengths of the cycles in the decomposition of the permutation nn.

Claim 1. The following statements hold for a random permutation nn with probability at leastl-o(n-°0b).

(i) \L-\(ii) if i 4=7, then Ct and Cj have no common divisors larger than n09.

Proof. The asymptotic behaviour of L is well studied (see [8] and also [9] and [10]) and (i)follows immediately from known estimates. To verify (ii) note that the probability that nn

has cycles of lengths kx and k2 is bounded from above by

n\(n-kMk1-l)\(k2-l)\(n-k1-k2)\ 1kj\ k2 ) n\ kxk2

Thus, the probability that -nn has two cycles of lengths k19 k2 such that n0'9 kx ^ k2, andfor some d n0'9 both d\kx and d\k2, is smaller than

n In0'1] ln(H] 1 1

ahah

On Random Generation of the Symmetric Group 465

Note that a random permutation nn can be generated in the following way. First chooseh = nrS\) uniformly at random from all elements of [«]. Then, for r ^ 2, pick at. randomir+1 = 7Tn(ir) uniformly from all elements of [n] — {/15 z2,..., ir} until ir = 1 for some rv Havingconstructed the cycle C(1) of nn containing 1, in the same way one may find the cycle C(2)

containing the smallest element of [ri\ — {ix, i2, ...,ir} and so on. Notice that for any /,1 ^ / < L, C(i), Oi+1\ ..., OL) can be viewed as cycles of a random permutation generatedon the set \Jf=iOj) = N\U5=i^ 0 ) ' *-e- before generating C{i) each permutation of the set[«]$JilJC0) is equally likely to appear.

Let C(<) denote the length of C{i) for / = 1,2, ...,L. Then, the probability that C(1) = rx isgiven by

.-mn-ry. 1r1—$ n\ n'

and, consequently, the probability that C{i) = rt is the same for every possible rt and equal

Claim 2. Let m0 = [Vtow]. Then, the probability that C(1), C(2), ..., C(m°} toe <2 commondivisor tends to 0 as n -> oo. Furthermore, with probability tending to 1 as n^ oo, we0i) > n0-99 y^r ^ ^ j / = 1, 2, ..., m0.

Proof. From (1) the probability that some given d divides C{i\ is less than

L

L'(j)

w / L rf

so the probability that d |Q for some 2 ^d^n and all / = 1,2, ...,m0 is less than

[n\

By Claim 1 (i), with probability tending to 1 as n-> oo, we have C(1) ^ 0-5«/ln«. Supposethat C{i) ^ «099 for some 1 ^ / ^ m0. This implies the existence of/, 1 ^ y ^ /— 1, for whichC0 )/C0 + 1 ) ^ exp(0-009 Vto.«). Thus, we have either

orL

^C( O/C( i ) ^ 1+ exp(-0004 VInn),

which, in turn, implies] T C ( O p l-exp(-0-004V

But the probability that for somey any of the above inequalities holds is, by (1), less than

2m0 e x p ( - 0-004 V In n) = 2[Vln«Jexp(-0-004 Vln«) = o(Y). •

Let us define the random variable M(n, k) as the number of cycles one can create beforethe number of 'unused' vertices drops down under k, i.e.

M = M(n,k) = mm\i: £ C ^ ^

Our next result states that, with large probability, Yjf=M(n,k)+iCU) *s n o t t o ° small.

Claim 3. For every k and n, 1 ^ fc1'011 ^ n, the probability that Ef=M(n,*:1011)+iCO) ^ 2£ is

smaller than 2k~001.

Proof. Set R = M(n,k1011). Then the formula for the total probability, together with (1),gives

=r+1) = E P r( E °j) < 2 k

/ £>A:1011 V=r+1

maxjprf f; CO) < 2*

E CO) =

C0) = / ) : / > ki=r

^ max{2k/i:i > k1011} ^ 2k-°'01. D

We conclude this section with an old result of Erdos and Turan [6].

Claim 4. Ifo)(n) -> oo, the probability that the largest prime that divides Y\t=i Ct is smaller than«exp( — G)(n)^Xnri) tends to 0 as n^cc.

3. An upper bound for i(n, k)

This part of the note will be entirely devoted to the proof of the following result.

Lemma. There exists an absolute constant A such that i(n,k) ^ Ak'001 for all 1 k ^ n/2.

Proof. Note first that one only has to show that the assertion holds for k,n>N, where Nis sufficiently large. We split the proof into two cases.

Case 1. n ° " ^ k ^ n / 2 .The proof of this part is based on a rather simple idea - by Claim 1 (i), most permutations

have less than 111 In n cycles, so, one may choose a subset of indices /ls /2, ..., ix in at most2i-iiinn ^ no 77 w a y S xhus, the probability that the sum £ j = 1 Ct attains a particular valuek should decrease as some power of n.

In the proof we shall need some more notation. Let si denote the family of allpermutations of the set [n] that contain less than H l l n / i cycles. Furthermore, for

n099 ^k^n/2 and t = 0,1,..., [nO8\, define 9(k, t) as the family of all permutations suchthat for some i19z2,..., it we have

We will show that for n large enough,

\@(k,0)n^\/\stf\^n-°-02. (2)

Suppose that (2) does not hold, i.e.

«-°-°2. (2)

Fix t and aestf 0 @(k,0). Choose /15/2,...,/, in such a way that XJ=1Q =n — k, andC< ^ Q for 2 ^ 7 ^ / . Let C, denote the length of the longest of the remaining cycles. Note,that because aes/, both Ct and Ci are larger than (k-t)/(l'll\nn) >n09. Below wedenote the cycles of indices ix and j by C and Ci respectively.

Now take t consecutive elements from Ch and add them to the cycle Cp i.e. choosemoeCt and m\eCp and define a new permutation r = r(m^m^ setting:

(i) r ' K ) = o-*K) for i = 1,2,..., r,(ii) r ' K ) = (r^K*) for i = r+ 1, ^ + 2, ...,t+Ci9

(iii) r ' K ) = ^ K ) for i = 1,2,..., C4i-f,(iv) r(m) = o-(m) for m$Ch\} Cr

In this way, for any cres/ n@(k,0), one can obtain ChCj different permutations fromstf fl (/r, 0- Clearly, each such permutation r is a result of the modification of at most(1-11 Inn)\Ch-t)(C5 + t) permutations a from J / n 2{k,Q). Hence

Note that as Ci, C ^ «099 and / ^ «08, for sufficiently large n we have

so, from (2), we get

[^8j \s/ n @(k, oi ^ r ^°8 1 ^ n @(k, 0)1 ^ ^ ft 782 J ^ 0*5 2 ^ 0*5/7 .

On the other hand, every single permutation crsstf can contribute at most 2 l l l l n n + 1 to thesum Yut W H @(k, 01. Hence

Thus, (2) leads to a contradiction and (2) holds.

To complete the proof of the first case it is enough to note that by Claim 1 (i)

) < \9{W)[\st\ n\-\s*\ o{n~^)

Case 2. l^k^ n°".As we have already mentioned, c(i°+1),C(i°+2\ ...,C(L) might be viewed as cycles of a

random permutation on m = YJi=i +1 ^ vertices, where, by Claim 3, with probability atleast 1 -2Ar001, we have 2k < m ^ V 0 1 1 . Thus,

i(n,k)-i(m,k)\ ^ 2/r001,

and, since m°" < k < ra/2, the assertion follows from the part of the Lemma we havealready shown. •

4. Proofs of Theorems 1 and 2

Proof of Theorem 1. Let us first look at primitive subgroups of Sn. Babai [1] (see also [12])showed that the minimal degree of a primitive subgroup not containing A n is at leastWn—1)/2. On the other hand, a well known result of Bovey [2] states that the probabilitythat the minimal degree of a subgroup generated by a random permutation nn is greaterthan na decreases with n as n~a+0{1\ Thus, the probability that nn belongs to a non-trivialprimitive subgroup is «~05+o(1), and, consequently, primitive subgroups contain not morethan n~°'5+0(1) elements of Sn combined.

Now consider permutations with proper blocks, i.e. those permutations a for which onecan find a proper divisor r of n and a partition of the set [n] into blocks A19 A2, ..., Ar suchthat

(i) \AJ = n/r for / = 1,2,..., r,(ii) for every / = 1,2, ...,r there exists an index j such that cr{A^ = Ar

Note that each cycle of a has the same number of elements in each block it intersects.We shall prove that, with large probability, for a random permutation nn such a block

system does not exist. We rule out all possible candidates for the number of blocks r in threesteps.

Case 1. 2 ^ r ^ r0 = exp (InIn« \/\nn).By Claim 2, for every r we can find a cycle C of nn whose length is not divisible by r. Let

B denote the union of all blocks that intersect C. Since r does not divide the length of C,B must be a proper invariant subset of [n] of sn/r elements, for some 1 s < r. By theLemma, the probability that such a subset exists is bounded from above by

] i(n,sn/r) ^ Ar2o(n/roy

001 = n'0009.r=2 s=l

Case 2. r0 = exp (In In n \n n) ^ r ^ n exp (— In In n \/\n n).Since each cycle of a permutation shares the same number of elements with each block

it intersects, Claim 4 implies that the probability that a random permutation has the aboveblock structure tends to 0 as n -> oo.

Case 3. n exp (— In In n ^/\n n) ^ r ^ n.Let C be a cycle longer than n0'99 whose length is not divisible by the block size s = n/r

(the existence of such a cycle is guaranteed by Claim 2), and let B be the union of blocksthat intersect C. Since s does not divide the length of C, C must be a proper subset of B.Thus, nn must contain another cycle C that intersects each block of B. Hence, both C andC should have length divisible by the number of blocks contained in 2?, which is greaterthan

\B\/s ^ \C\/s ^ n°"/s > n°\

contradicting Claim 1 (ii). •

Proof of Theorem 2. Let x1,x2,...,xc denote permutations chosen randomly andindependently from Sn. If xx,x2, ...,xc do not generate invariably Sn, then there existpermutations yvy2, -..,yc, with yt conjugate to xt, such that one of the following holds:

(i) there exists a proper subset of [n] invariant under yt for all / = 1,2,..., C;(ii) y19y2, ...,yc generate a non-trivial transitive subgroup of Sn;

(iii) yl9y2, ...,yc generate An.

However, the probability that (i) holds is bounded from above by Y}k=lKKnik))c, and,from the Lemma and the fact that i{n,k) ^ 2/3 (see [5]), can be made arbitrarily small bychoosing C large enough. Furthermore, by Theorem 1, the probability that yx is containedin some non-trivial transitive subgroup of Sn tends to 0 as «->oo, which rules out thesecond case. Finally, the probability that all permutations yx, y2,...,yc are even is less than2~c and tends to 0 with C. Thus, the assertion of Theorems 2 holds for C = C{e) such that^kZ\(i(n,k)yn < e/3 and 2"c° < e/3 whenever n is large enough to make the fraction ofelements of Sn belonging to non-trivial transitive groups smaller than e/3. •

5. Final remarks and comments

As a matter of fact, using our argument it is possible to show that the fraction of elementsof Sn that belong to non-trivial transitive subgroups decreases with n as n~", for someabsolute constant a > 0. (One only has to estimate more carefully how fast the probabilitiesin Claims 2 and 4 tend to 0. Since the proof is somewhat lengthy and not particularlyinteresting (though not very difficult), we decided to present Theorem 1 in a slightly weakerform). Nevertheless, we cannot prove that, for every e > 0, one can take oc = 0*5 — e, as wasrecently conjectured by Cameron. Indeed, if n is even, the number of possible splits of(1 + o(l)) In n cycles of a random permutation into two groups is at least 2(1+0(1))lnw, so, usingour argument, we cannot approximate the probability that the set can be divided intotwo blocks of equal size by anything better than 2(1+0(1))lnn/« = >r(1+0(1)a foroc = 1 — In2 = 0-30685.... Thus, a proof of the stronger version of Cameron's conjecturewould require either much more detailed analysis or a new method.

Some analogues of Dixon's theorem were recently obtained by Kantor and Lubotzky [7]for finite simple classical groups. This prompts us to ask whether analogues of our resultalso hold for families of groups of Lie type. In [12], Stong proved a number of results for


the statistical theory of the group GL(n,pa), which are somewhat similar to that obtained

by Erdos and Turan for the symmetric group. Thus, the first question to decide seems to

be the following:

Problem. Suppose that p is a fixed prime and n tends to infinity. Is it true that almost all

elements ofGL(n,p) do not belong to an irreducible subgroup not containing SL(n,p)l

Acknowledgements

We would like to thank Boris Pittel and Valentin F. Kolchin who kindly provided us with

suitable references to Claim 1 (i), and Peter J. Cameron for stimulating discussions.

References

[I] Babai, L. (1981) On the order of uniprimitive permutation groups. Ann. of Math 113, 553-568.[2] Bovey, J. D. (1980) The probability that some power of a permutation has small degree. Bull.

London Math. Soc. 12 47-51.[3] Cameron, P. J. (1992) Almost all quasigroups have rank 2. Discrete Math. 106/107, 111-115.[4] Dixon, J. D. (1969) The probability of generating the symmetric group. Math. Z. 110, 199-205.[5] Dixon, J. D. (1992) Random sets which invariably generate the symmetric group. Discrete

Math. 105, 25-39.[6] Erdos, P. and Turan P. (1967) On some problems of a statistical group-theory. II. Acta Math.

Acad. Sci. Hung. 18, 151-163.[7] Kantor, W. M. and Lubotzky, A. (1990) The probability of generating a finite classical group.

Geometriae Dedicata 36, 67-87.[8] Moser, L. and Wyman M. (1958) Asymptotic development of the Stirling numbers of the first

kind. / . London Math. Soc. 33, 133-146.[9] Pavlov, Yu. L. (1988) On random mappings with constraints on the number of cycles. Proc.

Steklov Inst. Math. Ill, 131-142. (Translated from Tr. Mat. Inst. Steklova 111 (1986), 122-132).[10] Pittel, B. (1984) On growing binary trees. / . Math. Anal. Appl. 103, 461^80.[II] Pyber, L. (1991/1992) Asymptotic results for permutation groups. In: Kantor, W. M. and

Finkelstein, L. (eds.) Groups and Computations, DIMACS Ser. Discrete Math. (And TheoreticalComp. Sci. (1993) 11 197-219.)

[12] Pyber, L. (to appear) The minimal degree of primitive permutation groups. Handbook ofCombinatorics.

[13] Stong, R. (1988) Some asymptotic results on finite vector spaces. Adv. Appl. Math. 9, 167-199.

On Vertex-Edge-Critically n-Connected Graphs

W. MADER

Institut fur Mathematik, Universitat Hanover, 30167 Hanover, Weifengarten 1, Germany

All digraphs are determined that have the property that when any vertex and any edgethat are not adjacent are deleted, the connectivity number decreases by two.

1. Introduction and notation

Whereas the characterization of all graphs having the property that the deletion of anytwo edges decreases the connectivity number by two is rather easy, and well known [6] (seeSection 2), the characterization of all graphs with the analogous property for the deletionof two vertices instead of two edges seems to be hopeless. So the following idea suggestsitself. A graph or digraph G is called vertex-edge-critically n-connected (abbreviated ton-ve-critical), if the deletion of any vertex v and any edge e not incident to v decreases theconnectivity number n of G by two (and such v and e exist). If we do not want to specifythe connectivity number, we write vertex-edge-critical or ve-critical. When I determinedthe minimum number of 1-factors of a (2/c)-connected graph containing a 1-factor, theve-critical graphs played an important role and all ve-critical undirected graphs werecharacterized there [2]. It was shown in [2] that every ve-critical undirected graph isobtained in the following way. For an integer m > 1, take vertex-disjoint circuits of lengthm + 2 and vertex-disjoint copies of Km (the complementary graph of the complete graphKm on m vertices) and take all edges between these vertex-disjoint graphs. We will givean easier proof of this characterization in Section 3 by using the characterization of allminimally n-connected graphs with exactly n + 1 vertices of degree n, given in [3]. Themain result of the paper is the characterization of all ve-critical digraphs in Sections 4and 5: every vertex-edge-critical digraph arises from a vertex-edge-critical undirected graphby replacing every edge with a pair of oppositely directed edges.

First we will put together our notation and definitions. A (directed) multigraph G =(V,E) consisting of the vertex set V(G) = V and the edge set E(G) = E may have

472 W. Mader

multiple edges, but no loops. A multidigraph is a directed multigraph. The set of edgesbetween the vertices x and y ( in the directed case, from x to y) in G is denoted by[x,y]G, and, for X, Y c V(G), [X, Y]G := \JxeX,yeY fo^G- Distinct edges from [x,y]G aredistinguished by an upper index, for instance [x,y]1. If G is directed and e G [x,y]G, thenx is the tail and y is the head of e. The set of edges in G with tail in x (head in x) isdenoted by £+(x;G) (E~(x;G)). A grap/z has no multiple edges and is undirected and adirected graph or digraph has no multiple edges of the same direction. For emphasis, wesometimes say undirected (multi-)graph for (multi-)graph. In a graph or digraph we write[x, y] for the edge from x to y. An edge [x,y] of a digraph D is symmetric, if [y, x] G £(D)also, and asymmetric, otherwise. If every edge of a digraph D is symmetric, we call Dsymmetric. In a drawing of a digraph, a pair of symmetric edges is displayed as a linewithout an arrow-head. Edges e G [x,y]G and e' G [ X ' , / ] G of a directed multigraph Gare consecutive if j ; = x' or / = x holds. For a multigraph G, the directed multigraph Garises from G by replacing every edge of G with a pair of oppositely directed edges. Fora directed or undirected multigraph G and a positive integer n, Gn is constructed fromG by replacing every edge of G with n edges. The dua/ of a digraph D arises from D byreversing the direction of every edge of D. The vertex number and the edge number of Gare denoted by \G\ and ||G||, respectively. For a vertex set A, we define AnG := An V(G),and x e G means x G V(G). For ,4 c V(G), the submultigraph of G spanned by A isG(^) := G — (V(G) — A). For undirected G and x G G, we use d(x;G) to denote the degreeof x in G, and N(x; G) is the set of neighbours of x in G. For directed G and x G G, we used+(x;G) (d~(x;G)) to denote the outdegree (indegree) of x in G, and N+(x;G) (N~(x;G))is the set of outneighbours (inneighbours) of x in G. For a digraph D and x G D, wedefine Ns(x;D) :=iV+(x;D)nr(x;D), iVf l

6 (x;D) := Ne(x;D) - Ns(x;D) and d*(x;Z)) :=|A^(x;D)| fore-G { + , - } , and Aa(D) := max{^(x;D) : x G D and e G {+,-}} . A directedmultigraph D is called n-regular if d+(x;D) = d~{x,D) = n for every x G D. If there isno doubt which graph is meant, we drop it in the above notation. N denotes the set ofpositive integers, n is always from N, and Nm := {n G N : n < m} for m > 0.

A path and a circuit in G pass through every vertex of G at most once. If G isdirected, they are continuously directed. For x,y G G, an x,y-path F is a path fromx to y, and for u,v e P such that w is before v on P in the directed case, P[u,v] isthe w,z;-path contained in P and P[M,U) := P[u,v] — {v} =: P[u,v] — v. We considerpaths and circuits as subgraphs, but write them as a sequence of their vertices in theorder passed through (for multidigraphs, in the direction of the path or the circuit). Wesay that the paths Pi , . . . ,P n in G cover G if \JieHl V(Pi) = V(G) holds. In a directedor undirected multigraph G, x,y-paths P,Q are openly disjoint if they are distinct andV(P) n V(Q) = {x,y} holds. The maximum number of pairwise openly disjoint x,y-paths in G is denoted by K(X, y,G). The connectivity number K(G) of G is defined byK(G) := minx^y K(X,y', G) for \G\ > 2 and /c(G) := \G\ — 1 for \G\ < 1. In an analogousmanner, the edge-connectivity number k(G) is defined by edge-disjoint paths. A directedor undirected multigraph G is k-minimally n-(edge-)connected, for k G N, iff ||G|| > k and,for all E' c E{G) with \Ef\ < k, we have K(G -E') = n- \E'\ (A(G -E') = n- \E'\). For

1-minimally n-connected we say minimally n-connected. Let us make precise the definition

On Vertex-Edge-Critically n-Connected Graphs 473

of ve-criticality: a (di-)graph G is vertex-edge-critically n-connected iff K(G) = n > 2, and

for every v G V(G) and e G E(G — v), K(G — v — e) = n — 2 holds.

A sequence v\,[vi,v\],v\, [v2,v\],V2, [v2,V2l,..., [vn,thi],ihi, [v\,v^\ of vertices v\,v\9...9v^

and distinct edges of a digraph D is called an alternating cycle in D. Normally, weomit the edges in the notation and write vf,v\9V29...9v^ for an alternating cycle, wherethe upper index + at vt means that the edges (cyclically) on either side of vt havetheir tails in vt. Sometimes we consider an alternating cycle as a subdigraph of D. IfGi,.. . ,Gn are graphs (digraphs) with V(Gt) D V(Gj) = 0 for i ^ j , the graph (digraph)]T^=1 G,•, = Gi + G2 + • • + Gn is defined as

/n n n I

| J V(Gi\ | J E(Gi) U (J I [x,y] : x G G,- and«=1 '=1 '=1 [

If all G, are isomorphic, we write nG\ := ^" = 1 G,-. If Gi,...,Gw are not vertex-disjoint,we define ^ " = 1 Gt by vertex-disjoint copies of Gu..., Gn. If G = H{ + H2 and E(H{) = 0,we also write G = l^(//i) + H^. For an integer m > 3, Cm denotes an undirected circuitof length m. For integers m ;> 3, k > 0, / > 0, the multidigraph D = CkJ is defined by

: = N m and | [ M ' + 1 ] D I =K | [i H-1, f]z>l = ' for /modulo m.

2. 2-minimally n(-edge)-connected graphs

B. Maurer and P. Slater determined in [6] all 2-minimally n-connected graphs and all2-minimally n-edge-connected multigraphs. We give a simpler proof of the latter result,and show that, in the proof of the former, it is not necessary to use the fact from [1] thatevery minimally n-connected graph has at least n + 1 vertices of degree n.

Let G = (V,E) be a 2-minimally n-connected multigraph. Consider any x G V. Thereare an e G E incident to x, say, e G [x,y]G and a system of n openly disjoint x,y-paths Pi , . . . ,Pn . From K(G — e) < n, it easily follows that jc(x,y;G) = n by Menger'sTheorem. Hence e G ( JU £(^i)- For every ef e E - {e}, K(G - {e,e'}) = n-2 holds andimplies K(x,y;G — {e,ef}) = n — 2 by Menger's Theorem. Hence e' G (J"=1 £(P/) and thus£ = U/Li ^(^/) an<^ ^(x) = n follow. Hence G is finite and n-regular. If \G\ > 3, there isa z G K — {x,y}, and z is on exactly one of the paths Pi , . . . ,Pn , since £ = |J"=1 £(P/)holds and Pi , . . . ,P n are openly disjoint. Hence £ = (J"=1 ^ W ) implies /(z) = 2, and G is2-regular. So we have somewhat generalized a result from Maurer and Slater.

Theorem 1. [6] The only 2-minimally n-connected multigraphs are K\ and, for n = 2, thecircuits Cm.

Let G = (V,E) now be a 2-minimally n-edge-connected multigraph, and choose x G Vand e G [x,y]G as above. Now there are edge-disjoint x,y-paths Pi , . . . ,Pn . As above,E = (J"=1 E(Pf) and rf(x) = n follow. Again, G is finite and n-regular. Let us assume\G\ > 3 and consider z e V — {x,y}. Then every edge incident to z belongs to exactlyone of Pi , . . . ,Pn . Hence n is even and exactly n/2 of Pi , . . . ,P n pass through z. SupposeAT(x) - {y, j ; i , . . . ,^} . Then |G| > 3 and E = \J=l E(Pt) imply {^i,...,^} + 0. Wedefine a directed multigraph D on the vertex set {y\,...,yk}- Every path P7 of lengthat least 2 generates the following edges of D (and there are no further edges in D)\

474 W. Mader

if z is the first vertex of Pj after x, then z G {)>i,...,)>&} and we add the edges [z,u]j

for all u G (Pj — z) n {)>i,... ,)>*}. We prove that D has no circuit. Suppose there is acircuit in D, and this may have the edges [z\,u\yi,..., [zm,um]jm in this cyclic order (henceut = Zj+i). By definition of D, j\9...9jm are distinct, since zi, . . . ,zm are. If we replace P,,with the x, y-path Pj := Pj. [x, zj U P;-_, [w;_i, y] for i = 1,..., m (i modulo m), we get fromPi , . . . , Pn a system P{,. . . , P'n of edge-disjoint x, y-paths in G. But (J"=1 E(P() £ |J"=i £ (^ ' )holds, contradicting the remarks above. Hence D is acyclic and there is a z € K(D) withd~(z\D) = 0. This means that all the n/2 paths Pt containing z have E(Pt) n [x,z]G ^ 0.But this implies |[X,Z]G| = n/2. Considering an e' G [X,Z]G instead of e, we get, in thesame way, a vertex zf ^ z with |[X,Z']G| = n/2. Since G is n-regular and finite, we getG = C"/2. The following theorem summarizes what we have proved.

Theorem 2. [6] The only 2-minimally n-edge-connected multigraphs are K2 and, for even

n>4, also CnJ2.

Obviously, the deletion of two consecutive edges of a digraph cannot decrease theconnectivity number or edge-connectivity number by two (cf. [6]). So it is natural toconsider only the deletion of non-consecutive edges in a digraph. Let us call a multidigraphD weakly 2-minimally n-connected (weakly 2-minimally n-edge-connected), if K(D) = n > 2(X(D) = n>2\ but for all non-consecutive e\ ^= ei from E(D), we have K(D — {ei,^}) =n-2(X(D-{eue2}) = n-2).

Let D = (V,E) be a weakly 2-minimally n-connected multigraph. Choosing [x,y]o i= 0and openly disjoint x,y-paths Pi , . . . ,Pn , we conclude, as above, E — (E~(x) U E+(y)) c(J"=1 E(Pi). Hence D is n-regular and finite. Let us assume \D\ > 3. Then D has no multipleedges, since D is n-regular and K(D) = n > 2. Since only one edge of |J"=1 £(P/) has its

head in z e V - {x,y}, we get n = 2 and \y9z] G E for all z e V - {x,y}. Now D =K3

follows easily.

Theorem Id. The only weakly 2-minimally n-connected multidigraphs are

K% and K3 for n = 2.

Let us now consider a weakly 2-minimally n-edge-connected multidigraph D = (V,E).If [x9y]D ^ 0 and Pi, . . . ,Pn are edge-disjoint x,y-paths, E -(E~(x)uE+(y)) c |J|J=1 E(Pf)follows as above. Hence D is n-regular and finite. Put m := maxxjey |[x,y]p\ and choosex9y G V such that m = \[x9y]D\ holds. If N+(x) = {y} holds, then m = n and E-(E~(x)UE+(x)) c [x, };]/>. But this implies D K% or D Cf. So we assume |N+(x)| > 2.Let Pi , . . . ,Pw be edge-disjoint x,^-paths. As in the proof of Theorem 2, we find a z GN+(x)- {y} such that z G P, implies [x,z]D n£(P,) ^ 0. Set k := |{/ G Nn : z G P/}|. Sincez G iV+(x) and d+(x) = n9 we have k = \[x9z]D\ > 1. Since E-(E~(x)UE+(y)) c y^=1 £(P,)and J~(z) = n, we conclude |Ly,z]^| = n — k. Since m -f /c < J+(x) = rc and n — k < mby choice of m, it follows that n — k = m. Since £ — (£~(x) U £+(z)) is contained inthe n-edge-disjoint x,z-paths of D({x,y,z}), we conclude E~(y) — [x,y]o = [z,y]o andE+(y)~ \y,z]D = \y9x]D9 hence \[z9y]D\ = k = \[y,x]D\. Furthermore, \D\ = 3 follows,since D is ^-regular and k > 1. So D = C^n~k, and we have proved the following theorem.


Theorem 2d. The only weakly 2-minimally n-edge-connected multidigraphs areK" and Ck^n~k

for k G Nn.

3. Vertex-edge-critical undirected graphs

First we will deduce some common properties of undirected and directed ve-criticalgraphs. Subsequently, we will determine all ve-critical undirected graphs.

Let G = (V,E) be an n-ve-critical graph or digraph. Consider an edge e = [x,y] G Eand openly disjoint x,y-paths P\,...,Pn in G. For v G V — {x,y}, K(G — v — e) = n — 2by assumption, hence K(X,y;G — v — e) = n — 2 follows easily from Menger's Theorem.But this implies v G (J?=i V(Pi\ hence V = (J|Li v(pi) a n d G i s finite- O n t h e o t h e r h a n d>if for every edge [x9y] of a graph or digraph G with K(G) = n > 2, every system of nopenly disjoint x,y-paths covers G, obviously G is n-ve-critical. We state this equivalenceformally.

Lemma 1. A graph or digraph G with K(G) = n > 2 is n-ve-critical iff for every edge [x,y]ofG, every system of openly disjoint x,y-paths P\,...,Pn covers G.

From this, the following property is easily deduced.

Lemma 2. Every n-ve-critical graph or digraph is finite and n-regular.

Proof. It remains to show that an n-ve-critical G is n-regular. Consider an edge [x,y]of G and openly disjoint x,y-paths P\,...,Pn in G. Suppose there is an edge [x,z] in Gthat is not on any Pt. Since P\,...,Pn cover G by Lemma 1, there is a Pt containing z,say, z G Pn. Then the openly disjoint x,y-paths Pi, . . . ,Pn_i,F^, where P'n := x,Pn[z,y], donot cover G, contradicting Lemma 1. Hence d(x) = n or d+(x) = n, respectively, and G isn-regular. •

If we delete a vertex v from an n-ve-critical graph or digraph G, then G — v is minimally(n — l)-connected. So one can apply the results on minimally n-connected graphs anddigraphs. By Lemma 2, G — v has exactly n vertices of degree n — 1 or n vertices ofoutdegree n — 1 and n vertices of indegree n — 1, respectively. On the other hand, it is wellknown [1] that a minimally (n— l)-connected graph has at least n vertices of degree n— 1,and in [3] even a characterization of all minimally (n — l)-connected graphs containingexactly n vertices of degree n — 1 was obtained. This permits a straightforward proof ofthe characterization theorem on n-ve-critical undirected graphs, which was first proved in[2] using the fact known from [1] that every circuit in a minimally n-connected graphcontains a vertex of degree n. First, we state the above mentioned result for minimallyn-connected graphs.

Theorem A. [3] For n>2, all minimally n-connected graphs containing exactly n+1 verticesof degree n are obtained in the following way.

(a) For an integer m G Nw U {0}, let H be an (n — m)-regular, (n — m)-connected graph onn + 1 vertices. Then Km + H is minimally n-connected, containing exactly n + 1 verticesof degree n.

476 W. Mader

(b) For an integer m with 4 < m < n, let H be an (n—m)-regular, (n—m)-connected graph onn—\ vertices, and let P be a path with \P\ = m. Then P +H is minimally n-connected,containing exactly n + 1 vertices of degree n.

For characterizing all ve-critical graphs, we need a further lemma.

Lemma 3. If G + H is a non-complete, ve-critical, undirected or directed graph, H is ve-critical or 11if|| = 0 .

Proof. Set n := K(G + if) and m := n — \G\. Using Lemma 2, we see that H is m-regularand m-connected. We assume ||if || > 0. Then ?c(if) = m > 0 holds. Suppose m = 1 andconsider an edge [x,y] G E(H) ^ 0. There are n openly disjoint x,y-paths in G + H({x,y}).This implies |G + if | = n + 1 by Lemma 1, hence G + H is complete. This contradictionshows K(H) > 2. Since for e G E(H), every separating vertex set S of (G + H) — e with\S\ = n — 1 must contain V(G), it is easy to see that H is ve-critical, since G + H is. •

Without difficulty, we now get the following result.

Theorem 3. [2] The vertex-edge-critical graphs are exactly the graphs Gm^j := kKm + lCm+2>where m> 1,/c, / are non-negative integers such that K(Gm^j) > 2 holds.

Proof. Suppose G is a ve-critical graph of the form YM=\ Kmi + Yl)=\ Cnj with m, G N.Since G is regular by Lemma 2, we get immediately that m\ = mi = • • • = m^ andm = m - • - = n\ and n\ = m\ + 2, if k > 0 and / > 0. This implies G = Gmukj- On theother hand, it is easy to check that the graphs Gm ,/ with K(Gm^i) > 2 are ve-critical. So itremains to show that every ve-critical graph has the form ^Kmi-\-^2 Cnj. We will provethis by induction on the connectivity number.

Let G be an (n + 1)-ve-critical graph. If n = 1, then G is a circuit by Lemma 2. Sosuppose n > 2 and choose v G V(G). Then G — v is minimally n-connected and hasexactly n + 1 vertices of degree n, namely N(v; G). So G — v has the structure describedin Theorem A. If G — v = Km + H, as in case (a) of Theorem A, then G = Xm+i + if,where V(Km+\) = V(Km)u{v}. If G is complete or ||ff || = 0, then G has the form wanted.Otherwise, Lemma 3 implies that H is ve-critical, and hence, by the induction hypothesis,H has the form ^Kmi + J2 CnP and hence G does as well. If G — v = P + H, as in case(b) of Theorem A, then G = C + H, where C is a circuit containing v with the propertyC — v = P. By an application of Lemma 3 and the induction hypothesis as in case (a),the proof is complete. •

4. Vertex-edge-critical directed graphs: introduction and preliminaries.

Of course, every n-ve-critical graph G provides an n-ve-critical digraph G. The aim of thispaper is to show that we get every ve-critical digraph in this way, i.e., that every ve-criticaldigraph is symmetric. With regard to Theorem 3, we will then have proved the followingtheorem.

Theorem 4. The vertex-edge-critical digraphs are exactly the digraphs Gm^j with

On Vertex-Edge-Critically n-Connected Graphs All

The proof of this theorem cannot be based on an analogue of Theorem A: only recently[5], I have shown that every minimally n-connected digraph has at least n + 1 vertices ofoutdegree n (that there are at least n such vertices was known before from [4]), but at themoment there is no hope to characterize all minimally n-connected digraphs containingexactly n + 1 vertices of outdegree n and n + 1 vertices of indegree n. However, there isan analogue in the directed case to the fact that a minimally n-connected graph does notcontain a circuit consisting only of vertices of degree exceeding n, which we will statenow.

Let D = (V,E) be a minimally n-connected digraph. Let Do be the subdigraph of Dgiven by V(D0) := {v G V : d+(v\D) > n or d~(v;D) > n) and E(D0) := {[x,y] e E :d+(x;D) > n and d~(y,D) > n}. It was proved in [4] that Do has no alternating cycle.

Theorem B. [4] For every minimally n-connected digraph D, Do does not contain an alter-nating cycle.

To every digraph D, we let correspond a bipartite undirected graph D as follows: take

vertices xr ± x" f o r e v e r v x e V(D) s o t h a t {x',*"} n {/>/'} = 0 n o l d s for x =£ y> a n d

define D by V(D) := \JxeD{x',x"} and E(D) := {[x',/ '] : [x,y] G £(D)}. The followingequivalence is easily seen and was shown in [4].

Lemma C. [4] A digraph D does not have an alternating cycle iff D is a forest.

In the following, D = (V,E) always denotes an n-ve-critical digraph containing anasymmetric edge that has a minimum number of vertices. Our aim is to show thatsuch a digraph cannot exist. By Lemma 2, D is finite and n-regular and \D\ > n + 2holds. Since the dual digraph of a ve-critical digraph is ve-critical again, for every resulton D, there is a dual one, which we will use, but, in general, not state explicitly. Forx G V, H := D — x is minimally (n — l)-connected. So HQ and Ho are defined, and weset Dx := Ho and Fx := Ho - ( { / ' : y G N+(x)} u { / : ^ N~(x)}). Defining R(x) :=V-(N+(x)UN-(x)U{x}l we observe Dx = (V-(Ns(x)U{x}\ [N+(x)UR(x), R(x)UN-(x)]D),since D is n-regular. Furthermore, Fx has the partition F'x := {y' : y G iVj~(x) U i^(x)},^x : = 1^" : ^ G ^ ( x ) u ^ ( x ) } m t 0 independent vertex sets. Since D is rc-regular by Lemma2, we have dj~(x) = d~(x) = : dfl(x), and hence |F^| = |F^|. By Theorem B and Lemma C,Fx is a forest. Theorem B implies the following important properties of D.

Lemma 4.

(a) Ifa~[,ai,a2,...,ak is an alternating cycle ofD, then for every x G V — { a i , . . . , ^ } , f/ierefs an i G N^ swc/z t/zat [a,, x] G £ or [x, a,-] G E holds.

(b ) / / z ^N + (x)UN + ( j ; ) /or distinct x,y,z G 7, r/zen |N+(z)nN+(x)nN+(>;) | > \N+(x)nN+(y)\ - 1.

Proof, (a) If [at,x] £ E and [x,at] £ E holds for all i G N&, then a^,a\9...9ak is analternating cycle in Dx, contradicting Theorem B.

(b) For M ^ v in N+(x) n N+(y), x+, M,y+,u is an alternating cycle in D. If z ^ N+(x) UAf+(.y)u{x,y}, we get [Z,M] G E or [z,i;] G £ by (a). This implies \N+(z)nN+(x)nN+(y)\ >\N+(x)nN+(y)\- 1. D

478 W. Mader

Lemma 5. For all vertices x =/= y of D, the following statements are true:

(a) if [x,y] G E, then \N+(x) n N~(y)\ <n-2;(b) if [x,y] G E, then |JV+(x) n N+(y)\ < n - 2;(c) N+(x) =/= N+(y).

Proof, (a) Suppose [x,y] G E and \N+(x) n N~(y)\ > n — 1. Then there are n openlydisjoint x,y-paths in D(N+(x)u{x}). These paths cover D by Lemma 1, which implies thecontradiction |D| = n + 1.(b) Suppose [x,y] G E and \N+(x)nN+(y)\ > n-\. Let z be the element of N+(y)-N+(x).Suppose z =/= x, and consider a system of n openly disjoint y,z-paths in D. Obviously,these paths cannot contain x. So Lemma 1 implies z = x. But then 5 := N+(x) — {y} —N+(y) — {x} with |S| = n — 1 is separating, since |D| > n + 2 holds. This contradictionproves (b).(c) We suppose there are vertices x =/= y in D with N+(x) = N+(y) = : N. For z GV - (N U {x,y}), we get |N+(z) n N\ > n - 1 by Lemma 4 (b). Hence (b) implies[z,x] ^ E and [z,y] £ E and, therefore, N~(x) = N = N~(y) holds. Suppose thereis an edge [z,z] G E(D — (N U {x,j;})) and consider n openly disjoint z,z-paths in D.Since \N+(z) n N\ > n — 1 and N+(x) = N+(y) = N, these paths cannot contain boththe vertices x and y. So Lemma 1 implies \\D — (N U {x,y})|| = 0. In particular, forz G 7 - (N U {x,y}), we get AT+(z) = N and so also AT(z) = N. Altogether, we haveshown D = (V-N) + D(N). Hence ||D(N)|| = 0 holds or D(N) is ve-critical by Lemma 3.If ||D(N)|| = 0 holds, D is symmetric, contrary to our assumption. So D(N) is ve-critical.But then D(N) is symmetric by choice of D as a minimal counterexample, hence D issymmetric as well. This contradiction proves (c). •

Lemma 5 (a) and (b) mean that for every x G V, the maximum outdegree and maximumindegree of D(N+(x)) are at most n — 2, and, dually, the same holds for D(N~(x)). Wenow deduce some properties of Fx from Lemma 5.

Lemma 6.

(a) For every v G Fx, d(v;Fx) > 1 and for every v G Fx — \JyeR(x){y'9y"}, d(v;Fx) > 2 holds.(b) For F = Ff

xandF = F'xf, \{v G F : d(v;Fx) = 1}| = c + E ( ^ ; ^ x ) - 2)

c denotes the number of components of Fx.

i€F

d(v,Fx)>3

Proof, (a) By duality, it suffices to consider a v e Fx. Then there is a z G Nj~(x) U R(x)such that zf = v holds. Since [z,x] ^ £, we have d(z'\Fx) = n - \N+(z;D) n N+(x;D)\.So Lemma 5 (c) implies d{z'\Fx) > 1. Assume z G iV+(x). Then d(zf;Fx) > 2 by Lemma5(b).(b) Since F_£, F^ is a bipartition of the forest Fx into independent sets F . and Fx of thesame cardinality, we get

J 2 d(v; Fx) = \\FX\\ = \ F x \ - c = 2\F'X\ - c = ( ^ 2) - c.veF'x veF'x

This proves assertion (b), since there are no isolated vertices in Fx by (a) and the caseF = F'l is dual. •

Lemma 6 (b) provides at least one vertex z G N+(x) U R(x) ^ 0 with d(z'\Fx) = 1,and by Lemma 6 (a), every such vertex is in R(x). So there is a vertex z G R(x) with|N+(z) 0 N+(x)| = n - 1, and we define K+(x) := {z € #(x) : |JV+(z) n N+(x)| = w - 1}.R~(x) is defined dually as {z G R(x) : \N~(z)nN~(x)\ = n—1}. We emphasize once againthat R+(x) ^ 0 and R~(x) ± 0.

We need a series of preliminary lemmas. Herein, xo always denotes any vertex of D.

Lemma 7.

(a) Ifxe N+(x0) such that R~(x0) <£ N+(x) holds, then |AT(x) n AT(xo)| = n-2 and\N-(x)n(N+(xo)UR(xo))\ = L

(b) For all x G N+(x0) bat at mosf on^, |N~(x) Pi (N+(x0) U R(XQ))\ = 1 holds.

Proof, (a) Suppose there is z G i?~(xo) — N+(x). Since [x,xo] ^ ^? w^ can apply the dual of

Lemma 4 (b) for x0, z, x, and get |N"(x) nN"(x 0 )nN"(z) | > |iV"(x0)nN"(z)| - 1 = n- 2,by definition of R~(xo). This implies \N~(x)C\N~(xo)\ = n — 2 by the dual of Lemma 5(b),so |AT(x) n (N+(x0) UK(xo))| =n- \N~(x) nN"(xo) | - 1 = 1 follows.

Since for every z G K~(xo), the definition of R~(xo) implies that \N~(z) O Nj"(xo)| < 1holds, (b) follows from (a), since R~(xo) = 0. D

Lemma 8.

(a) Ifve R+(x0), x G N+(x0) - N+(v), and y G N~(x) n (N"(x0) U K(x0)), f/zen [u,y] GE or y e R+(x0) holds.

(b) / / x G N+(x0) such that \R+(x0) - N~(x)\ > 2 holds, then N~(x) n N"(x0) = 0.

Proof, (a) Suppose [v,y] £ E. Since [xo,.y] ^ E also, we can apply Lemma 4 (b) toxo,v,y and get \N+(y) n N+(x0) n N+(t;)| > |N+(x0) n N+(t;)| - 1 = n - 2. This implies\N+(y) n N+(x0)| > n - 1, since x G N+(y) - N+(t;) holds. Hence, by Lemma 5 (b),y £ N~(xo) holds, so y G R(xo) and even y G i^+(xo) by Lemma 5 (c).(b) Suppose there are v\ ± vi in K+(xo) — N~(x) for an x G N+(xo), and there is ay G i\T(x) n N"(xo). Then (a) implies [vi9y] G £ for i = 1,2. Since N+fe) n N+(x0) =N+(xo) — {x} for / = 1,2, we get N+(i?i) = N+(vi), which contradicts Lemma 5 (c). •

For every [x,y] G £, we have |N+(x) n N~(y)\ <n — 2by Lemma 5 (a). Let us assumeequality holds. Then (D — [x,y]) — (N+(x) Pi N~(y)) has exactly one x,y-path, since everysuch path does contain V — (N+(x)nN~(y)) by Lemma 1. This path has length at least 3.

Lemma 9. Let [x,y] G E be such that S := N+(x) n N~(y) has exactly n — 2 vertices, andlet P : x,Xi,...,Xfc,Xfc+i be the x,y-path in (D — [x,y]) — S. Then the following statementsare true.

(a) N+(xi) = {x,x2} US and N~(xk) = {xk-Uy}vS;(b) S <= N~(x)-+ [y,x] G£.

480 W. Mader

Proof. Since, by Lemma 1, Af+(xi) n {x3,...,X£+i} = 0 holds, we must have N+(x\) cV — {x3,...,Xfc+i}, hence Af+(xi) = {x,X2} US. The other claim in (a) follows by duality.Now suppose S c N~(x). Then there are n— 1 openly disjoint xi,x-paths in D({xi,x}uS)by (a). There i s a z E N~(x)D {x2,...,Xfc+i}, and by Lemma 1, P[xi,z],x does contain

}, which implies z = Xk+\ = y, and hence (b). •

If we assume that [x, y] G E is asymmetric, and that Aa(D) = 1 holds, then S :=JV+(x) n N~(y) is a subset of N~(x), and Lemma 9 (b) implies |S| < n — 3. It is possibleto improve this result.

Lemma 10. Let [x,y] G E be asymmetric and assume Aa(D) = 1. Then

(a) \N+(x) n AT Ml < n - 4, am/(b) |N+(x) nN+(y)\ <n-3 hold.

Proof, (a) We suppose S := N+(x) n N~(y) has at least n — 3 elements. Since Aa(D) = 1holds, [x,y] is the only asymmetric edge with tail in x. Hence, there is exactly oneasymmetric edge in D with head in x, say [/ ,x]. In particular, we see S c N~~(x) and,dually, S c N+(y). Then Lemma 9 (b) implies \S\ = n—3, since [x,y] is asymmetric. Hence,there are two openly disjoint x,y-paths Pt : x,x'1?...,x^+1 (i = 1,2) in (D — [x,y]) — S.Furthermore, &,- > 2 holds, and Pi,P2 cover D — S by Lemma 1, in particular, / ePiix2>y) ^Pi[x\->y)' First, we prove a few properties.

(1) S U {xi+1} £ JV+(xj) for i = 1,2 (mod 2).

Suppose Su{xf} c Af+(x}). Then Su{xf} c N+(x})nAT-(x) holds. Applying Lemma 9to [x},x] e E, we get from the second equality in Lemma 9 (a) the contradiction that[/, x] is symmetric.

(2) F o r / = 1,2 (mod 2), [xi,x^+1] <E £ and |N+(xi) n (S U {xi+1})| = n - 3 hold.

By (1), there is at least one edge from x\ to Pi[xl3,y] U Pi+i[x^\y] for / = 1,2. By

Lemma 1, this can be only the edge [x^x^1] , since kt > 2. Hence (2) follows.Dually, we get [ x [ r l , x ^ ] G E for j = 1,2 (see Figure 1).

(3) S ^N+(x\) for i= 1,2.

Suppose, for instance, S £ N+(x{), say s e S - N+(x\). Then Sr := (S - {s}) U {xf} cN+(xj) holds by (2). Set Df := (D - [xj,x]) - S;. If / ^ {x^xj^}, we can easily findopenly disjoint x},x-paths Q\ with / e Q\ and ^2 with [ ,5] e £(62) in I>/ such thatQi n {X^,XJT2} = 0 and \Q2 n {x^,x^2}| = 1, contradicting Lemma 1. Hence, y' e {x^,x^}holds. Suppose there is a z ^ / in"N~(s) n ( P i [ x ^ ) U P 2 [x^ ) ) . If {/,z} ^ K(Pi) and{/,z} ^ I/(P2), using [x},X2] G £, we get, obviously, openly disjoint x},x-paths 2i,Q2 in

Df with y ^ F(gi) U V(Qi), contradicting Lemma 1. So {/,z} c K(Pi) or {/,z} c F(P2)holds. Then we find, again, two openly disjoint x},x-paths in Df — y, namely in the formercase (then x^ = / ) the paths x},P2[x2,x^2_1],/,x and Pi[x},z],5,x, and in the lattercase (then XJ?2 = / ) the paths x},P2[x2,z],5,x and P\[x\,xl

ki_{],y',x. (Note that ki > 3in the former case, since in this case k\ > 3, hence X2 ^ x\ holds, but there is onlythe edge [x^x^] from x^ to Pitx^,}7] by (2)). This contradiction with Lemma 1 shows


Figure 1

N (s) = (S — {s}) U {x,y,y',x2{}. Now it is easy to find in D n openly disjoint x,s-paths

not containing {XJ^XJ^} — {/}. This contradiction to Lemma 1 proves (3).We may assume y' e Pi, hence y' G Pi[x\,y). If [x^xj] G E holds, by using (3) and

(2), it is easy to find n openly disjoint xf,x-paths in D — y, contradicting Lemma 1.So [x},*^] G E is asymmetric. Hence [x^x^] G E is symmetric, since Aa(D) = 1, thatis [x2,Xj] G E holds. By using (3) and (2) again, we now easily find n openly disjointx},x-paths in D — y. This contradiction to Lemma 1 proves (a).(b) If \N+(x)nN+(y)\ > n-2 holds, then \N+(x)nN-(y)\ > n-3 follows, since Aa(D) = 1,thus at most one of the edges [y,z] for z G N+(x)nN+(y) is asymmetric. Hence (b) followsfrom (a). •

Lemma 11. D > n + 4 holds.

Proof. If Aa(D) > 2, we choose an x0 with da(x0) > 2 and get \N+(x0) U N~(xo)\ > n + 2,hence \D\ > n + 4, since R(xo) ^= 0. If Aa(D) = 1, we choose an asymmetric edge [x,y] G £and get |N+(x) U N+(y)\ > n -h 3 by Lemma 10 (b), hence \D\ > n + 4. •

Lemma 12. If da(x0) > 0,

(b) > 2

Proof. Since |Af+(i;)nN+(xo)\ = n—\ for v G K+(xo), it suffices to prove (a). Suppose thereis an x G N+(XO)-\JVGR+{XO) N+(V). There is a v e K+(x0), and N+(v) — N+(xo) has exactlyone element, say, z. Consider any y G K — (JV+(xo)U{xo,^,z}) and suppose [j;,x] G E. ThenLemma 8 (a) implies y G K+(XQ), contradicting the choice of x. This contradiction shows

482 W. Mader

N (x) c JV+(x0) U {x0, z}, hence |AT(x) n N+(x0)| = n - 2 and [2, x] e E by Lemma 5 (a).Set S := AT(x) n N+(x0), and let y be the vertex of N+(x0) - (S U {x}) (see Figure 2).We apply Lemma 9 (a) to [xo,x] G E and to the only xo,x-path P : xo,j>,...,z,x in(D - [xo,x]) - S, and get S <= AT(Z) and [y,x0] € E, [x,z] G £. Since [i?,z] G £ andS c AT+(t;) 0 AT(z), the path i>,y,xo,x,z must cover D - S by Lemma 1. But this implies\D\ = n + 3, contradicting Lemma 11. Q


As in section 3, let D = (V, E) be a minimal counterexample to Theorem 4, and n := K(D).We will show that D cannot have an asymmetric edge. First we prove that D must containmany symmetric edges.

(1) Aa(D) < 2.

Suppose there is an x0 G V with dfl(x0) > 3. Since \R+(x0)\ > 2 by Lemma 12 (b), Lemma7 (b) implies da(x0) = 3 and \R+(x0)\ = 2, since \N+(v) n N+(M) n N+(xo)| > da(x0) - 2 foru £ v in K+(x0). Set JV+ = {xi,x2,x3} and choose v G K~(x0) ^ 0. Then |N~(i;)nNj"(xo)| <1, say, [xi9v] <£ E for i = 1,2. Hence, Lemma 7 (a) implies |N-(x,) n AT(xo)| = n - 2 andthus |#-(x,-) n K+(xo)| = 1 by Lemma 7 (b) (or 12 (a)) for i = 1,2. Therefore, wehave d~(Xi;D(N+(xo)» = 0 for i = 1,2. Furthermore, \N~(xi) n R+(x0)| = 1 for / = 1,2implies #+(x0) c JV""(x3). Since [x3,xj] ^ £, and [x3,x0] <£ £, we get from the dualof Lemma 4 (b) that |AT(x3) n N~(x0) n N~(xi)| > |AT-(x0) n N~(xi)| - 1 = ^ - 3 ,so N~(x3) c AT-(xo) U {x0} U R+(x0). Together, we have seen ||Z>(N+(xo))|| = 0 and sod(x;;FX0) > 3 for i G N3. But this implies |^+(x0) | > 4 by Lemma 6 (b) and (a). Thiscontradiction proves (1). •

In the next step we show the following.

(2) Aa(D) = 1.

Suppose Aa{D) > 2, hence Aa(D) = 2 by (1). Let x0 G K with dfl(x0) = 2, say,

Figure 3

N+(x0) = {xx2} and N~(x0) = {yi,y2}. By Lemma 12 (b), |#+(x0)| ^ 2 holds. Considerany v G R~(x0) + 0- Since \N-(v)nN+(xo)\ < 1, say, [x2,v] <£ E. Then |Ar(x2)nAr(x0) | >n — 2 by Lemma 4 (b). Since |JV~(x2) n JVs(xo)| > rc — 2 is not possible by Lemma 9 (b),iV~(x2) n Af-(xo) 7 0 follows, say, [y2,x2] G £. Since |N"(x2) n (N"(x0) U {xo})| > n - 1and ^2 € N~(x2), Lemma 8 (b) implies |#+(x0)| = 2, say, R+(x0) = {v\,v2} and |N~(x2) nK+(x0)| = 1, say, t;2 G N"(x2). Hence [n,x2] £ £, so N+(t;i) n JV+(x0) = Ns(x0) U {xi},and since y2 G N~~(x2), we get [tfi,)>2] G £ from Lemma 8 (a). Hence [ui,yi] i E, andLemma 8 (a) implies [yi,x2] £ E, hence \N~(x2)nNs(xo)\ = n—3. Furthermore, [ 1, 1] ^ Eimplies \N+(y{) n N+(v{) D N+(xo)\ > n - 2 by Lemma 4 (b). Since Ns(x0) £ iV+(y1)is not possible by the dual of Lemma 9 (b), the last inequality implies [yi,xi] G £.(So far, we have got the edges without or with one arrow-head in Figure 3.) Since[xi,x2] £ E and |AT(x2) n iV(xo)| = n - 2, we get |N~(xi) D N~(x2) n N~(xo)| > w - 3from Lemma 4 (b). Since {xo,i^i,yi} ^ N~(x\)9 but {xo,^i,yi} H N~(x2) n N~(x0) = 0,we conclude [i;2,xi] ^ £, hence Ns(xo) c N+(v2). So Lemma 8 (a) implies, as above, that[i;2, j>i] G £ and [y2,*i] ^ £, hence |JV"(xi) n Ns(x0)| =? w - 3.

Since |K~(xo)| = 2 holds by duality, there are only two z G F"o with d(z;FXo) = 1 byLemma 6 (a), and so Lemma 6 (b) implies d(y";FXo) < 2 for 1 = 1 or i — 2. Supposed(y";FXo) < 2. This means \N~(yi) D N~(XQ)\ > n — 2. Now we will point out n openlydisjoint i;i,xi-paths that do not cover D. If z denotes the vertex of NS(XQ) — N~(x\), then

2 hold, since N~(y\) Pi {z,y2} =/= 0. So we get n openly disjoint t;i,xi-paths that do not

484 W. Mader

contain x2 (and v2). This contradiction to Lemma 1 proves Aa(D) < 1, since the cased(y2',FXo) < 2 is analogous. Since D is not symmetric, Aa(D) = 1 follows.

By (2), there is an x0 G V with da(x0) = 1, say, JV+(x0) = {x} and N~(x0) = {y}. Forsuch an xo, we now prove the following.

(3) R+(x0) = R-(x0) = R(x0), \\D(R(xo))\\ = 0, and R(x0) <= N+(x) n ATM hold.

Choose t; G K+(x0). If [v9y] £ £, then \N+(y) n N+(xo)l > n - 2 by Lemma 4 (b),which contradicts Lemma 10 (b). Hence y G N+(v) and N+(v) c AT+(xo)UiV~(xo) follows.Consider z G #(xo) — {v}. Since z ^ N+(v), Lemma 4 (b) implies

|N+(z) n N+(x0) n N+(t;)| > n - 2. (a)

Let us suppose [z,v] G £. Then [z,v] is an asymmetric edge with \N+(z)nN+(v)\ > n—2,contradicting Lemma 10 (b). Hence N~(v) c N+(xo) U N~(XQ) follows. This impliesi; G #~(xo) a n d t x ^ ] G £ by Lemma 5 (c). So we have shown R+(xo) c K~(xo). Since theother inclusion is dual, we get R+(XQ) = R~(XQ). Furthermore, we have seen

N+(v)UN~(v) c N+(x0)UN-(x0) for all v G K+(x0), (j8)

hence ||D(K+(x0))|| = 0, and R+(x0) £ N+(x) n N"(>;).So it only remains to prove R(xo) = R+(xo). Suppose R := R(xo) — R+(xo) ^ 0. First

we show

N+(Zl) n N+(x0) = N+(z2) n N+(x0) for all z h z2 G R (y)

We can choose v\ =/= v2 from R+(XQ), since |K+(xo)| > 2 by Lemma 12 (b). Sincey G i V > i ) n N+(v2\ Lemma 5 (c) implies N+(v{) n N+(x0) ^ N+(i;2) n N+(x0). Using (a)for t;i and r2, we conclude N+(z) n N+(xo) = N+(^i) H N+(i;2) n N+(xo) for every z E K ,since |N+(z) n N+(x0)| < n - 2. This implies (y).

Let us now consider D := DXo - K+(x0) and F := FXo - \JveR+{xQ){v'>v")- S i n c e \N+(Z)n

N+(x0)| < n - 2 and |N~(z) n N-(xo)| < n - 2 for Z G I = R(X0) - R~(x0), using ()8), wesee d(z;T) > 2 for all z G 7(F) - {*',/'}. Since F is a forest with |F| > 4, it must be anx',/ '-path. There are zi?z2 G K, such that [x',z'{] G £(F) and [zf

2,y"] G £(F) hold. Using(a) and (y), Lemma 10 (b) implies, that DXo(R) is symmetric. This implies z\ = z2, sinced+(z;D) = d~(z;D)(= 2) holds for every z e R. Then the undirected graph G with theproperty G = DXo(R) has one vertex of degree 1, namely z\, and all the other vertices ofdegree 2. This contradiction shows R = 0, and (3) is proved.

By Lemma 12 (b) and (3), r := \R(xo)\ = |#+(x0)| ^ 2 h o l d s - Define S := N~(y) nN+(xo) and T := N+(x0) - S. Since K(x0) c N-(y) by (3), ATM = K(x0) U S, hence|S| = n - r and |T| = r hold. Since 4(y) = 1, and thus N+(y) - N~(y) = {x0}, weconclude R(y) = T. Since ||D(T)|| = 0 by (3), using Lemma 5 (c), we immediately havethe following.

(4) For every t G T, \N+(t) n(SU R(xo))\ >n-\ and for all t\ ± t2 from T, S U R(x0) ^

holds.

Now we will complete the proof of Theorem 4 by constructing for an edge [w, v] in D nopenly disjoint u,t;-paths, that do not cover D. This contradicts Lemma 1.


First, we assume there is a zo G K(xo) with N+(zo) ^ 5. Then [zo,y] G £ by (3).Since z0 G R+(xo) by (3), there is only one vertex in S — N+(z0), say so, and T ciV+(z0) holds. Then K(zo,y;D({zo,y} U (S - {s0}))) = n-r holds. By (4), it is easyto find r disjoint edges in [T,(R(xo) — {ZQ}) U {SO}]D> since r > 2 holds. This impliesK(zo,y;D({y,so} U R(xo) U T) — [zo,y]) = r. Together, this gives n openly disjoint zo,y-paths not containing x0- Since the asymmetric edge [y,xo] was arbitrary, this contradictionestablishes our next claim.

(5) For every asymmetric edge [u,v] G E and every t G R(v), N~(u)nN+(v) c N+(f) holds.

For every z G R(XQ), therefore, S c N+(z) and thus K(z,y;D({z,y}uS)) = n—r + 1 holds.Choose z0 G R(x0) and define V := iV+(zo)nT. Then |T' | = r - 1 holds. If r > 3, we get, asabove, r - 1 disjoint edges in [T',R(xo)—{zo}]D, hence K(zo,y;D({y}uR(xo)UT')-[zo,y]) =r— 1, so there are w openly disjoint zo,y-paths not containing xo. This contradiction showsr = 2.

So we have \S\ = n — 2, and Lemma 10 (b) shows that N+(y) 3 S is impossible. Hencethere is an asymmetric edge [so,j>] G [S, {>>}]D- Since da(y) = 1, all the edges [£(xo), {y}]Dare symmetric, hence .R(xo) ^ N+(y) holds by (3). So we have R(xo) c N~(so) n N+(y),and (5) implies i^(xo) ^ N+(t) for every t G K(y) = T. So we see fc(zo,y;/) — xo) = n, sincefc(zo, >';/)({}'} U R(xo) U Tr)) = 2. This contradiction to Lemma 1 completes the proof ofTheorem 4.

References

[1] Mader, W. (1972) Ecken vom Grad n in minimalen rc-fach zusammenhangenden Graphen.Archiv der Mathematik 23, 219-224.

[2] Mader, W. (1973) 1-Faktoren von Graphen. Math. Ann. 201, 269-282.[3] Mader, W. (1979) Zur Struktur minimal rc-fach zusammenhangender Graphen. Abh. Math. Sem.

Universitat Hamburg 49, 49-69.[4] Mader, W. (1985) Minimal n-fach zusammenhangende Digraphen. J. Combinatorial Theory (B)

38, 102-117.[5] Mader, W. (in preparation) On vertices of outdegree n in n-minimal digraphs.[6] Maurer, St. B. and Slater, P. J. (1978) On /c-minimally n-edge-connected graphs. Discrete

Mathematics 24, 185-195.

V

On a Conjecture of Erdos and Cudakov

A. R. D. MATHIAS

Peterhouse, Cambridge

Let / be an arbitrary function from the set of positive integers to the set {—1,+1}, Ca negative integer and D a positive one. We write / < D if for all positive m and d,Y^k=i f(kd^ < ^ a n d c « / if f o r a11 positive m and d, C < Y^=i f(kd).In this terminology, a question popularised by Erdos runs:

Are there /,C,D with C < / < D ?

The purpose of this note is to prove that for D = 2, no such / and C exist:

Proposition. Let f: {1,2,3,...} -> {-1,+1} be such that for all m,d, Y!k=\f(kd) < *•Then for all C < 0, there are m, d with

i

k=\

Proof. For each x ^ 1, f(x)+f(2x) = —2,0 or 2, but the last possibility is excluded by thecondition on f, so / (x)+/(2x) ^ 0. Thus (j)(n) = S"= 1( /(x)+/(2x)) *s a weakly decreasingfunction of n. If for some n, </>(ri) < 2C, then either YH=i f(x) < C or £ " = 1 /(2x) < C.We may therefore suppose that for all n, (j>(n) ^ 2C, and hence that 0(n) is eventuallyconstant. Thus there is a d such that for all x ^ d, /(x) + /(2x) = 0, and because / takesthe value —1 infinitely often, we may, without loss of generality, assume that f(d) = — l.Wenow consider only multiples of d, and since the original condition on / , together with thefact that for all /c, f(2kd) = —f(kd), imply that for each m ^ 1 and each multiple d' of<*> - 1 < Y!k=\f(kd') < +1» we find that the values of/(fcd) and /(2W) for k = 1,..., 12must be as in the following table; but then Ylt=\ fi^kd) = +2, a contradiction.

k 1 2 3 4 5 6 7 8 9 10 11 12f(kd) - + + - + - - + + - - +

/ ( 2 W ) + _ _ + _ + + _ _ + + _

Remark 1. By compactness the problem may be stated in terms of finite sequences, as isdone in [2] Probleme 49, a negative answer to the above question being equivalent to theassertion that for all D > 0 there is an JV > 0 such that for all / : {1,2,..., N} - • {-1,1}

488 A. R. D. Mathias

there are m > 1, d ^ 1 with md ^ N and |XX=i / ( ^ ) | ^ D. In [2] reference is made tothe paper [1] which studies related questions.

Remark 2. Repeated attempts to improve the proposition to the case D = 3 have failed.Indeed the following is still open:

Is there an / with - 3 < / < +3 ?

References

[1] Cudakov, N.G. (1956) Theory of the characters of number semigroups, Journal Ind. Math. Soc.20 11-15.

[2] Erdos, P. (1963) Quelques Problemes de la Theorie des Nombres. Monographies del'Enseignement mathematique, No 6, Geneve.

A Random Recolouring Method for Graphs andHypergraphs

COLIN McDIARMID

Department of Statistics, University of Oxford

We consider a simple randomised algorithm that seeks a weak 2-colouring of a hypergraphH; that is, it tries to 2-colour the points of H so that no edge is monochromatic. If H hasa particular well-behaved form of such a colouring, then the method is successful withinexpected number of iterations O(n3) when H has n points. In particular, when applied toa graph G with n nodes and chromatic number 3, the method yields a 2-colouring of thevertices such that no triangle is monochromatic in expected time O(n4).

A hypergraph i / o n a set of points V is simply a collection of subsets E of V, the edgesof H. A d-graph is a hypergraph in which each edge has size d. A weak 2-colouring of ahypergraph is a partition of the points into two 'colour' sets A and B such that each edgeE meets both A and B. Deciding if a 3-graph has a weak 2-colouring is NP-complete[6, 4].

The following simple randomised recolouring method attempts to find a weak 2-colouring of a hypergraph H. It is assumed that we have a subroutine SEEK, which oninput of a 2-colouring of the points outputs a monochromatic edge if there is one, andotherwise reports that there are none.

RECOLOUR

start with an arbitrary 2-colouring of the points

while SEEK returns a monochromatic edge Epick a random point in E and change its colour.

If H has a weak 2-colouring, the method will ultimately find one with probability 1:indeed if if has n points and maximum edge size d, at any stage the probability of successwithin the next n/2 steps is at least d~n^2. What is of interest is the expected number of

490 C. McDiarmid

iterations needed. A fair partial 2-colouring of a hypergraph is a pair of disjoint 'colour'sets A and B of points such that for each edge E we have

\EC\A\ = \EHB\>0.

Theorem 1.Let H be a hypergraph with maximal edge size d, and suppose that H has a fair par-tial 2-colouring that colours m points. Then RECOLOUR returns a weak 2-colouring withexpected number of iterations at most \dm2.

Proof. By assumption there is a (fixed) pair of disjoint subsets Ao, Bo of points such that\A0\ 4- \Bo\ = m and \E n AQ\ = \E n BQ\ > 0 for each edge E of H. Suppose that at somestage we have a 2-colouring / of if, that is a partition of the points into two sets A andB. Define the agreement number N(f) of / to be \A n AQ\ + \B n BQ\.

Suppose that SEEK returns a monochromatic edge E. Note that we must have 0 <N(f) < m. Let / be the random colouring obtained from / by changing the colour ofa point picked uniformly from £, as in the algorithm RECOLOUR. Then, if we letk = \EHA0\ = \EnB0\,

{ - 1 with probability k/\E\+1 with probability k/\E\0 with probability 1 -2k/\E\.

Denote the colourings produced by the algorithm by /o , / i , . . . . Then we see thatN(fo),N(f{)9... is a symmetric random walk with 'holding', up to a stopping time.Further, this stopping time is no later than when such a walk would be absorbed at 0 orm. But the expected number of unit steps to absorption here is at most m2/4, and eachexpected holding time is at most d/2, so the expected number of iterations is at most\dm2 (see for example [3]). •

Comments

The algorithm RECOLOUR is similar in spirit to the randomised method proposedby Petford and Welsh [7] for seeking a proper 3-colouring of a graph (see also [1, 2, 8]).A strong k-colouring of a hypergraph is a colouring of the points with k colours suchthat in each edge the points all receive distinct colours. Let d > 2, and let H bea d-graph with n points and with a strong d-colouring. Then H has a fair partial2-colouring with at most 2n/d points coloured, and so RECOLOUR will yield a weak2-colouring in expected number of iterations at most n2/2d. We consider the particularcases d = 2 and d = 3 below.Let G be a connected bipartite graph with n nodes. It is of course easy to find a proper2-colouring of G. However, suppose that we do apply RECOLOUR to G to obtain aproper 2-colouring, starting from a random 2-colouring. Then the expected numberof iterations is exactly \n(n — 1).Let G be a graph with n nodes that has a proper 3-colouring. Then we may use

A Random Recolouring Method for Graphs and Hypergraphs 491

RECOLOUR to obtain a 2-colouring of the nodes such that no triangle is monochro-

matic, in expected number of iterations at most n2/6 and thus in expected time O(n4).

Of course, if we could find a proper 4-colouring of G, we could amalgamate colours to

obtain a 2-colouring without monochromatic triangles. However, it has recently been

shown [5] that it is NP-hard to find a proper 4-colouring in a graph with chromatic

number 3.

5 We could allow SEEK to be an oracle or adversary, as long as future coin tosses

cannot be seen.

6 (added in proof) For a deterministic approach to problems as considered above, see

McDiarmid, C. (1993). On 2-colouring a 3-colourable graph to avoid monochromatic

triangles (manuscript).

Acknowledgements

I would like to thank Noga Alon and Nathan Linial for helpful discussions.

References

[1] Donnelly, P. and Welsh, D. J. A. (1983) Finite particle systems and infection models. Math.Proc. Cambridge Philosophical Society 94 167-182.

[2] Donnelly, P. and Welsh, D. J. A. (1984) The antivoter problem: random 2-colourings of graphs.In: Bollobas, B. (ed.) Graph Theory and Combinatorics, Proc. Conference in honour of PaulErdos, Cambridge, 1983, Academic Press, 133-144.

[3] Feller, W. (1968) An Introduction to Probability Theory and its Applications, Volume 1, 3rdedition, Wiley, New York.

[4] Garey, M. R. and Johnson, D. S. (1979) Computers and Intractability, WH. Freeman & Co,San Francisco.

[5] Khanna, S., Linial, N. and Safra, S. (1993) On the hardness of approximating the chromaticnumber. (Manuscript.)

[6] Lovasz, L. (1973) Coverings and colorings of hypergraphs. Proc. 4th S.E. Conference on Com-binatorics, Graph Theory and Computing, Utilitas Mathematica Publishing, Winnipeg, 3-12.

[7] Petford, A.D. and Welsh, D. J. A. (1989) A randomised 3-colouring algorithm. Discrete Math-ematics 74 253-261.

[8] J. Zerovnik (1987) A randomised heuristical algorithm for graph colouring. Proc. 8th YugoslavSeminar on Graph Theory, Novi Sad.

Obstructions for the Disk and the CylinderEmbedding Extension Problems

BOJAN MOHARt

Department of Mathematics, University of Ljubljana, Jadranska 19, 61111 Ljubljana, Sloveniaemail: [email protected]

Let S be a closed surface with boundary dS and let G be a graph. Let K ^ G be a subgraphembedded in S such that dS c K. An embedding extension of K to G is an embedding ofG in S that coincides on X with the given embedding of K. Minimal obstructions for theexistence of embedding extensions are classified in cases when S is the disk or the cylinder.Linear time algorithms are presented that either find an embedding extension, or return anobstruction to the existence of extensions. These results are to be used as the corner stonesin the design of linear time algorithms for the embeddability of graphs in an arbitrarysurface and for solving more general embedding extension problems.

1. Introduction

Let K be a subgraph of G. A K-component or a K-bridge in G is a subgraph of G that iseither an edge e € E(G)\E(K) (together with its endpoints) that has both endpoints in K,or it is a connected component of G— V(K) together with all edges (and their endpoints)between this component and K. Each edge of a K -component R having an endpoint inK is a foot of R. The vertices of R n K are the vertices of attachment of R. A vertex ofK of degree in K different from 2 is a main vertex of K. For convenience, if a connectedcomponent of K is a cycle, we choose an arbitrary vertex of it and declare it to be amain vertex of K as well. A branch of K is any path in K whose endpoints are mainvertices and such that no internal vertex on this path is a main vertex. If a K -componentis attached at a single branch of X, it is said to be local. The number of branches of K iscalled the branch size of K.

Let K c G, and suppose that we are given an embedding of K into a (closed) surface Z.The embedding extension problem asks whether it is possible to extend the given embeddingof K to an embedding of G, and any such embedding is said to be an embedding extension

t Supported in part by the Ministry of Science and Technology of Slovenia, Research Project Pl-0210-101-92.

494 B. Mohar

of K to G. Let Z be the (closed) disk or the cylinder. Let K be embedded in Z such that<3Z c X. An obstruction for embedding extensions is a subgraph Q of G — £(K) such thatthe embedding of K cannot be extended t o X u Q . The obstruction is small if K UQ hasbounded branch size. If Q is small, it is easy to verify that no embedding extension toK U Q exists, and hence that Q is a good verifier that there are no embedding extensionsof K to G as well. In this paper, minimal obstructions for embedding extension problemsin the disk and the cylinder are classified for several 'canonical' choices of K. Althoughmuch work has been done on 'embedding obstructions', our results seem to be new, apartfrom the case of the disk (cf. [20]; see also Section 3) or the case when K = 0 and Z is aclosed surface [19]. It is interesting that minimal obstructions are not always small. Theycan be arbitrarily large but their structure is easily described. We also present linear timealgorithms to either find an embedding extension, or return a (minimal) obstruction tothe existence of extensions.

The basic results of this paper (Theorems 3.1, 4.3, 5.3, and 6.2) are to be used asthe basic building stones in the design of linear time algorithms for embedding graphsin general surfaces [10, 12, 16, 11]. Moreover, we are able to solve even more generalembedding extension problems in linear time.

Robertson and Seymour (cf. [19] and the graph minors papers preceding it) proveda Kuratowski theorem for general surfaces. In our further project [17], results of thispaper are used to obtain a reasonably short proof of Robertson and Seymour's result. Itis worth mentioning that all our results are direct and constructive, in the tradition ofArchdeacon and Huneke [1]. (Recently, Seymour [23] also obtained a constructive proofby using graph minors and tree-width techniques.)

Embeddings in orientable surfaces can be described combinatorially [6] by specifyinga rotation system: for each vertex v of the graph G we have the cyclic permutation nv

of its neighbors, representing their circular order around v on the surface. In order tomake a clear presentation of our algorithm, we have decided to use this description onlyimplicitly. Whenever we say that we have an embedding, we mean such a combinatorialdescription. Whenever used, it is easy to see how one can combine the embeddings ofsome parts of the graph described this way into the embedding of larger species.

In discussing the time complexity of our algorithms, we assume a random-accessmachine (RAM) model with unit cost for basic operations. This model was introduced byCook and Reckhow [4]. More precisely, our model is the unit-cost RAM where operationson integers, whose values are O(n), need only constant time (n is the order of the givengraph).

2. Basic definitions

Let G and K be graphs (both subgraphs of some graph H). Then we denote by G — Kthe graph obtained from G by deleting all vertices of GnK and all their incident edges. IfF ^ E(G), then G — F denotes the graph obtained from G by deleting all edges in F. If Kand L are subgraphs of G, then we say that a path P in G joins K and L if P is internallydisjoint from K U L and one of its ends is in K and the other end is in L. Moreover, ifan end of P is in both K and L, then P is a trivial path.

Obstructions for the Disk and the Cylinder Embedding Extension Problems 495

A block or 2-connected component of a graph G is either an isolated vertex, a loop, abond of G, or a maximal 2-connected subgraph of G. One can also define the concept of3-connected components. A graph G is said to be k-separable if it can be written as a unionG = H U K of (non-empty) edge-disjoint graphs H and K that have exactly k verticesin common, and each of them contains at least k edges. Such a pair {H,K} is called akseparation of G. A graph is nonseparable if it has no 0- or 1-separations. Let G be anonseparable graph and let {H,K} be a 2-separation of G. Let x, y denote the vertices ofV(H)DV(K). The 2-separation is elementary if either H — {x,y} or K — {x,y} is non-emptyand connected, and either H or K is nonseparable. It turns out [26] that nonseparablegraphs without elementary 2-separations are either 3-connected graphs, cycles Cn (n > 3),p > 1 parallel edges, K\, or a loop. Assume now that the 2-separation {H,K} of G iselementary. Denote by H' and K' the graphs obtained from H and K, respectively, byadding to each of them a new edge between the vertices of H 0 K. The added edgesare called virtual edges. It is easy to verify that H' and Kf are both nonseparable, andwe may repeat the process on their elementary 2-separations (if there are any) until nofurther elementary 2-separations are possible. As mentioned above, the obtained graphsare either 3-connected, cycles, edges in parallel, or rather small. Each of the graphsobtained this way is called a 3-connected component of G. It was shown by MacLane [14](cf. also [26]) that the set of 3-connected components of the graph is uniquely determined,although different choices of the 2-separations may have been used during the processof constructing them. Every 3-connected component consists of several edges of G andseveral virtual edges. It is obvious by construction that each edge of G belongs to exactlyone 3-connected component, and each virtual edge has a corresponding virtual edge insome other 3-connected component. The 3-connected components of G may be viewed assubgraphs of G, where each virtual edge corresponds to a path in G. These subgraphs arepositioned in G in a tree-like way [26]. We also speak of 3-connected components whenthe graph is separable. In that case we define them to be the 3-connected components ofthe blocks of the graph.

A linear time algorithm for obtaining the 3-connected components of a graph wasdevised by Hopcroft and Tarjan [7].

There are very efficient (linear time) algorithms that for a given graph determinewhether the graph is planar or not. The first such algorithm was obtained by Hopcroftand Tarjan [8] back in 1974. There are several other linear time planarity algorithms(Booth and Lueker [2], de Fraysseix and Rosenstiehl [5], Williamson [27, 28]). Extensionsof the original algorithms also produce an embedding (rotation system) whenever the givengraph is found to be planar [3], or find a small obstruction - a subgraph homeomorphic toK$ or X33 - if the graph is non-planar [27, 28] (see also [13]). The subgraph homeomorphicto Ks or X33 is called a Kuratowski subgraph of G.

Lemma 2.1. There is a linear time algorithm that, given a graph G, either exhibits anembedding of G in the plane, or finds a Kuratowski subgraph of G.

We will refer to the algorithm mentioned in Lemma 2.1 as testing for planarity. This

496 B. Mohar

b

(a) (b)

Figure 1 Disjoint crossing paths and a tripod

procedure not only checks the planarity of the given graph but also takes care of exhibitingan embedding, or finding a Kuratowski subgraph.

Let C be a cycle of a graph G. Two C-components B\ and B2 overlay if either B\ andB2 have three vertices of attachment in common, or there are four distinct vertices a, b,c, d that appear in this order on C and such that a and c are vertices of attachment of B\,and fc, d are vertices of attachment of B2. In the latter case, B\ and B2 contain disjointpaths Pi and P2 whose ends a, c and b, d, respectively, interlace on C. Such paths willbe referred to as disjoint crossing paths - see Figure l(a). We will need another type ofsubgraph of G that is attached to C. A tripod is a subgraph T of G that consists of twomain vertices v\, v2 of degree 3, whose branches join them with the same triple of verticesMi, u2, M3, together with three vertex disjoint paths n\, n2, 713 joining «i, u2, and M3 withC. Moreover, T intersects C only at the ends of 711, 712, and 7C3 - see Figure l(b). Oneor more of the paths 7r, are allowed to be trivial, in which case ux e C. If all three paths7ii, n2, and 7C3 are trivial (just vertices), then the tripod is said to be degenerate. We usethe same name for attachments of the tripod in the case when the corresponding path istrivial.

3. The disk

Let D be the closed unit disk in the euclidean plane. Given a graph G and a cycle Cin G, we would like to find an embedding of G in D so that C is embedded on dD. Ofcourse, this is a case of the embedding extension problem for which an easy answer is athand. First, we construct the auxiliary graph G = Aux(G, C), which is obtained from Gby adding a new vertex v (called the auxiliary vertex) and joining it to all vertices on C.It is easy to see that an embedding extension of C on dD to G exists if and only if theauxiliary graph G is planar. Its plane embedding also determines an embedding extension.In the case of non-planar G, a Kuratowski subgraph K of G determines the subgraphK = K — v of G that is an obstruction for the embedding extension in the disk. AlthoughK U C can have arbitrarily large branch size, it can easily be modified to an obstructionQ for which ftuC has bounded branch size. Our answer seems to solve the questionreasonably well. However, there is a better solution. Namely, it is known that when G


is 3-connected, a pair (G,C) for which there is no disk embedding extension necessarilycontains either a pair of disjoint crossing paths or a tripod. This simple but useful resultwas 'in the air' for quite some time. It seems to have appeared for the first time in apaper by Jung [9] in a slightly weaker version. It also appeared in a paper by Seymour[21] (with the complete proof in [22]), Shiloach [24] and Thomassen [25], all in relationto the non-existence of two disjoint paths between specified vertices. This result recentlyappeared in a more explicit form in Robertson and Seymour's work on graph minors [20].In this section we will prove a slightly more specific result by also taking care of the casewhen G is not 3-connected. Moreover, we will show how to obtain such an obstructionin linear time.

Theorem 3.1. Let G, C, D be as above. Let G = Aux(G, C). There is a linear time algorithmthat either finds an embedding of G in D with C on dD, or returns a small obstruction Q.In the latter case, Q is one of the following types of subgraphs of G — E(C):

(a) a pair of disjoint crossing paths,(b) a tripod, or(c) a Kuratowski subgraph contained in a 3-connected component of G distinct from the

3-connected component of G containing C.

Before giving the proof of Theorem 3.1 we state a lemma whose easy proof is left tothe reader.

Lemma 3.2. Let H be a graph with a cycle C and let e be an edge of H that is not achord of C. If the edge-contracted graph H/e contains a tripod or a pair of disjoint crossingpaths with respect to C (or C/e if e E E(C)), then H also contains a tripod or a pair ofdisjoint crossing paths.

Proof of Theorem 3.1. By testing G for planarity, we can check if G is planar. If yes, wealso get a required disk embedding of G.

Suppose now that G is non-planar. Determine the 3-connected components of G, forexample, by using the linear time algorithm of Hopcroft and Tarjan [7]. Note that C (withsome of its edges having possibly become virtual) and the auxiliary vertex are in the same3-connected component. Denote this 3-connected component by R. If R is planar, thenthe obstruction to the planarity of Q lies in one of the other 3-connected components.We get (c) in one of the planarity tests. From now on we may thus assume that R isnon-planar. Let us show how to get disjoint crossing paths or a tripod.

Let K be a Kuratowski subgraph of R found by a planarity test on R. Denote by Hthe graph (K U C) — w, where w is the auxiliary vertex of G. Note that the branch sizeof H is not necessarily small. We will first try to find disjoint crossing paths or a tripodin H. Consider the C-components in H. First of all we check if two of them overlap. Inorder to perform efficient checking, we split the bridges into two classes: the bridges thatcontain main vertices of K (possibly as their vertices of attachment) are main bridges ofC in //, and the remaining bridges are called chords, since they are just paths joining

498 B. Mohar

two distinct vertices of C. There are at most 20 main bridges. To be more efficient in thefollowing, we can temporarily replace every bridge by a single vertex joined to all of itsvertices of attachment on C.

Step 1: Are there two main bridges that overlap ?If yes, we either have a (degenerate) tripod or disjoint crossing paths. If not,proceed with the next step. Since only the main bridges have to be considered,this step can be carried out in constant time.

Step 2: Is there a main bridge that overlaps with a chord?If yes, we have disjoint crossing paths. Otherwise continue with Step 3. Thisquestion can easily be answered in linear time. Observe that the number ofcandidates for one of the disjoint crossing paths in the fhain bridges is small.

Step 3: Are there two overlapping chords?Note that no chord contains a main vertex. Thus, at most two chords are attachedat the same vertex. A simple way to find overlapping chords is to start building astack by traversing C once around, starting at an arbitrary vertex of C. If thereare two chords at the same point on C, we first consider chords that have alreadybeen met during the traversal. If both chords are not new, we give priority tothe one that is on the top of the stack (if none is on the top, their order is notimportant). Then we process the new chords. If both are new, we give priority tothe one whose other attachment is further away in the direction of the traversal.Every new chord met during the traversal is put to the top of the stack. Meetingthe chord for the second time, we check if the top element in the stack is the samechord. If yes, the chord is removed from the stack and the traversal is continued.If not, then we have another chord at the top. It is easy to see that these twochords overlap and they give rise to disjoint crossing paths.

We may now assume that no two distinct bridges of C in H overlap. This means thatthe obstruction is in one of the main bridges. Such a bridge B can be discovered in 0(1)time since the number of main bridges is small, and each of them has small branch size.We will also assume that B is minimal in the sense that for every branch e of £, thegraph (B — e) U C is planar (if riot, we can remove e and repeat the above procedure inorder to get (a), (b), or a new bridge B with a smaller number of branches). Note thatB U C is non-planar and has small branch size. Therefore B can be used as a legitimateobstruction in some applications. However, our goal is to show more: we want a tripodor disjoint crossing paths.

Since B U C has constant branch size, it is easy to find a tripod or disjoint crossingpaths in B whenever B contains one of them. Assume from now on that this is not thecase. We will prove that under this assumption, B has at most two vertices of attachmenton C. Let K' be a Kuratowski subgraph of B U C. By the minimality property of B, K'contains the whole of B plus, possibly, some parts on C. If two main vertices of K' lieon C, they are either non-adjacent in K\ or connected by a branch that is containedin C. Therefore it is easy to see that at most three main vertices of K' lie on C (thecase of four vertices of K33 forming a cycle on C is the only possibility, but they giverise to disjoint crossing paths). Similarly, we can exclude three main vertices of K' being

on C. (In the case analysis for the last claim, an application of Lemma 3.2, using the'contraction' argument as also used below, makes the number of cases much smaller.)

Now, if B has a vertex of attachment on C that is not a main vertex of K\ wemay contract the corresponding branch e of B U C and obtain the non-planar graph(B/e) U C = ( B u C)/e with more main vertices of K'/e (« K') on C Inductively, wehave a tripod or disjoint crossing paths. Also, by Lemma 3.2, B U C contains a tripod ordisjoint crossing paths.

Suppose noW that B has t < 2 vertices of attachment on C, and recall that we knowhow to get in linear time a tripod or disjoint crossing paths in the case of three or morevertices of attachment. Let B be the C-corfipofieht in R that contains B as a subgraph.Since R is 3-connected, there are disjoint paths e\, ;••, e*$-t in B, starting at C — B andterminating in B — C, whose only vertices in B are their endpoints. Such paths can be foundin linear time by applying, for example, the appropriate modification of the augmentedpaths method used to test connectivity of graphs [18, Chapter 9]. The connectivity testshould be applied on the graph BU C with the t attachments of B removed. Since t < 2,the graph H = BuCUe\U'-U e^-t contains a copy of K' that does not contain theendpoints of e\, . . . , ei-t on C (this fact is really needed only when t = 2). Therefore, thegraph H' = H/(e\U- • -Ue^-t) also contains a copy of K'. Note that the only 3-separationsin Kuratowski subgraphs intersect at the three vertices of the same color class in X33.Therefore Hf is equal to C plus a single bridge (plus, possibly, a branch between twovertices on C that can be replaced by a segment on C), except when the three vertices ofK' c U' that lie on C are the three vertices of the same color class of X33. In the lattercase, we clearly have a tripod in Hf. In the other cases, we can apply the results fromabove, since H' is of appropriate form and has three attachments on C, we can find atripod in it. By Lemma 3.2, We have a tripod or disjoint crossing paths in H. •

4. The cylinder

In this section we will consider the embedding extension problems in the cylinder. LetC\ and C2 be disjoint cycles in the graph G, and for an integer k > 0, let Pi, P2, . . . ,P/c be vertex disjoint paths in G joining C\ and C2 (with no interior points on C\ U C2).Suppose, moreover, that the endpoints of the paths Pt appear on both cycles C\, C2 in thesame (cyclic) order. The embedding extension problem in the cylinder with respect to thesubgraph K = C\ U C2 U Pi U • • •. U P/c, where K is embedded in such a way that C\ and C2cover the boundary, will be referred to as the k-prism embedding extension problem. Notethat when k < 2, we have two essentially different problems depending on the embeddingsof C\ U C2 on the boundary of the cylinder.

In testing for the /c-prism embedding extension of K to G, we make use of the auxiliarygraph G, which is obtained from G by adding two new auxiliary vertices v\ and V2, and fori = 1, 2, joining vt to all vertices of Q. If k > 3, an embedding extension of K to G existsif and only if G is planar, and a planar embedding of G determines a cylinder extension.Something similar holds; also when k < 2. More details will be provided later. Note thatin the cylinder case, the auxiliary graph contains two auxiliary vertices, while the auxiliarygraph for disk embeddability has just one. Although we are using the same name and

500 B. Mohar

pl F P2 F'

(a) (b)

Figure 2 The 1- and 2-prism embedding extension problem

Figure 3 A large obstruction using local bridges

notation, there will be no confusion, since it will always be clear from the context whichcase is applied.

If there are local bridges attached to one or more of the paths P,, we may getarbitrarily long chains of successively overlapping local bridges on P, (see Figure 3).There are examples where, after eliminating any of the branches, there exists an embeddingextension. So we can have arbitrarily large minimal obstructions. On the other hand, inapplications using the obstructions, certain connectivity conditions on the involved graphscan be achieved. In that case, the local bridges can be eliminated efficiently (in linear time:see [15] and [10] for more details). Since we are usually allowed to change the paths P,during the pre-processing time, it makes sense to assume that there are no local bridgesattached to any of the paths Pi, . . . , P*.

Obstructions for the /c-prism embedding extension problem with k > 3 are easy to find- they are not much more complicated than the closed disk obstructions classified intheorem 3.1. Besides the disjoint crossing paths and the tripods, we get another type ofobstruction. A dipod (with respect to the cycle C) is a subgraph H of G consisting ofdistinct vertices a, b, c, d € V(C) that appear on C in that order, distinct vertices v, uwhere v £ V(C), and u £ V(C) unless u = b, and branches va, vc, vu, ub, and ud (Figure 4).

d c

(a) (b)

Figure 4 A dipod

The branches are internally disjoint from C. If u = b, the branch ub vanishes (see Figure4(b)). We also define a triad (with respect to a subgraph K of G) as a subgraph of Gconsisting of a vertex x £ V(K) and three paths joining x with K that are pairwise disjointexcept at their common end x.

For K = C\ UC2UP1U • • -UPk embedded in the cylinder with C\ and C2 on its boundary,let Fi, . . . , Fk be the faces of K. We suppose that for i = 1, 2, .. . , fc, dFt contains P, andPi+i (index modulo fc).

Theorem 4.1. Let K ^ G be the subgraph of G for the k-prism embedding extensionproblem, where k > 3. Suppose that no K-component of G is attached to just one of thepaths Pi of K, 1 < i < k. Then there is no embedding extension of K to G if and only ifG — E(K) contains a subgraph Q of one of the following types:

(a) A path joining two vertices of K that do not lie on the boundary of a common face ofK, or (with k = 3) a triad attached to Pi, P2, and to P3.

(b) A tripod attached to the boundary of one of the faces Ff. Not all three attachments ofthe tripod lie on just one of the paths Pi, PI+i on dFi.

(c) A pair of disjoint crossing paths with respect to the boundary of one of the faces Fi.None of the two paths is attached to just one of the paths Pt, Pi+\ on dFt.

(d) A dipod with respect to the boundary cycle of some Fi. In this case, the vertices a, c,and d from the definition of the dipods all lie on one of the paths Pif or P,+i, whileb G dFt does not lie on the same path.

(e) A Kuratowski subgraph contained in a ^-connected component L of the auxiliary graphG of G, where L is such that it does not contain auxiliary vertices of G.

There is a linear time algorithm that either finds an embedding extension of K to G, orreturns an obstruction Q which fits one of the above cases.

Proof. We can find embedding extensions, if they exist at all, by testing the auxiliarygraph G for planarity. Suppose now that embedding extensions do not exist. Our goal isto show how to find the required obstruction Q.

Since k > 3 and there are no local bridges at the paths P;, every X-component isembeddable in at most one of the faces Ft. If one of the bridges contains a path whose

502 B- Mohar

ends do not belong to the boundary of the same face, this path is clearly an obstructionfor the embedding extendibility. If a bridge B of K does not have all of its vertices ofattachment on the boundary of a single face i7,, then B either contains such a path, or itcontains a triad attached to Pi, P2, and P3. (Note that the latter case is needed only whenk = 3.) So, we have (a). Otherwise, every K-component is attached to dFt for exactly one i,1 < i < k. Therefore, there is no embedding extension if and only if for some i, 1 < i < k,we have a closed disk obstruction (cf. Theorem 3.1) in the subgraph Gt consisting ofC = dFi and all the K-components attached to C. By Theorem 3.1, an obstruction tothe (d, C) disk embeddability is either a pair of disjoint crossing paths, or a tripod, ora Kuratowski subgraph in a 3-connected component of G, not containing the auxiliaryvertex. In the last case, G, is the auxiliary graph of G, with respect to C for the diskembedding extension problem. Since there are no local bridges attached to the paths P,and Pj+i, the 3-connected components of G, not containing the auxiliary vertex are also3-connected components of G. Consequently, the Kuratowski subgraph obstruction in G,gives (e).

Suppose now that in G, we have a tripod T. If T is not local on P; and not local onPj+i, we have (b). Otherwise, assume all three attachments of T are on Pt. Denote byv\, V2, Mi, M2, M3, 7ii, 7i2, 7E3 the elements of T as they are shown on Figure 1, and supposethat 7T2 is attached at P, between n\ and 713. Construct a path P, internally disjoint fromC, that connects C — Pt with an interior vertex x of T. The existence of P is guaranteedsince the bridges containing T are not local on P,. If x is on ns for some s £ {1,2,3}, wecan replace the segment of ns from x to P, by P and get a tripod satisfying (b). If x isan interior vertex of the branch U2VU then T U P contains a dipod satisfying (d). By thesymmetries of T, the only essentially different remaining case is when x is on the branchu\V\, where x =/= u\ but possibly x = v\. Let Q\ be the path Pxv\U27i2 and let Q2 be thepath 711U1V2U3713 in T UP. If Q\ and Q2 are in the same X-component of G, we can find apath P' from Q\ to Q2 that is disjoint from C, and Q\ U Q2 U Pr is a dipod satisfying (d).On the other hand, if Q\ and Q2 are in different bridges B\, B2 of K, respectively, let P'be a path from the interior of Q2 to C that is disjoint from P,. Such a path exists, again,because B2 is not local on P,. Now, Qi UQ2UPf contains disjoint crossing paths satisfying(c), unless the endpoints of P and P' on C coincide. But in this case, Q\ U Q2 U P' is adegenerate dipod with the attachment /? (see Figure 4(b)) corresponding to the commonpoint of P and P'.

It remains to consider the case of disjoint crossing paths, say Q\ and Q2, obtained asan obstruction in G,. If both Q\ and Q2 are attached locally to P,, we change one of themso that it has an attachment on C — Pt. For this purpose, the same method as above canbe applied. If just one of the paths (possibly after the previous change) is local on P,, thesame procedure can be applied, as above, with the paths Qi and Q2 in the case of thetripods. We either get a dipod or disjoint crossing paths satisfying (d) or (c), respectively.

It is easy to perform the above construction in linear time. To find disk obstructions,we use Theorem 3.1, and to find paths P, P', etc., we can use standard graph search. •

Once we know how the case k > 3 works, we can also solve the 0-prism embeddingextension problem. If C\, C2 are cycles of G embedded on the boundary of the cylinder,

an orientation of the cylinder yields consistent orientations of C\ and Ci. If P i , . . . , Pu aredisjoint (Ci,C2)-paths, they are said to be attached consistently if their ends on C\ followeach other in the inverse cyclic order to that on Ci, i.e., the embedding of C\ U Ci can beextended to d U C2 U Pi U • • • U Pk. Note that for k < 2, the paths are always attachedconsistently.

Before stating our next result on obstructions, let us formulate a lemma that will beused in its proof.

Lemma 4.2. Let G be a 3-connected graph, C a cycle of G, and B a C-component in G. LetG(B, C) be the graph obtained from B U C by adding a new vertex adjacent to all verticeson C. Then G(B, C) is 3-connected.

Proof. The graph is clearly 2-connected. It is also easy to see that it has no 2-separations.

•Theorem 4.3. Let C\ and Ci be disjoint cycles of a graph G that are embedded on theboundary of the cylinder. There is no embedding extension to G if and only if G — E(C\) —E(Ci) contains a subgraph Q of one of the following types:

(a) Three disjoint paths from C\ to Ci that are not attached consistently on C\ and Ci.(b) Disjoint paths Pi, P2, P3, where P\, Pi join C\ with Ci, both endpoints of P3 are on

C\ (respectively, on C2) and the endpoints of P^ interlace with the endpoints of P\ andPi on C\ (respectively, on Ci).

(c) A tripod or a pair of disjoint crossing paths with respect to C\ (respectively, Ci). IfG is not 3-connected, this obstruction may have a vertex, two vertices, or a segment ofone of its branches contained in Ci (respectively, in C\).

(d) A path P from C\ to Ci together with a tripod T with respect to C\ U Ci disjoint fromP that has two attachments on C\ and one on Ci, or vice versa.

(e) Disjoint paths P\, Pi, P3 from C\ to Ci attached consistently on C\ and Ci, togetherwith a triad attached to P\, Pi, and to P3.

(f) A Kuratowski subgraph contained in a 3-connected component L of the auxiliary graphG of G, where L is such that it does not contain auxiliary vertices of G.

Moreover, there is a linear time algorithm that either finds an embedding extension ofC\UCito G, or returns an obstruction Q for the embedding extendibility. In the latter case, Q fitsone of the above cases (a)-(f).

Proof. First of all we try to find three disjoint (Ci,C2)-paths in G. If such paths exist,let k = 3, and let Pi, P2, P3 be the paths. Otherwise, let k < 2 be the maximal numberof disjoint paths from C\ to C2. All these can be obtained in linear time by standardconnectivity algorithms using flow techniques [18, Chapter 9].

Let us first consider the case when k = 3. If Pi, P2, P3 are not attached consistently atC\ and C2, then Q = Pi U P2 U P3 is a small obstruction satisfying (a). Otherwise, we firstreduce the problem to the 3-connected case. Without loss of generality, we can remove the

504 B. Mohar

(a) (b) (cl)

(c2) (d) (e)

Figure 5 0-prism obstructions

(a) (b) (c)

(d) (e)

Figure 6 Some obstructions of type (c) meeting both cycles


3-connected components of the auxiliary graph that do not contain the auxiliary vertices(or we get (f)). So, we assume from now on that G is 3-connected. Next we try to replacethe paths Pi, P2, and P3 by disjoint paths joining the same pairs of endpoints so that nolocal bridge of Kf = C\ UC2UPi UP2UP3 will be attached to some P, only. It can be shownthat this is always possible, since G is 3-connected, but it is not entirely obvious how toperform it in linear time. For i = 1, 2, 3, let G, be the graph consisting of P, together withall its local bridges and with an additional edge joining the ends of Pt. If Gt is planar, analgorithm from [15] replaces P, with a new path P/, joining the same endvertices, that isinternally disjoint from K' — Pt, and such that no local bridge of (Kr — P,) U P( is attachedto P(. So, we either achieve our goal, or get one of G,-, say G\, to be non-planar. Let usfirst deal with the latter possibility. Let C be the cycle composed of the paths P2 and P3together with the segments on C\ from P3 to Pi and from Pi to P2, and the segments onC2 from P2 to Pi and from Pi to P3. Denote by B the (Ci U C2 U P2 U P3)-componentin G that contains Pi. If B contains a vertex of (C\ U C2) — C, then a path in B fromthat vertex to an end of Pi, together with P2 and P3 determines three non-consistentlyattached paths from C\ to C2, and so we have case (a). Therefore, we may assume that Bis attached to C only. Let H — BuC. It is clear that H is 2-connected, and, by Lemma 4.2,its auxiliary graph H with respect to C is 3-connected. Moreover, H is non-planar, sinceGi is contained in H (with the edge joining the ends of Pi replaced by a path in C).By Theorem 3.1, we can find in if a tripod T or disjoint crossing paths Q\ and Q2 withrespect to C. Let us first consider the case when we have disjoint paths Q\, Q2. For 7 = 1,2, denote by e, the foot of Pi on C7, and let C° be the open segment of C7 obtained fromCj n C by removing its endpoints. If Q\ U Q2 is not attached to CJ5, take a path P inB — C from e\ to Q\ U g2. Such a path clearly exists, since Qu Q2 are both contained inthe same bridge. Using this path, we can change Q\ or Q2 to get disjoint crossing pathsthat are attached to C\. We repeat the same procedure at C\. Up to symmetries, there arethree possible outcomes:

(i) g i joins C[ and C2°:

If Q2 is attached on P2 and P3, take a path P in B — C joining Q\ and g2. Now, thepaths Qi, P2, P3 and the triad Q2 U P satisfy (e). Otherwise, it is easy to see that weget a subgraph of type (a), or (b) contained in Q\ U g2 U P2 U P3.

(ii) Qi is attached to C{ and Q2 is attached to C\\Excluding the above possibility (i), we may assume that the other attachment of Q\is on P2 — C\. Then Q\ U Q2 U P2 contains disjoint crossing paths between C\ and C2.Together with P3 they determine a subgraph of type (a).

(iii) Qi is attached to C\ and both endpoints of Q2 are on P2:Let P be a path in B - C joining Q{ and Q2- Then Q{ U Q2 U P U P2 is a tripod onC\ U C2, and, together with P3, we have (d).

Suppose now that T is a tripod with respect to C that is contained in B. If T is notattached on CJ5, let P be a path in B — C from e\ to T. Then T UP either contains a pairof disjoint crossing paths (which we have already covered above), or a tripod T' with anattachment on C°v If T is not attached to P3, then T u P2 contains a tripod T" withrespect to C\ U C2 that is either attached to C\ only (case (c)), or is attached to C\ and

506 B. Mohar

C2 (in this case T" U P3 satisfies (d)). Similarly, if T is attached only to C - P2. We areleft with the case when V is attached to C\ and to P2 and P3. In this case we constructa path in B — C from e2 to T". It gives rise to disjoint crossing paths, or to a tripod thatare disjoint from P2 or P3, and both of these cases have already been covered above.

From now on we may assume that we have Pi, P2, P3 without local bridges. LetK' = C\ U C2 U Pi U P2 U P3. By Theorem 4.1, we either extend the embedding of K' toG, or find an obstruction. The first outcome is fine, while in the second case we get oneof the obstructions (a)-(d) of Theorem 4.1. Obstruction (a) of Theorem 4.1, together withPi, P2, P3, necessarily contains one of our cases (a), (b), or (e). Case (b) of Theorem 4.1,together with Pi, P2, P3, implies our cases (c) or (d). The possibility (c) of Theorem 4.1yields either (a), (b), or (c). Finally, a dipod D of type (d) in Theorem 4.1 attached threetimes on Pi, say, gives rise to a tripod with respect to C\ U C2 contained in D \JP\ (plus asegment on P2 (or P3) if D is attached to P2 (respectively, on P3)). This tripod is disjointfrom one of the paths, and fits our case (d).

Finally, we have reached the cases k = 0, 1, 2. The first two (k = 0, 1) are easy. We arefaced with two disk embedding extension problems, and to solve each of them, we applyTheorem 3.1. The resulting obstruction fits (c). If a cutvertex v of G separating C\ andC2 is on C2 (assuming that the block containing C\ is non-planar), the obstruction maycontain v. (The possible cases are shown in Figure 6(a), (c), and (d).) Note that this is thefirst time that disjoint crossing paths or a tripod with respect to C\ have a vertex on C2.

Suppose now that k = 2. Let Q be the block of G containing Ci and C2. If theembedding of C\ U C2 extends to Q, we test the other blocks of G for planarity. We eitherget an embedding extension to G, or one of the blocks is non-planar. In the latter case,we have (f). So we may assume from now on that G is 2-connected and that there is noembedding extension. Since k = 2, Menger's theorem guarantees that C\ and C2 are indistinct 3-connected components of the auxiliary graph G. If all 3-connected componentsof G are planar, G is also planar. (This is easily seen by constructing a plane embeddingof a graph by using plane embeddings of graphs forming its 2-separation.) Unfortunately,it may happen that the plane embedding of G obtained in this way will not determine anembedding extension, since C\ and C2 may not be oriented consistently. In this case, letQ\ => C\ and Q2 => C2 be the graphs used in merging at the time when C\ and C2 mergein the same part, and let e be the corresponding virtual edge. Fixing the embedding ofQi9 there are two possibilities for the embedding of Q2 that differ from each other onlyby the choice of orientation. One of them gives the consistent orientation of C\ and C2.

We may assume now that one of the 3-connected components of G is non-planar. Ifthe two 3-connected components containing C\ and C2, respectively, are planar, we get(f). Suppose now that the 3-connected component Q\ => C\ of G is non-planar. This isequivalent to the property that Q\ minus the auxiliary vertex has no embedding into thedisk having C\ on its boundary. By Theorem 3.1, we know how to handle this case. SinceQ\ is 3-connected, we get disjoint crossing paths or a tripod in it. This almost alwaysgives rise to a subgraph of G satisfying (c). The only trouble may arise if our obstructionin Q\ contains the virtual edge e having its pair in the 3-connected component Q2 of Gthat contains C2. In this case, the replacement of e by a path P in Q2 — e should be donecarefully so that P n C2 is either empty, a vertex, or a segment on C2. Since this is easy

to achieve, we are through with our case analysis. It is worth remarking that some of thepossibilities when P n Ci is non-empty lead to cases (b) or (d). Some of the really newcases are shown in Figure 6, where each of the bold segments can be contracted to apoint. The cases shown in Figure 6 include all possibilities that arise when the obstructionin Q\ is either a pair of disjoint crossing paths, or a tripod whose intersection with Ci isa vertex, or a segment.

Finally, we remark that all the steps of the algorithm that follows the above proof areeasy to implement in linear time. •

It is worth remarking that all cases of Theorem 4.3 are indeed obstructions for the0-prism embedding extension problem, and that they are minimal (except in some casesof (c) when the intersection with the other cycle is non-empty) in the sense that if any ofthe branches is removed from such an obstruction, there exists an embedding extension.Note that the branch size of minimal 0-prism obstructions is at most 12. The obstructions(a)-(e) (without showing their 'degenerate' versions) are presented in Figure 5.

5. The 2-prism embedding extension problem

It may happen that minimal obstructions for the /c-prism embedding extension problemsare arbitrarily large. However, under the additional assumption that there are no localbridges attached to the paths Pl (1 < i < /c), large minimal obstructions are unavoidableonly for the /c-prism embedding extension problem with fc = 1 or 2. An example of suchan obstruction is shown in Figure 7. Since the general case of large minimal obstructionslook like our example in Figure 7, we use the name millipede. More precisely, a millipedeM for the 2-prism embedding extension problem is a subgraph of G — E(K) that can beexpressed as M = B°{ U B^ U • • • U B°m (m > 2), where B\, . . . , B°m are subgraphs of distinctK-bridges Bu #2, ••-, Bm (respectively) and satisfy the following conditions.

(1) Each of #i and B^ is embeddable in exactly one of the faces of K. If m is even, then2?i and B^ are embeddable in the same face of K. If m is odd, then B\ and B^ areembeddable in distinct faces of K.

(2) For 2 < i < m — 1, B° is embeddable in both faces of K.(3) For each i = 1, 2, . . . , m — 1, B° and B°+l cannot be embedded simultaneously in the

same face of K.(4) No other pair B°, B° (1 <i<i + 2<j<m) interferes with each other, i.e., for any

embedding of £°, there is an embedding of B° in the same face of K unless such anembedding is not possible by (1) (when i = 1 or j = m).

(5) B° (1 < i < m) are minimal in the sense that the removal of any branch from Bfdestroys either (1), or (3).

It is easy to see that a millipede is a minimal obstruction for embedding extendibility. Itfollows from the minimality property (5) that each Bf (1 < i < m) contains at most 6 feet(at most a triple for overlapping with B°_{ and possibly another triple for overlapping withB°+l) and has at most 11 branches. (We will see that it suffices to consider only millipedesin which every B° contains at most 4 feet.) Let us remark that the millipedes constructed

508 B. Mohar

c,Figure 7 A millipede

by our succeeding theorems will satisfy an even stronger 'minimality' condition: Properties(1), (2) and (4) will hold not only for the subgraphs B° but also for their 'master-bridges' Bt.

Given the 2-prism embedding extension problem with C\9 C2, Pi, P2, F, Ff, as inFigure 2(b), and K = C\ U C2 UP\ UP2, we define the overlap graph 0(G,K) of X-bridgesin G as follows. Its vertices are the K-bridges, and two of them are adjacent in 0(G,K) ifthey overlap in one of the faces, i.e., they can be embedded in the same face, say F, buttheir union cannot be embedded in F. The extended overlap graph A0(G,K) is obtainedfrom the overlap graph by adding two new vertices, w and w', that are adjacent to eachother. Moreover, w is adjacent to all bridges of K that are not embeddable in F, and w'is adjacent to all bridges that are not embeddable in F'.

Lemma 5.1. The embedding of K for the 2-prism embedding extension problem has anextension to G if and only if the extended overlap graph A0(G,K) is bipartite.

Proof. If a K -bridge B cannot be embedded in any of the faces F, Fr, then B togetherwith w and W determines a triangle, and the extended overlap graph is not bipartite.Therefore we may assume from now on that every bridge can be embedded in at leastone of the faces, F, or F'.

Suppose now that we have an embedding extension. Color the bridges that are em-bedded in F using color 1, and color the bridges in F' using color 2. Moreover, let w becolored 1, and let W be colored 2. It is easy to see that this determines a 2-coloring ofAO(G,K), so the extended overlap graph is bipartite.

Conversely, if the extended overlap graph is bipartite, choose one of its 2-coloringshaving w and W colored 1 and 2, respectively. Consider the bridges colored 1. Each ofthem can be embedded in F, since otherwise it would be adjacent in A0(G,K) to w thatalso has color 1. Moreover, all these bridges can be embedded in F simultaneously, sinceno two of them overlap. Similarly, the bridges colored 2 have an embedding in F , andwe get a required embedding extension. •


P P

(a) (b) (c) (d)

Figure 8 H-graphs

(e) (f)

The above lemma provides a simple answer for the 2-prism embedding extensionproblem. It also yields an algorithm that is linear in the number of edges of A0(G,K).Having a 2-coloring, we easily get an embedding extension. Otherwise, an odd cycle inA0(G,K) determines an obstruction. Unfortunately, the usual 2-coloring algorithm canbe of quadratic complexity in terms of the size of G, since the number of edges ofA0(G,K) may be quadratic in terms of the number of bridges, and this number can belinear in terms of |£(G)|. Therefore, we have to solve the biparticity problem of A0(G,K)with some additional care in order to fulfil our linearity goal. One possible approach isexplained in more detail in [12].

In the following results, we will use some special subgraphs of K -bridges. Let B be aK-bridge in G. For each branch e of K that B is attached to, let e\ and e2 be feet of Battached as close as possible on e to one and the other end of e (including the possibilityof being attached to the end). Furthermore, let these feet be chosen in such a way thattheir total number is as small as possible, i.e., if there is just one attachment on e, weselect e\ = e2, and similarly when different branches of K share an attachment of B.Let H = H(B) be a minimal subtree of B that contains all chosen feet. The obtainedgraph H is said to be an H-graph of B. Suppose now that B is attached only to Pi U P2.Then H contains at most 4 feet. If there are three or just two distinct feet in H, then His unique up to homeomorphisms. But in the case of four distinct edges, there are fourhomeomorphically distinct cases for H (see Figure 8). Let us remark that the last caseof Figure 8 is excluded if B can be embedded in F, since it contains disjoint crossingpaths. Note that H-graphs can be constructed in linear time by standard graph searchalgorithms. The following simple result justifies the introduction of H-graphs.

Lemma 5.2. Let G be a graph and K be a subgraph that is 2-cell embedded in some surface.Let B and Bf be K-bridges in G that can be embedded in the same face FofK.IfdFisacycle in G and neither B nor B' is a local bridge, then B and B' overlap in F if and only iftheir H-graphs overlap in F.

Theorem 5.3. Let K c G be the subgraph of G for the 2-prism embedding extensionproblem, and let F, F' be the faces of K. Suppose that no K-component of G is attachedjust to one of the paths P\, Pi of K. Then there is no embedding extension of K to G if andonly if G — E(K) contains a subgraph of one of the following types:

(a) A path that is internally disjoint from K and connects a vertex of dF — (P\ UP2)a vertex ofdF'-(P{ U P2).

510 B. Mohar

(b) A tripod T with respect to the boundary of one of the faces F, F'. Not all threeattachments of T lie on just one of the paths P\, Pi, and if all are in P1UP1, then thetripod is non-degenerate.

(c) A pair of disjoint crossing paths with respect to the boundary of one of the faces F, F''.Each of the two paths is attached to a vertex of K — (P\ U Pi).

(d) A non-degenerate dipod with respect to the boundary cycle of F or Ff. The verticesa, c from the definition of a dipod lie on P1UP1, and not all attachments of the dipodlie on just one of the paths Pi or Pi.

(e) Internally disjoint triads T\, T2 attached to the same triple of vertices on P\ U P2,together with a path joining the main vertices of T\ and Ti. Not all three attachmentsof T\ U T2 are on just one of the paths Pi, or P2.

(f) Subgraphs H\, Hi, #3 that are pairwise overlapping in F or in F'. They are minimalpairwise overlapping subgraphs of H-graphs of distinct K-bridges. Hi and H3 areattached to Pi U P2 only.

(g) A millipede.(h) A Kuratowski subgraph contained in a 3-connected component L of the auxiliary graph

G of G, where L is such that it does not contain auxiliary vertices of G.

Moreover, there is a linear time algorithm that either finds an embedding extension of K toG, or returns an obstruction Q of one of the above types.

Proof. We may check the embedding extendibility by testing the planarity of the auxiliarygraph G (cf the k = 2 case in the proof of Theorem 4.3 for details). Moreover, if there isno embedding extension, we can reduce the problem to the case where G is 2-connected(or we get (h)). Note that C\ and C2 are in the same block of G. If C\ and Ci are in thesame 3-connected component of G, we can also reduce the problem to the case where G is3-connected (or we get (h)). If this is not the case, let G' be the graph obtained from G byadding the edge joining the auxiliary vertices. By considering the 3-connected componentsof G', we can also in this case either get (h) (since the 3-connected components of G' arealso 3-connected components of G except for the one containing Ci and Ci), or reduce theproblem to the case when & is 3-connected. The latter case will be assumed henceforth.

Consider the K -components of G. Suppose that B is one of them, and that theembedding of K cannot be extended to K u B. We either have (a), or B is attached tothe boundary of one of the faces of K, say to dF. In the latter case, let L = dF U B.Since G' is 3-connected, the auxiliary graph L of L (with respect to dF) is 3-connected byLemma 4.2. Clearly, L is non-planar, and by Theorem 3.1, B contains a tripod or a pairof disjoint crossing paths with respect to dF. Since B is not local on Pi or Pi, we canuse the same strategy as in the proof of Theorem 4.1 to change the obtained obstacle sothat not all of its attachments are on just one of Pi or Pi. Usually we will get case (b),or (c), but there are also two exceptions. The first possibility is when we get a degeneratetripod with all attachments on Pi U Pi. Add a path P in B between the two triads in thetripod. If P joins the two main vertices of the tripod, then we have case (e). In all othercases, the union of P and the tripod contains a non-degenerate tripod, i.e., a subgraphsatisfying (b). The other case is when we have disjoint crossing paths that do not satisfy

(c). One of the paths is then attached only to Pi U ?i. Hence, by adding a path in B thatjoins the two paths, we get a non-degenerate dipod satisfying (d).

From now on we may assume that every K -component is embeddable either in F, orin F' (or both). A linear time algorithm of [12] shows how to solve this problem. Thatalgorithm finds an induced odd cycle F in the extended overlap graph AO(G,K). Thereare 4 cases to be distinguished.

(i) Both vertices w and W lie on F:In this case, the edge wwf is on F, since F is an induced cycle. Let B\, . . . , Bm bethe K -bridges corresponding to the sequence of vertices of F from w to W (but notincluding these two). By our assumptions, m > 1. By the definition of AO(G,K), B\cannot be embedded in F, and Bm cannot be embedded in F'. Note that m is odd,so the conclusion here fits condition (1) of the definition of a millipede. Next wedescribe how to get the subgraph Bf of Bi, for i = 1, 2, . . . , m. Since the cycle F ofAO(G,K) is induced, no bridge Bi9 2 < i < m — 1, is adjacent to w, or w'. This meansthat Bt itself is embeddable in F and in Fr. Therefore, arbitrary subgraphs B° of Bt

also satisfy (2). The bridges Bi and 2?,+i (i < m) cannot be simultaneously embeddedin the same face, and at least one of them is embeddable in both faces of K. ByLemma 5.2, their H-subgraphs (which are easy to find) overlap in the same way asthe bridges themselves. By taking such obstructions for all bridges Bi9 we get smallsubgraphs of B\, . . . , Bm satisfying (1) and (3). Since these subgraphs are small, wecan check whether each of them satisfies the minimality requirement (5), and removethe superfluous branches whenever necessary. Finally, (4) is satisfied automatically,since F is an induced cycle of AO(G,K). Therefore we have a millipede.

(ii) w e V(T):We get a millipede in the same way as in Case (i), except that m is even.

(iii)w'e F(F):Same as Case (ii).

(iv) w, W # V(T): We will show that in this case the length m of F is rather small. LetB\, . . . , Bm be the K-bridges corresponding to the successive vertices on F. Supposefirst that m = 3. If two of the bridges, say B\9 #2, are adjacent in AO(G,K) to w(or w'), we replace F by the triangle B\, Bi, w (respectively, B\, Bi, wr), and by (ii)(respectively, (iii)) we get a millipede. If one of them is adjacent to w, another to w',we get a millipede of length 3, as was the case in (i). (This works even though thecorresponding cycle in F obtained by replacing the edge B\Bi by the path B\\vWBi isnot induced.) We may therefore assume that Bi and B^ are embeddable in F and F'and that B\ is embeddable at least in F. For / = 1, 2, 3, let Ht be an H-graph of B\.By Lemma 5.2, the H-graphs overlap as much as the original bridges. We thereforehave (f).Suppose now that m > 5. As above, the cases when two of the bridges Bt are adjacentto w or W (possibly one to w, another to W) can be reduced to the previously treatedcases. We may thus assume that at most one of the bridges is not embeddable inboth F and in F'. If there is a bridge that cannot be embedded in one of the faces,we will assume that this is Bu and that this bridge can be embedded in F. Let us

512 B. Mohar

Figure 9 One-sided millipedes

write Bt < Bj if Bt U Bj can be embedded in F and Bt is embedded closer to C\ thanBj. Since none of the bridges is local on Pi or P2, the relation < is well defined. Therelation < is transitive, and since Gr is 3-connected, it is also asymmetric. Therefore ithas minimal elements. We may assume that B\ is a minimal element for this relation.(If B\ is only embeddable in F, we thus assume that it is attached to C\ — (Pi U P2),and then B\ is clearly minimal by the definition of the relation <.) We claim that fori = 1, 2, . . . , m — 2, Bx < Bi+2> Suppose that this is not true. Let i be the smallestindex for which Bi+2 < Bt. Since Bj and Bj+2 do not overlap, they are <-comparablefor every j and thus such an index i exists. By our choice of B\, we have i > 1.Since Bt+2 is attached closer to C\ than £,-, and Bt overlaps with J3,_i, J5,+2 has anattachment on Pi or P2 that is closer to C\ than one of the attachments of £,_i onthe same path. Similarly, since Bt+2 overlaps with Bt+\ and Bi+\ > £,_{, the bridgeBi+2 has an attachment that is further away from C\ than an attachment of £;_i onthe same path Pi, or P2. This implies that Bi+2 overlaps with B,-_i. But this is notpossible since m > 5. This proves the claim. In particular, we know that Bm-2 < Bm.Since B\ is ^-minimal and ^-comparable with Bj if 7 =£ 1, 2, m, we have B\ < Bm-2.By transitivity we have B\ < Bm. This contradicts the fact that B\ and Bm overlap.The proof is thus complete. Q

In the last part of the above proof, we have learned even more than needed. Astraightforward extension gives the following result. Let us call a millipede two-sided ifit is attached to C\ - (Pi U P2) and to C2 - (Pi U P2). Otherwise it is one-sided. Someone-sided millipedes are shown in Figure 9.

Proposition 5.4. If M is a one-sided millipede, the number of K-bridges it includes is atmost 4. In particular, M is a small obstruction.

6. The 1-prism embedding extension problem

It remains to determine minimal obstructions for the 1-prism embedding extension prob-lem. Let us first extend a few definitions used in previous sections for the purpose ofthis section. If F is a face of an embedded graph K c G, and P, P ' are paths in Gwith endpoints on dF but otherwise disjoint from K, they are said to be disjoint crossingpaths with respect to F if they cannot be simultaneously embedded in F. The essentiallydifferent cases of disjoint crossing paths with respect to the face F of a 1-prism embeddingextension problem are shown in Figure 10 (up to symmetries). The only case where we

(a) (b) (c)

Figure 10 Disjoint crossing paths with respect to F

(d)

have one of the paths attached to Pi is (d), which also includes the possibility of theattachment at C\ n Pi or C2 n Pi. Tripods with respect to the face F are another kind ofobstruction for the 1-prism embedding extension problem. They are defined in the sameway as in the case when the face is bounded by a cycle, with the additional requirementthat it should be an obstruction. It turns out that tripods with respect to F can be dividedinto four classes as follows.

1 Attached twice to C\ — P\ or twice to C2 — P\ with the third attachment anywhere elseon dF and with no restrictions on non-degeneracy.

2 Attached to C\ — Pi, to Ci — Pi, and to Pi. The attachment on Pi is non-degenerate.3 Attached once to C\ — P\ (or to Ci — Pi) and twice to Pi. The attachment on Pi that

is closer to C\ (respectively, closer to C2) is non-degenerate.4 Attached only to Pi. The middle attachment on Pi is non-degenerate.

The 1-millipedes are another type of obstruction for the 1-prism embedding extensionproblem. These obstructions are of the same type as the millipedes are for the 2-prismembedding extension problem, and they can be arbitrarily large, though minimal, as well.A 1-millipede is a subgraph consisting of a path P2 joining C\ and C2 and disjoint fromPi, together with a millipede for the 2-prism problem with respect to K U Pi- Moreover,the following additional requirement is imposed on 1-millipedes.

(6) For j = 3, 4, . . . , m — 2, denote by /~ and r~ the vertices of attachment of B°_{ U B°_2

on K closest to C\ and C2, respectively. Similarly, let /+ and r+ be the extremevertices of attachment of B°+l U B°i+2. Then rf is strictly closer to Ci than r~, and /~is strictly closer to C\ than If.

Note that (6) is void if m < 5. It should also be pointed out that we assume thatl^ G C\ — P\ (an attachment of B\), and this is considered as being strictly closer to C\than any vertex on Pi. Similarly on the other side, where r+_2 £ Ci — P\-

Yet another type of obstruction will be needed. Let x\, xi £ V(P\) and suppose that xi

is closer on Pi to C\ than x\ is. A subgraph Q of G — E(K) is a left side obstruction withrespect to xy and X2 if it satisfies:(i) Q contains a path joining C\ — P\ with x\ (respectively, a path joining Ci — P\ with

(ii) No attachment of Q to K is closer to Ci than x\ (respectively, closer to C\ than X2),and no attachment of Q is on C2 — Pi (respectively, C\ — P\).

514 B. Mohar

>

(a) (b) (c)

Figure 11 The minimal left side obstructions

(iii) Q can be embedded in F in such a way that all feet of Q attached to Pi (strictly)between x\ and X2 are touching Pi at the right side of F. (The left and the right arewell defined with respect to Figure 2.)

(iv) Q cannot be embedded in F in such a way that all feet of Q attached to Pi (strictly)between x\ and xi are touching Pi at the left side of F.

We define the right side obstructions similarly. Their attachments on Pi between xi andX2 can be embedded on the left side of F, but cannot be embedded on the right side.

Examples of left side obstructions are given in Figure 11. We will prove in Theorem 6.3that Figure 11 contains all minimal left side obstructions attached to C\ —Pi, wherecase (c) of Figure 11 represents arbitrary two-sided millipedes for the following 2-prismembedding extension problem. Add the edge x\X2 {x\ and X2 are formerly non-adjacent)and embed it across the face F so that it is attached to xi on the left and to X2 on theright. Add also a path ?i from C\ — Pi to xi. Let P[ be the segment of Pi from C\till X2 and let C'2 be the cycle consisting of the segment X2X1 on Pi together with thenew edge. Then we consider the 2-prism problem with respect to K' = C\ U C2 U P[ U P2embedded in the cylinder as described above. Note that the first bridge in a two-sidedmillipede for this problem will be attached between Pi and P2 on C\, while the last onewill be attached to the segment of Pi between X2 and xi. It is easy to see that such amillipede is a left side obstruction. Note that one-sided millipedes do not give rise to leftside obstructions.

Having a left side obstruction Q.\ attached to C\ — P\ and a left side obstruction Q2attached to C2 — Pi (with respect to the same pair xi, X2), which do not intersect outof Pi, their union Qi U Q2 cannot be embedded in F. This way we get a rich family of1-prism embedding obstructions.

Before stating the main result of this section, we will prove a lemma about 3-connectedsubgraphs that will be needed in the proof. Let us recall that a graph H is nodally3-connected if the graph obtained from H by replacing each branch with an edge betweenthe corresponding main vertices is 3-connected.

Lemma 6.1. Let G be a graph with disjoint nodally 3-connected subgraphs K and L. Let n\,ni, iti be disjoint paths in G joining three main vertices of K with a triple of main verticesof L. Let J = X U L U TTI U 7i2 U 713. If for every branch e of K U L, no two consecutive (on


e) connected components of eC\ (n\ U ni U n^) belong to the same path 7i,, then J is nodally^-connected.

Proof. J is clearly 2-connected. Suppose now that there is a 2-separation J = J\ U J2,j j n Ji = {x,y}, where x and y are main vertices of J and each of J\, J2 contains two ormore branches of J. One of the paths TT, is disjoint from x, y and is thus totally containedin Ji, say. Since K, L are nodally 3-connected, J2 contains all main vertices of K U L.By our assumption, J has no parallel branches. Thus J\ contains a main vertex z of J.Clearly, z is obtained as the intersection of one of the paths, say n\, with a branch e inK, say. Since J2 contains all main vertices of K U L, x, y both lie on e and both lie onn\. Follow 7Ti from z in a direction out of the branch e. The first intersection with K U Lmust be on e. Otherwise we could reach a main vertex of K U L different from x, y. Byour assumption on J, there is a main vertex w between the two intersections of n\ withe such that w ^ 7C1. That vertex is neither x nor y and belongs to J\. By repeating theabove arguments with w, we see that x, y also belong to another path, 712, or n^. This is acontradiction. •

Our next result describes minimal obstructions for 1-prism embedding extension prob-lems. In cases (f)-(h) of Theorem 6.2, obstructions (and, in particular, 1-millipedes) aredefined with respect to the following 2-prism embedding extension problem. Suppose thatin G — Pi, there is no path from C\ — P\ to Ci — Pi. Let P2 be a path from C\ — Pi to avertex x on Pi such that x is as close as possible to C2. Let y be the neighbor of x onPi that is closer to C\ than x. Then let P[ be the segment of Pi from C\ to y, and let C2

be a cycle yx\xx2y, where xi and X2 are new vertices. We consider the 2-prism problemfor the subgraph K' = C{ U C2 U P[ U P2 of the graph G obtained from K1 by addingall K-bridges in G with an attachment on C\ — Pi. (In particular, no attachment on Piof these X-bridges is closer to Ci than x.) In case (g) (and (h)) an additional edge xz isadded into G'. The vertex z € V(P[) has the property that in G there is a path internallydisjoint from Pi joining Ci — P\ with z. The 2-prism problems for (f)-(h) in the casewhen P2 joins Ci — P\ with Pi are defined similarly. It should also be mentioned thatthe millipedes appearing in (h) are defined with respect to the above 2-prism embeddingextension problems.

Theorem 6.2. Let G and K = C\ U C2 U Pi ^ G be graphs for a 1-prism embeddingextension problem. Suppose that no K-component of G is attached just to P\. Then there isno embedding extension to G if and only if G — E(K) contains a subgraph Q of one of thefollowing types:

(a) Disjoint crossing paths with respect to the face of K.(b) A tripod with respect to the face ofK. If G — P\ contains a path joining C\ and C2, at

least one attachment of the tripod is not on P\.(c) A path P2 joining C\ and C2 and disjoint from P\ together with a dipod attached three

times to P\ and once to (C\ UC2) — (Pi UP2). If the dipod is degenerate, its degenerateattachment is not on P\.

516 B. Mohar

(d) A path Pi from C\ to Ci disjoint from Pi together with a 2-prism embedding extensionobstruction of type (b)-(f) of Theorem 5.3 with respect to K UP2. (It way happen thatsuch an obstruction is not minimal. In this case, a part of P2 can be removed.)

(e) A 1-millipede.(f) Same as (d) but with P2 joining C\ — P\ (or Ci — P\) with a vertex x E P\. No other

attachment of the obstruction is closer to C2 (respectively, to C\) than x.(g) Same as (f) where Pi joins C\ — P\ with x G Pi (respectively, C2 — P\ with x € P\)

where one of the branches of the obstruction joins x with another vertex z € P\. In Q,this branch is replaced by a branch joining C2 — P\ (respectively, C\ — P\) with z.

(h) A one-sided 1-millipede attached to C\ (or C2) and to a segment of P\. The pathP2 of the I-millipede joins C\ — P\ (respectively, C2 — P\) with the attachment x onP\ closest to C2 (respectively, C\). If the 1-millipede contains a branch joining x withanother vertex z on P\, this branch is possibly replaced in Qby a branch joining C2 — P1(respectively, C\ —P\) with z.

(i) Union Q = Qi U fi2> where Qi Pi Q2 ^ Pi. For i = 1, 2, Q, contains a path 7c, joiningQ — P\ with P\. The end X2 of 712 on P\ is closer to C\ than the end x\ ofn\. Moreover,Qi and Q2 are both left side, or both right side obstructions with respect to X\ and %i.

(j) A Kuratowski subgraph contained in a 3-connected component of the auxiliary graphG. The 3-connected component does not contain auxiliary vertices.

Cases (f), (g), (h), and (i) appear only when in G — P\ there is no path from C\ to C2.Moreover, there is a linear time algorithm that either finds an embedding extension of K toG, or returns an obstruction Qfor the embedding extendibility. In the latter case, Qfits oneof the above cases (a)-(j).

Proof. First of all, we can check the embedding extendibility by applying Theorem 4.3.Suppose now that there is no embedding extension of K to G. Consider the block(s) ofG containing C\ and C2. If there is an embedding extension of K to this (these) block(s),there is also an embedding extension to G, unless we have (j). If C\ and C2 are in differentblocks of G, there is no embedding extension if and only if one of them, say the onecontaining C\, has no embedding extension in the disk with C\ on the boundary. We leavethis case to the end of the proof, since we will reduce it to the 2-connected case. Supposenow that C\ and C2 are in the same block of G. To simplify notation, we will assume thatthis block is G itself, i.e., G is 2-connected. Then there is no embedding extension if andonly if one of the 3-connected components of the cylinder auxiliary graph G is non-planar.(See the details in the proof of Theorem 4.3, case k = 2.) If the 3-connected component(s)of G containing the auxiliary vertices is (are) planar, then we have (j). Otherwise, let Gbe the graph obtained from G by adding an edge between the auxiliary vertices. SinceG is 2-connected, the cycles C\ and C2, and the auxiliary vertices will be in the same3-connected component of G. The other 3-connected components of this graph are also3-connected components of G, and can thus be eliminated.

We assume from now on that G is 3-connected. First of all, we try to find two disjointpaths in G — Pi joining C\ and C2. The search for such paths can be performed in lineartime by standard flow techniques, which were also used in our previous results. Suppose


first that we have found such paths Pi and P3. If Pi, P2, P3 are not attached consistentlyon C\ and Ci, then we have (a). Suppose now that this is not the case. Then we try tochange Pi and P3 in such a way that there are no local bridges of K U Pi U P3 attachedonly to Pi or only to P3. To achieve this goal, we use the same technique as in the proofof Theorem 4.3. Let Kf =KUP2U P3. For i = 2, 3, let Gt be the graph consisting of P,together with all its local K'-bridges and with an additional edge joining the ends of P,. IfGt is planar, an algorithm from [15] replaces P, with a new path that has no local bridgesattached to it. So, we either achieve our goal, or get one of G,, say G3, to be non-planar.Let C be the cycle composed of the paths Pi and Pi together with the segments on C\from Pi to P3 and from P3 to Pi, and the segments on Ci from Pi to P3 and from P3 toPi. Denote by B the (K U P2)-component in G that contains P3. If B contains a vertexof (Ci U Ci) — C, then a path in B from that vertex to an end of P3 together with Pidetermines an obstruction of type (a). Therefore, we may assume that B is attached onlyto C. Since G3 is non-planar, C U B is non-planar. By Lemma 4.2, the auxiliary graph ofCUB with respect to C is 3-connected. By Theorem 3.1, we can find in B a pair of disjointcrossing paths or a tripod with respect to C. We can change the obtained obstruction asin the proof of Theorem 5.3 to get a 2-prism obstruction in the graph H = K U P2 U Bwith respect to its subgraph K U P2. It follows from the proof of Theorem 5.3 that theonly obstructions that appear in this case are types (b)-(e) of Theorem 5.3. Thus we haveour case (d).

Suppose now that P2 and P3 do not have local K '-bridges. By Theorem 4.1, theobstructions to the non-extendibility of the embedding of K' to G are rather simple. SinceG is 3-connected and in G there are three disjoint paths from C\ to Ci, the auxiliarygraph G of G is 3-connected also. Therefore, we need only consider cases (a)-(d) ofTheorem 4.1. Case (a) of Theorem 4.1 gives our case (a) or (d) (the latter one being case(d) of Theorem 5.3). Case (b) yields case (b) of Theorem 5.3, thus our case (d). (Here weneed to be careful in selecting two paths among Pi, P2, P3 for which we get the 2-prismobstruction. We need to take Pi. The other path is P3 if the tripod obstruction of case(b) is attached to P2. Otherwise, we take P2.) Case (c) of Theorem 4.1 yields case (c) ofTheorem 5.3, thus our case (d). Finally, in case (d) of Theorem 4.1, we either have cases(b) or (d) of Theorem 5.3 (thus our case (d)), or we have a degenerate dipod attachedthree times to Pi and disjoint from P2, say. In the latter case, we get our possibility (c).(Note that (c) is contained in (d) if the dipod is non-degenerate.)

Next we suppose that there are no two paths P2, P3 as asked for above. Suppose thatthere is a path P2 disjoint from Pi joining C\ and Ci. Let B be the K-bridge containingP2. We will show that B either contains a subgraph of type (a) or (b), or P2 can bechanged so that the local bridges on P2 will disappear. In the latter case, we will be ableto get an obstruction of type (a), (d), or (e).

For i = 1, 2, let St be the segment on C, between the leftmost' and the 'rightmost'attachment of B to Ct - P{. Let B' = (B - P{) U S{ U S2. For i = 1, 2, if S,- is not just avertex, let xt be the end of P2 on S,-. Otherwise, we add to B' a pendant edge attachedto the vertex S,, and we let xt be the new vertex on this edge. Let Qo, Qi, . . . , Qt be theshortest sequence of blocks of Bf satisfying the following conditions:

518 B. Mohar

(i) X! € V(Qo), x2 € V(Qt), and(ii) for i = 1, 2, . . . , t, Qi-\ and Qt intersect at a cutvertex w,- of B'.

By the minimality requirement for t, the blocks Qt and the cutvertices w, are all distinctand uniquely determined. We also define wo = x\ and wt+\ = xi. Let us mention that thesequence wo, Qo, wi, Qi, . . . , wf, Qr, wt+i can be determined in linear time by standardbiconnectivity algorithms. We also note that wi, H>2, . . . , wt are vertices of P2. By ourassumption, Bf contains no two disjoint paths from C\ to C2. Thus, by Menger's Theorem,we have t>\.

Suppose that one of the blocks Q, (1 < f < t) cannot be embedded in the plane withwt and wi+\ on the boundary of the outer face. Let K, be its subgraph obtained from theKuratowski subgraph of Qt + w,-Wj+i by deleting its edge w/w,-+i if necessary. Next, findin G three disjoint paths n\, 712, 713 from K to the main vertices of Kt (in linear time byusing flow techniques). This is possible since G is 3-connected. We will show next that wecan change the paths Uj in such a way that one of them will be attached to the end ofP2 on C\ and one of them to the end of P2 on C2. Since Qt is 2-connected, there are twodisjoint paths in Qt from {wi9 wi+\} to main vertices of Kt. Let n[ and n2 be obtained fromthese paths by adding a segment of P2 from w, to C\ and from w,+i to C2, respectively. Ifn[ is disjoint from n\, 7U2, 713, it can replace n\. Otherwise, suppose that n[ first meets n$in the direction from C$. Do the same with n2: if it does not intersect any of the paths,it can replace 712. If its first intersection from C2 with n\ U 712 U 713 is on %2 (or similarly713), we replace n\ with the segment of n[ up to its intersection with n\ followed by theremaining segment of n\, and we replace 7C2 with a path consisting of the initial segmentof nf

2 and the terminal segment of 712. Our goal for the paths to attach to C\ and to C2 isthen satisfied. The remaining possibility is when n2 first intersects n\ as well as n\ does.In this case, let y\, yi,... be the consecutive intersections of n[ with n\ U712 Un^. Similarly,let zi, Z2, . . .be the intersections of n2 with the union of the paths. By our assumption,yu z\ G V(n\). Suppose that p, q are the largest indices such that all y\, . . . , yp and z\,. . . , zq belong to TII. Let y be the vertex among y\, ..., yp, zi, . . . , zq that is closest to theend of 7Ci in Kf. Suppose that y e n[. Now replace n\ by the segment of n[ till y and thesegment of n\ from y to its end at a main vertex of Kt. Now, n2 either does not intersectthe paths 7ij at all, or intersects first a path distinct from n\, and we can apply the aboveprocedure to fulfil our task.

Now we have three disjoint paths nu n2, 713 joining K with three of the main verticesOf Kh where one of the paths starts at C\ n P2 and another starts at C2 n P2. Note thatthese two paths pass through w, and w,+i, respectively. Let H be the graph obtained fromKt by adding the three paths and the cycle C obtained as follows. Let e\ and e2 be edgesof C\ and C2, respectively, that are adjacent to Pi (both at the same side of Pi withrespect to the given embedding of K in the cylinder). Then let C be the cycle obtainedfrom K — e\ — e2 by adding a new edge between the two vertices of degree one. We canchange n\, 7C2,713 so that the graph H = C UKt^ U n\ U112 U 7C3 has no parallel branches. ByLemma 6.1, the auxiliary graph of// with respect to the cycle C is nodally 3-connected.By Theorem 3.1, H contains a tripod T (since there are only three attachments on C). Byconstruction of//, the tripod T is also a tripod in G with respect to the face of K, since

it is attached twice to K — Pi, and if the third attachment lies on Pi, it is non-degenerate.Thus we have our case (b).

Consider now Qo and suppose that it is non-trivial, i.e., S\ is not just a vertex. Let pand q be the endpoints of the segment Si, and let Qf

0 be the graph obtained from Qoby adding the edges pw\ and qw\ to it. If Qf

0 has no embedding in the plane such thatthe cycle Si + pw\ + w\q bounds the outer face, we can get an obstruction Qo by usingTheorem 3.1. If Qo is attached to the cycle at wi, we add to it the segment on Pi fromwi to Ci. Now Qo is either a pair of disjoint crossing paths with respect to the face of K(case (a)) or a tripod attached to X — Pi (case (b)).

We perform the similar procedure with Qt. Not having obtained an obstruction, weknow that the graph Q = g0 U Q\ U • • \U Qt can be embedded in the face F of K. SincePi ^ Q, every (K U P2)-bridge in G that is locally attached to P2 is totally contained inQ. Now, since Q can be embedded in F, the algorithm of [15] enables us to remove thelocal bridges at P2 in linear time. Note also that after a possible change of P2 by anotherpath in Q with the same endpoints, P2 still passes through wi, .. . , wf, and that for every(K U P2)-bridge B, there is some i, 0 < i < t, such that B is attached to P2 only betweenw, and Wj+i.

Now we can apply Theorem 5.3 for the subgraph K' = K U P2 of G. We note thatP2 plus a 2-prism embedding obstruction of Theorem 5.3 is not necessarily a minimalobstruction for our embedding extension problem. Clearly, the deletion of any branchesnot in P2 is not possible, since we have a minimal obstruction for the embedding extensionof K'. However, P2, or a part of it, may be superfluous in the obstruction and may thenbe omitted. Note that case (a) of Theorem 5.3 gives our case (a), and cases (b)-(f) give (d).Case (h) of Theorem 5.3 can be excluded because of our initial connectivity reductions.In the remaining case (g) of millipedes, we claim that we really get a 1-millipede. Weneed to show that the corresponding millipede for K' satisfies (6). In order to achieve thisproperty, we change P2 before applying Theorem 5.3, as explained in the next paragraph.

For i = 0, 1, . . . , t, let Q- be K' together with all K'-bridges attached to the segmentw/Wf+i of P2, except for those K'-bridges whose only attachment on P2 is one of w,, w,+i.If the embedding of K' cannot be extended to the obtained subgraph Qf

t of G, we getan obstruction of type (a), (d), or a millipede. Having a 1-sided millipede, its length mis at most 4 (Proposition 5.4), and thus it is clear that it satisfies the required property(6). Two-sided millipedes are excluded, since in Qf

t, bridges of Kf are either not attachedto C\ — P\ (if / 7= 0), or are not attached to Ci — P\ (if i ^ t), since t > 1. Thus wemay assume that we have an extension of the embedding of K' to Q\. Suppose first thati i= 0, t. Consider the induced embedding of Qt c Q'. and change the segment w/w,+i ofP2 to be the leftmost path in Qt from w, to w,-+i. After this change of P2, there is justone bridge Ri attached to the right side of the segment (with respect to the embeddingof Qi) that has an attachment on Pi, since Qt is 2-connected. Unfortunately, some localbridges attached at the right side of the segment may arise. In such a case, replace theobtained segment of P2 by the rightmost path through such local bridges. It is easy to seethat, because of the 3-connectivity, local bridges disappear after this change. On the rightside of the new segment, the same bridge Rt remains as the only bridge on the right of it,while on the left, we can get more than one bridge. No two of the left bridges overlap.

520 B. Mohar

Moreover, every left bridge overlaps with Rt since Rt is attached to wt and to w,-+i. Weperform similar changes for i = 0, t (and possibly change the ends of Pi on C\ and Ci).

Suppose that, after the above change of P2, we get a millipede M when applyingTheorem 5.3. If M is one sided, it satisfies (6) since m < 4. If M is two sided, it cancontain at most three bridges from Q't. If it contains two or three such bridges, Rt isamong them. Suppose now that for some j , 3 < j < m — 4, r+ < rj (i.e., (6) is violated).Notation < means being closer to C\ than to C2 on Pi. Then rj = rj. By (4), r = r t isthe only attachment of £°+ 1 U B°j+2 on Pi. Since BJ < B°+2, £°+1 overlaps with Bj andwith B°+2 on Pi. This implies that B°, B°+{, B°+2 are the three bridges of some Q\ andthat B°j+l = Rt. Since £°+ 2 is attached only to Pi U P2, we have ; + 2 < m. Thus, there is abridge £°+ 3 overlapping with B°+2 and not overlapping with £°+1. But clearly, this is notpossible. We have a contradiction, so rj <r^. The proof that Ij < /+ is similar. Hence,(6) holds.

We have covered the case when, in G — Pi, there is a path from C\ to Ci. Suppose nowthat this is not the case. Then the K -bridges can be partitioned into classes ^ 1 , ^ 2 suchthat 0&[ (i = 1, 2) contains exactly those bridges that are attached to Q — P\. Let x\ be thevertex of Pi as close to Ci as possible such that there is a bridge in 3&\ that is attachedto xi. (If none of the bridges in $\ is attached to Pi, we let xi be the end of Pi on C\.)Define xi similarly for the bridges in ^ 2 . For i= 1, 2, we let G, be the graph consisting ofCt, the segment of Pi from C, to x,-, and the bridges in ^St. We will use the same notationlater when providing details for the case when C\ and Ci are in distinct blocks of G.Clearly, this is not the case if and only if on Pi, xi is strictly closer to Ci than xi.

Let 7Ci be a path in Gi joining C\ with xi such that n\ Pi Pi = {xi}. Define ni similarlyin Gi. Suppose that we have an embedding extension of K to G. If n\ is attached to xi atthe left side of F (with the obvious meaning of the 'left' with respect to Figure 2), then niis attached at the right side. Then all the attachments of Gi to Pi at xi and between xiand xi are also on the left. Thus we say that Gi has the left side embedding and, similarly,Gi has the right side embedding with respect to xi and xi. There are two possibilities forthe non-existence of embedding extensions: either Gi (or Gi) admits neither the left northe right side embedding, or each of Gi and G2 does not admit the left side embedding(respectively, the right side embedding), but each of them admits the right (left) sideembedding.

Suppose first that Gi admits neither side embeddings. What are the possible obstruc-tions? Define the graph G\ as follows. Let y e V(P\) be the neighbor of xi that is closerto C\ than xi. Replace the edge yx\ of G\ by a pair of paths of length two between thesetwo vertices and denote by C'2 the obtained cycle of length 4. In addition to this, add theedge X1X2. It is easy to see that Gi has neither side embeddings if and only if the obtainedgraph G\ has no embedding extending the 1-prism embedding of K' = C\ UC2U(P\ CiG\)with C\ and C2 on the boundary. (Note: the two embeddings of K' are really equivalentto each other.) This type of 1-prism embedding extension problem has been covered above- the case when there is a path n\ disjoint from Pi joining the two cycles. Since C2 isjoined to C\ only through y and xi, we have just one such path. It is easy to see thatG\ is 2-connected and that the auxiliary graph G\ with the auxiliary vertices joined toeach other is 3-connected (since there are no local bridges on Pi). As shown above, this


gives rise to obstructions of types (a), (b), (d), or (e) for the extension problem of G\. Theobtained obstruction possibly contains the new edge X1X2, which is not present in G. Byreplacing this edge by 712, we get an obstruction Q for our original 1-prism embeddingextension problem. If Q is of type (a) (with respect to G\\ it is easy to see that 712 Q,and that Q fits our case (a) as well. If Q is of type (b), then, again, 712 is not a part ofQ. Thus Q fits case (b). Note that in this case Q can be a tripod attached only to Pi. Inthe case of type (d), the path appearing in (d) is n\ (which may have been changed whenconstructing the obstruction). We have our case (f) or (g) with P2 = n\. In case (e) in G\,note that Q cannot be attached to C2 — (P\ U P2). Thus the corresponding 1-millipede isone-sided, and we have (h).

Up to symmetries, the only remaining case is when neither G\ nor G2 admits the leftside embedding. We may as well assume that Gi and G2 admit the right side embeddings.Then we are looking for minimal left side obstructions. It is clear that we get case (i).

It remains to consider the case when C\ and C2 are in distinct blocks of G. Let Gi andG2 be the corresponding blocks. We suppose that Gi has no embedding in the plane withC\ bounding a face. An obstruction for this will obstruct the original embedding extensionproblem. Let us remark that using Theorem 3.1 is not a straightforward success since anobstruction obtained by using that result could intersect Pi too many times. However, theapplication of Theorem 3.1 is possible if Gi Pi Pi is just the end of Pi on C\. Then theobstruction is either our case (a), or (b) (or (j)). Thus we assume that this is not the case.

Note that Gi n Pi is a segment of Pi from C\ to xi. Let y be the neighbor of xi that iscloser to Pi than xi. By the assumption made above, y is well-defined. Define G\ to be thegraph obtained from Gi by replacing the edge yx\ with two paths of length two betweenxi and y. Denote by C2 the obtained cycle of length four consisting of these two paths.Clearly, G\ has an embedding in the plane with C\ and C2 bounding faces if and only ifGi has an embedding with C\ bounding a face. By our assumption, this is not the case.Thus G\ has no 1-prism embedding extension with respect to K' = C\ U C2 U (Pi Pi Gi).Possible obstructions have been classified above since in this problem we have a pathdisjoint from the first one. We get obstructions of types (a), (b), (f), or (h). •

Case (i) of the previous theorem is well-described if we know what the minimal leftside obstructions are and how we get them in linear time. Their discovery was covered inthe theorem. The proof of the last theorem was also detailed enough to yield a simpleclassification of these obstructions.

Theorem 6.3. Let Q be a minimal left side obstruction with respect to x\ and xi. Then Qis one of the graphs shown in Figure 11, where case (c) represents an arbitrary two-sidedmillipede for the 2-prism embedding extension problem described before Lemma 6.1.

Proof. We will use all the notation and assumptions introduced in the preceding proofup to the point where we encountered the case (i). We suppose that there is a left sideobstruction in G\.

Let P[ be the segment of Pi from C\ to X2, and let C2 be the segment of Pi from xi toxi together with an additional edge X1X2. (If *2 *s Ju s t preceding xi on Pi, then we add a

522 B. Mohar

path of length two in order not to get parallel edges.) Define G\ to be the graph obtainedfrom C\ U P[ U C2 by adding all bridges from $&\. Then G\ has no left side embedding ifand only if G\ has no embedding in the cylinder with C\ and C2 on the boundary (withC\ at the bottom and C2 with its new edge on the right side). Thus we are looking for1-prism embedding extension obstructions in G\ with respect to K' = C\ U C2 U P[. Sincein 08 \ there is a bridge joining C\ — P[ with x\ e C2 — P{, we get the case with two orthree disjoint paths from C\ to C2 (counting also the path P[). If there are three disjointpaths, they obstruct the right side embeddings of G\. By applying Theorem 6.2, we get anobstruction Qi of type (a), (b), (d), or (e) with respect to G[, since these are the possiblecases that arise when in addition to P{, there is only one path. In cases (d) and (e), we maysuppose that the corresponding path Pi is %\ (which has possibly been changed duringthe procedure of constructing the obstruction, but its end xi has remained unchanged).We know, moreover, that Qi has right side embeddings, since G\ has such an embeddingin F. Let us now consider particular cases for the resulting left side obstruction.

If Qi is of type (a) (in G\), we claim that the disjoint crossing paths Q\, Qi can bechanged in such a way that one of them is attached to xi. If this is not already the case inQi, add the path n\. Note that each of Qi, Qi is attached to C\ — P\ and to the segmentof Pi between %2 and xi, since otherwise they also obstruct the right side embeddings. Ifn\ intersects Q\ or Qi in an internal vertex, we can change Q\ or Q2 so that one of themis attached to xi. Otherwise, we can either replace one of the paths by n\, or Q\ Un\obstructs the right side embeddings. Hence the claim. Consequently, we have a left sideobstruction represented in Figure 11 (a).

If Qi is of type (b), it is also the right side obstruction. If it is of type (d), correspondingto one of the cases (b), (d), (e), or (f) of Theorem 5.3, it is a right side obstruction aswell. Type (d), case (c) is a right side obstruction except in two cases. One of them isrepresented in Figure ll(b), while the other contains case (a) of Figure 11 after removingthe middle part of n\. Finally, if Qi is a millipede, it does not obstruct the right sideembeddings if and only if it is two-sided. So, the last type of minimal left side obstructionis as claimed. •

7. Conclusion

There is an additional property that the millipedes may be assumed to have. This propertyis essential for our further applications of the results of this paper and will be stated inour last results.

An extended millipede (or an extended 1-millipede) is defined by the same conditions(l)-(4) (and (6)) as the millipede, but (5) is replaced by the requirement that B° (1 < i < m)is an H-graph of a K -bridge in G. This ensures, in particular, that Bf (2 < i < m — 1)are attached to Pi and to P2, but we lose the minimality property of millipedes asobstructions.

For Q c G — E(K), we define b(Q) to be the number of branches of Q where all verticesof attachment of Q (including those of degree 2 in Q) are considered to be main verticesof 0.

Theorem 7.1. Let K c G be a subgraph for a 2-prism embedding extension problem. Thereis a linear time algorithm that either finds an embedding extension ofK to G, or returns anobstruction Q c G — E(K) for such extensions. In the latter case, the obstruction Q satisfiesone of the following conditions:

(a) Q is small, b(Q) < 20, K U Q has at most four K-bridges, and at most 8 vertices ofQare on Pi.

(b) Q = B\ U B\ U • • • U B°m is an extended millipede of length m>5. Then b(Q) < 5m andat most 2m vertices o/Q are on Pi. Moreover, if D f=2 B% U • • • U B ^ j J is the union ofall K-bridges in G that are attached only to P\ U Pi, there is an embedding extensionofK to KuD.

Proof. Consider first the 2-prism problem for K U D. By applying Theorem 5.3, weeither get an embedding extension t o X u D or a small obstruction, since millipedes haveattachments out of Pi UP2. In the latter case we have (a). Otherwise, we apply Theorem 5.3again, this time for the original 2-prism problem. What we have gained, is that in the caseof millipedes, we can guarantee the property stated in (b). The stated bounds follow fromTheorem 5.3. Note that the case of millipedes having m < 5 is hidden in our case (a). •

Theorem 7.2. Let K = CiUCiU P\ <= G be a subgraph for a 1-prism embedding extensionproblem. Suppose, moreover, that there is a path in G disjoint from P\ that joins C\ andCi. There is a linear time algorithm that either finds an embedding extension of K to G, orreturns an obstruction Q £= G — E(K) for such extensions. In the latter case, the obstructionQ satisfies one of the following conditions.

(a) Q is small, and b(Q) < 29.(b) Q = PiUB^UB^U- • -UJ5° is an extended 1-millipede of length m > 5. Then b(Q) < 7m.

Moreover, if D (^ B^ U • • • U B°m_x) is the union of all (K U P2)-bridges in G that areattached only to P\ U Pi, there is an embedding extension ofK to KU PiU D.

Proof. Apply the algorithm described in the proof of Theorem 6.2 except that instead ofusing Theorem 5.3 within that proof, we use Theorem 7.1 instead. The stated bounds onthe branch size also follow from Theorem 7.1. •

Acknowledgement

We are greatly indebted to the referee for a very detailed checking of the manuscript andfor his numerous comments that improved the presentation.

References

[1] Archdeacon, D. and Huneke, P. (1989) A Kuratowski theorem for non-orientable surfaces. J.Combin. Theory, Ser. B 46 173-231.

[2] Booth, K. S. and Lueker, G. S. (1976) Testing for the consecutive ones property, interval graphs,and graph planarity using Pg-tree algorithms. J. Comput. System Sci. 13 335-379.

[3] Chiba, N., Nishizeki, T, Abe, S. and Ozawa, T. (1985) A linear algorithm for embedding planargraphs using PQ-tYQQS. J. Comput. System Sci. 30 54-76.

524 B. Mohar

[4] Cook, S. A. and Reckhow, R. A. (1973) Time bounded random access machines. J. Comput.System Sci. 7 354-375.

[5] de Fraysseix, H. and Rosenstiehl, P. (1982) A depth-first search characterization of planarity.Ann. Discrete Math. 13 75-80.

[6] Gross, J. L. and Tucker, T. W. (1987) Topological Graph Theory, Wiley-Interscience.[7] Hopcroft, J. E. and Tarjan, R. E. (1973) Dividing a graph into triconnected components. SI AM

J. Comput. 2 135-158.[8] Hopcroft, J. E. and Tarjan, R. E. (1974) Efficient planarity testing. J. Assoc. Comput. Mach. 21

549-568.[9] Jung, H. A. (1970) Eine Verallgemeinerung des n-fachen Zusammenhangs fiir Graphen. Math.

Ann. 187 95-103.[10] Juvan, M., Marincek, J. and Mohar, B. (submitted) Embedding graphs in the torus in linear

time.[11] Juvan, M , Marincek, J. and Mohar, B. (in preparation) Efficient algorithm for embedding graphs

in arbitrary surfaces.[12] Juvan, M. and Mohar, B. (submitted) A linear time algorithm for the 2-restricted embedding

extension problem.[13] Karabeg, A. (1990) Classification and detection of obstructions to planarity. Linear and Multi-

linear Algebra 26 15-38.[14] MacLane, S. (1937) A structural characterization of planar combinatorial graphs. Duke Math.

J. 3 460-472.[15] Mohar, B. (1993) Projective planarity in linear time. J. Algorithms 15 482-502.[16] Mohar, B. (submitted) Universal obstructions for embedding extension problems.[17] Mohar, B. ( in preparation) A Kuratowski theorem for general surfaces.[18] Papadimitriou, C. H. and Steiglitz, K. (1982) Combinatorial Optimization: Algorithms and Com-

plexity, Prentice-Hall.[19] Robertson, N. and Seymour, P. D. (1990) Graph minors. VIII. A Kuratowski theorem for

general surfaces. J. Combin. Theory, Ser. B 48 255-288.[20] Robertson, N. and Seymour, P. D. (1990) Graph minors. IX. Disjoint crossed paths. J. Combin.

Theory, Ser. B 49 40-77.[21] Seymour, P. D. (1980) Disjoint paths in graphs. Discrete Math. 29 293-309.[22] Seymour, P. D. (1986) Adjacency in binary matroids. European J. Combin. 7 171-176.[23] Seymour, P. D. (submitted) A bound on the excluded minors for a surface.[24] Shiloach, Y. (1980) A polynomial solution to the undirected two paths problem. J. Assoc.

Comput. Mach. 27 445-456.[25] Thomassen, C. (1980) 2-linked graphs. European J. Combin. 1 371-378.[26] Tutte, W. T. (1966) Connectivity in Graphs, Univ. Toronto Press, Toronto, Ontario; Oxford Univ.

Press, London.[27] Williamson, S. G. (1980) Embedding graphs in the plane - algorithmic aspects. Ann. Discrete

Math. 6 349-384.[28] Williamson, S. G. (1984) Depth-first search and Kuratowski subgraphs. J. Assoc. Comput. Mach.

31 681-693.

A Ramsey-Type Theorem in the Plane

JAROSLAV NESETRIL+ and PAVEL VALTR*

^Department of Applied Mathematics, Charles University,Malostranske nam. 25, 118 00 Praha 1, Czech Republic

JGraduiertenkolleg 'Algorithmische Diskrete Mathematik',

Fachbereich Mathematik, Freie Universitat Berlin,Takustrasse 9, 14194 Berlin, Germany

We show that, for any finite set P of points in the plane and for any integer k > 2, thereis a finite set R = R{P,k) with the following property: for any ^-colouring of R there is amonochromatic set P,P ^ R, such that P is combinatorial^ equivalent to the set P, andthe convex hull of P contains no point of R \ P. We also consider related questions forcolourings of p-element subsets of R (p > 1), and show that these analogues have negativesolutions.

1. Introduction and statement of results

In this paper we investigate geometrical Ramsey-type results that are related to thecelebrated Erdos-Szekeres Theorem.

Theorem 1. (Erdos-Szekeres Theorem) For every positive integer n there exists a positiveinteger ES(n) such that any set X of ES(n) points in general position in the plane (i.e., nothree lie on a line) contains vertices of a convex n-gon.

The Erdos-Szekeres Theorem is one of the original gems of Ramsey theory. Bycombining it with Ramsey's theorem [17] (see also [5] or [14]) itself we get the followingcorollary.

Corollary 2. For every choice of positive integers /?, /c, n, there exists a positive integerES(p,k,n) with the following property: for any set X of at least ES(p,k,ri) points in general

* The author was supported by 'Deutsche Forschungsgemeinschaft', grant We 1265/2-1.

526 J. Nesetfil and P. Valtr

position in the plane, and for any partition (*) = C\ U ... U Q , there exists a set Y cX,\Y\ = n9 such that all psubsets of Y belong to one class Cio of the partition and Y isthe set of vertices of a convex n-gon.

To verify the corollary, one simply puts ES(p,k,n) = r(p,k,ES(n)), where r(p,k,N) isthe usual Ramsey number for p-tuples and k colours.

We are interested in a generalization of Corollary 2 to sets Y of a given configuration.Somewhat surprisingly, this generalization can be made for p = 1 (i.e., for partition ofpoints), while in general (p > 1) a similar statement is not true.

Here is the key concept of this paper: two finite planar point sets P and Q are calledcombinatorially equivalent if there exists a bijection i : P —• Q such that p G conv(P') ifand only if i(p) G conv(f(P')) for any p G P and P' c p. Here conv(AT) is the convex hullof the set X and i(X) denotes the set {i(x) : x G X}.

A finite planar point set X is said to be convex independent if conv(X') j= conv(X) forevery proper subset X' of X (or, equivalently, if points of X are vertices of a convexpolygon). Otherwise X is said to be convex dependent. It is easy to see that two sets arecombinatorially equivalent if and only if there is a bijection between them that preservesboth convex dependent and convex independent sets.

Combinatorially equivalent sets and similar concepts have already been studied explic-itly in several papers (see [3] or [4] for a survey). Implicitly they play a very importantrole, mainly in discrete and computational geometry. See [10] also for a survey on moregeneral structures. In Section 2 we prove the following Ramsey-type theorem.

Theorem 3. For any finite set P of points in the plane, and for any integer k > 2, there isa finite set R = R(P,k) of points in the plane such that for any partition of R into k colourclasses there is a subset P of R with the following three properties:(i) P is monochromatic (i.e., it is a subset of one of the colour classes),(ii) P is combinatorially equivalent to P,(Hi) conv(P)nR = P.

Moreover, as we shall see from the proof, the set P in Theorem 3 may be required tobe an affine transform of P.

A subset I of a finite planar point set P is called a hole in P, or simply a P-hole, oran empty polygon, if X is convex independent and conv(X) HP = X. Horton [8] (see also[1] or [19]) proved that, for n > 7, the Erdos-Szekeres Theorem cannot be strengthenedto guarantee the existence of an n-hole. The existence of a 5-hole in any set of 10 pointsin general position in the plane was shown by Harborth [7], while the case n = 6 is stillopen.

Now Theorem 3 can be rephrased as follows.

Theorem 4. For any finite set P of points in the plane and for any integer k > 2, there isa finite set R = R(P,k) of points in the plane such that for any partition R = C\ U... U Qthere is an injection i : P —• R with the following three properties:(i) i(P) c Ciofor some i0 G {l,...,/c},

A Ramsey-Type Theorem in the Plane 527

(ii) i preserves convex independent and convex dependent sets,(Hi) i maps P-holes to R-holes.

Motivated by Corollary 2, one can also consider a higher-order Ramsey-type theorem(for p > 1). If we are partitioning the set (2) into k colour classes, a result analogous toTheorem 3 cannot be valid in general: the pairs of any finite set R can be coloured asfollows. On every line with at least two points of R we colour pairs of consecutive pointsalternately by two colours while all the other pairs are coloured arbitrarily. In this waywe avoid a monochromatic triple of collinear points of R containing no other point of Rin the convex hull. It follows that if we are partitioning the set (2), a result analogous toTheorem 3 cannot be valid for planar point sets P that are not in general position. Moregenerally, for p > 2, if we are partitioning the set (K), a result analogous to Theorem 3cannot be valid for planar point sets P with p+1 points on a line.

However, the situation is more difficult if we restrict our attention to point sets P ingeneral position. It turns out that Theorem 3 has no higher-order analogues even in thiscase. In fact, this remains true even if we drop the hole-preserving condition:

Theorem 5. For every p>2, there exists a finite planar point set P(p) in general positionwith the following property: there exists a partition (R ) = C1UC2 of all p-element subsets ofthe plane into two colour classes such that no monochromatic subset oflR2 is combinatoriallyequivalent to P(p).

Thus Theorem 3 fails to have a higher-order analogue in general. However, such ananalogue holds in some particular cases. Apart from Corollary 2 (which yields such ananalogue for any finite convex independent set P), we have the following result, whichdeals with the configuration Q containing the three vertices and an inner point of atriangle.

Theorem 6. Let Q be a convex dependent set of four points in general position in the plane.Then, for every integer k > 1, there exists a finite planar point set R = R(k) such that forevery partition of pairs of R into k classes, one of the classes contains all 6 pairs of pointsof some 4-point set combinatorially equivalent to Q.

The paper is divided into sections as follows: Section 2 contains the proof of Theorem 3;Section 3 contains the proofs of Theorems 5 and 6; and Section 4 contains concludingremarks.

2. Proof of the main result

We begin with a short outline of the proof of Theorem 3. Given a planar point set, we findan equivalent set P whose points are placed inside a small neighborhood of a line. Thenwe construct a set R containing many subsets combinatorially equivalent to P and applythe Hales-Jewett Theorem to show that at least one of these subsets is monochromatic.

Fix a set P in the plane, and let n = \P\. Without loss of generality (or by a slight

rotation of P), we may assume that all x-coordinates of the set P are different. Let thepoints of P, ordered according to their x-coordinate, be p(\),...,p(n).

Let M > 0 be any number such that all points of the set P lie inside the M-neighborhoodof the x-axis (i.e., the ^-coordinates of the points in P lie in the interval (—M, M)). For anys > 0, let Pe be the set obtained from the set P by replacing each point (x,y) by the point(x, j^y). If £ 7 0, the set PE is equivalent to P and is placed inside the ^neighborhoodof the x-axis. Otherwise (if s = 0) the points of Po are collinear. The points of Pe, listedaccording to increasing x-coordinate, will be denoted by pe(l),...,pe(n).

Let a G [O,TT) and let ra be the rotation of the plane by the angle a around the origin.Thus ra(x) denotes the point that we get by rotating x by ra. Put p£,a(0 = ra(pE(i)) (fors > 0,i = l, . . . ,n) and P£a = {/7£a(l),...,p£a(n)} (for e > 0). Thus we could also writeP£,a = ra(Pe).

To invoke the Hales-Jewett Theorem we need some notation. Put A = {l,...,w} (andthink of A as an alphabet). Given a positive integer N, we consider the set AN of allmappings {1,...,JV} -» A One can also think of AN as the N-dimensional cube over A.A (combinatorial) line L in AN is defined as an n-element subset of AN satisfying thefollowing condition: there exists a proper subset co a {1, . . . , N} and a mapping fo:a>-*Asuch that the line L is formed by all mappings / : {1, . . . , N} —• A that satisfy f(i) = fo(i)for i e co and f(i) = f(j) whenever ij £ co. More explicitly, L = {x(l),...,x(n)}, where

and

/o(0 for i G coi for i ^ co.

Now we can formulate the Hales-Jewett Theorem [6] (see also [5] or [14]).

Theorem 7. (Hales-Jewett Theorem) For any two positive integers nyk > 2, there existsan integer N = N(n,k) such that for any k-colouring of the points of the cube AN,A ={1, . . . , n}, there exists a monochromatic line.

Denote by J£(AN) the set of all (combinatorial) lines in AN. For two points x =(x\,X2),y = {yuyi) in the plane, we define x + y = (x\ + y\,X2 + yi).

We shall embed the cube AN into the plane using appropriately chosen rotationsdetermined by 'independent' angles. For simplicity, for s > 0, a = (a,)f, and x = (x,)f GAN, put g£,a(x) = YliLi Pe,oLi(xi) a n d <7aM = ^o,a(^)- Now let L G y(AN) be a combinatorialline such that the points qa(x) (x G L) are distinct. It is easy to check that the points qa(x)(x G L) all lie on a straight line on the plane. Let us denote this line by qa(L). We saythat a = (ai,...,ajv) above is P-independent if the points qa(x) (x G AN) are distinct, andq*(x) £ qa(L) for all lines L G ^(AN) and all xeAN\L.

Lemma 8. For any finite planar point set P with different x-coordinates, and for anypositive integer N, there is a P-independent N-tuple a = (ai , . . . ,a^).

Proof. Let a = (ai, . . . ,a^) be an arbitrary tuple of N angles. We shall show that onecan get a P-independent N-tuple by a small change of the angles a,-. Suppose two pointsga(*i)5ga(*2) (*i,X2 G AN,X\ ^= X2) coincide. Let xi and xi differ in the i-th coordinate. Fors > 0, set a = (ai,...,a/_i,a,- + £,a/+i,...,a#). The points <?a(*i) a n d ^(^2) are different,and, if £ > 0 is sufficiently small, any two points ^(x) ,^(x ' ) ,x ,x ' G AN, are differentwhenever the points q^(x) and ga(x') are different. Thus, if a is replaced by a (with £ > 0sufficiently small), the number of coincidences between the points qa(x) (x G AN) drops.Repeating this procedure, we obtain a tuple a such that the points qa(x) (x G AN) aredistinct.

Suppose now that, for Lo G J£(AN) and for xo G >4N\Lo, the point ga(*o) lies on the linega(Lo). Let i G {1, . . . , N} be such that the n points of Lo have distinct i-th coordinates (i.e.,their i-th coordinates are 1 through n). For s > 0, set a = (ai,...,a,-_i,a,- + £,a,-+i,...,aw)as above.

Suppose ga(Lo) determines the angle /} with the positive x-axis. Picking small enough £ >0, we may assume that (a, + (a, + e))/2 ^ fi-n/2 and (a, + (a, + e))/2 ^ j? + TT/2. Then,fa(Lo) and q%(L§) are not parallel. Let x be the point of Lo whose i-th coordinate equals

the i-th coordinate of xo- Then, the points q^(x) and ^(xo) lie on a line that is parallelto qa(Lo) and, hence, not parallel to qx(Lo). Note that the point q^(x) lies on the lineqx(Lo) by definition. It follows that the point ^(^0) does not lie on the line q-^Lo) (sinceotherwise ^(^0) = ^a(^) a nd , consequently, ga(xo) = q^x)). If £ > 0 is sufficiently small,the points q^(x) (x e AN) are distinct and, for any L G (AN) and x G AN \ L, the point<fc(x) does not lie on the line q^(L) whenever the point qa(x) does not lie on the lineqa(L). Thus, if a is replaced by a (with & > 0 sufficiently small), the number of line-pointincidences drops.

Repeating the procedure described in the above two paragraphs, we finally obtain aP -independent tuple a. •

Now we are in a position to conclude the proof of Theorem 3.

Proof of Theorem 3. Let P be a set of n points in the plane with distinct x-coordinates.Put A = {1, . . . , n}. Let N = N(n, k) be a number guaranteed by the Hales-Jewett Theorem.Let a be a P -independent N-tuple. Thus for any line L G J£(AN), and any x G AN \ L,we have ga(x) ^ ga(L). By an obvious continuity argument, there is a sufficiently small£ > 0 such that, for any line L G <£(AN), the convex hull of the points qz^(x) (x € ^)contains no point q&#(x) with x e AN\L. For any line L G 5£(AN), the set {g£,a(x) : X G L }is combinatorially equivalent to the set P. Now we are ready to show that the setR = {qea(x) : x G AN} has all the required properties.

Let R = Ci U . . . U Q be a /c-colouring of R. It induces a /c-colouring of the setAN. According to the Hales-Jewett Theorem, there exists a monochromatic line Lo G££(AN). The set P = ge,a(Lo) = {ge,aM • x G Lo} is a monochromatic subset of R, it iscombinatorially equivalent to P, and its convex hull contains no point of R \ P. •

Note that the set P in the above proof is actually an affine transform of the set P.

3. Proofs of related results

In this section we prove Theorems 5 and 6. The proof of Theorem 5 follows from thefollowing geometric result, which we proved in [15].

Theorem 9. ([15]) For any integer k > 0 and for any /c +1 positive real numbers e, ru r2,...,r/c > 0, there exists a finite planar point set P in general position such that any set combinato-rially equivalent to P determines k + 1 distances dt (i = 0, 1,..., k) such that \rt — dt/do\ < sfor any i — 1,2, ...,fc. Moreover, one may require the distances d[ (i = 0, l,...,fe) to bedetermined by pairwise disjoint pairs of points,

Proof of Theorem 5. First we prove Theorem 5 for p = 2. Let P be a set satisfyingTheorem 9 for k = 2, e = 0.01, r\ = 1.9, and r2 = 2.5. We find a 2-colouring of all pairs ofpoints of the plane such that no set combinatorially equivalent to P has all pairs colouredby the same colour. Any pair (x,y) e (^ ) of points of the plane with Euclidean distance|xy| > 0 will be coloured blue if |_log2 I JVlJ *s a n e v e n integer. Otherwise (x,y) will becoloured red. In other words, a pair of points is coloured blue if and only if the distancebetween the two points belongs to some interval [2r,2r+1), where t is an even integer.

Now let Pf be a set combinatorially equivalent to P and let dtj = 0,1,2, be the threedistances in Pr ensured by Theorem 9. Thus \d\/do — 1.9| < 0.01. If the two pairs of pointsdetermining do and d\ are coloured by the same colour, the numbers do and d\ belongto the same interval [2r, 2r+1), r e Z, and, consequently, d2 belongs to the next interval[2r+1,2r+2). It follows that all the three pairs determining the distances do,d\Ai cannot becoloured by the same colour. Theorem 5 for p = 2 follows.. Now let p > 2. Fix an arbitrary linear order < of the points of the plane, and colourevery p-tuple of points of the plane by the colour in which the pair of the two smallest(in the order <) points of the p-tuple was coloured above. A short argument showsthat Theorem 5 holds for this 2-colouring and for the set P(p) = P obtained fromTheorem 9 for k = 3p — 4, e = 0.01, r\ = ... = rp_2 = 1, rp-\ = ... = r2p-3 = 1.9,and r2p_2 = ... = r3p_4 = 2.5. •

Thus, an analogue of Theorem 3 fails to be true for partitions of p-tuples in a verystrong sense. However, some particular cases are valid. One such example is providedby Theorem 6. We could denote the statement of Theorem 6 by R —• (Q)l. Despite thesimplicity of the configuration Q, our proof is quite involved, and, in particular, it makesuse of Theorem 3. We shall only sketch the proof here as we intend to return to this topicelsewhere.

Proof of Theorem 6 (Sketch). First we prove the following lemma.

Lemma 10. For every given point sets Pi,P2 in general position, and for every k > 1, thereare point sets R\,R2 with the following two properties:

(i) R\ U Ri is in general position,

(ii) for every partition C\ U . . . U Q of all pairs (xi,X2), x\ G R\,X2 G R2, there exist two

sets P[ <= R\,Pi — ^2? P'i combinatorially equivalent to Pi for i = 1,2, swc/i f/zat all

pairs (x\,X2),x\ G #i,X2 G #2, flre monochromatic.

Proof. We apply Theorem 3. Put R{ - • (P i^ , / ^ - • (^2)*, where K = /c|Rl1. By a standardRamsey theory argument we get the statement. •

Clearly Lemma 10 may be generalized (from bipartite to multipartite graphs with moresets Pi,P2,P3,...)« We shall use this for r-partite graphs, where r = ^(3) is the classicalRamsey number for a monochromatic triangle in any /c-colouring of the edges of thecomplete graph.

Somewhat surprisingly, we shall prove Theorem 6 by induction on k. For k = 1Theorem 6 trivially holds. Let us assume that we have already found a planar point set 5such that IS —> (Q)l-\- Now let R\,...,Rrber planar point sets such that for any partitiono f a l l p a i r s ( x , y ) , x G R t , y G RjJ =£ 7, t h e r e a r e r s e t s S i , . . . , S r , S \ ^ R \ , . . . , S r ^ R r ,equivalent to S such that for every choice of indices /, j9 i 7 7, all pairs between S, and Sjare coloured by the same colour c(ij). Now we can suppose that the set Ri is placed in avery small neighborhood of the vertical line L, = {(Uy) : y G 1R} with all its ^-coordinatesdistinct. Assume that the y-coordinates of all the points in Ri are in the interval (i2, i2 -f 1).According to Ramsey's Theorem (r = ^(3)), there are three indices Uj,lJ < j < /, suchthat all pairs x,y between St,Sj,Si are coloured by the same colour c (in the abovenotation c = c(ij) = c(ul) = c(jj)). If no pair of distinct points of Sj is coloured bythe colour c, we can use the inductive assumption Sj —• (Q)l_{ to get a copy of Q withall pairs of points coloured by the same colour. Thus we may assume that there exists apair {xj9x

fj) of points of Sj coloured by the colour c. Choose x, G 5,-,x/ G S/ arbitrarily.By our construction, both x7 and x lie below the line x,x/ and, if Ri is in a small enoughneighborhood of L,-, the line x7x^ separates the points x,- and x/. Thus {x,,x7,x^,x/} is ahomogeneous set equivalent to Q. •


1. Theorem 3 may be generalized to any fixed finite dimension. Since Theorem 9 holds ina higher dimension (see [15]), Theorem 5 may also be generalized to a higher dimension.

2. Another way of rephrasing Theorem 3 (for sets in general position) is by means ofgeometric graphs (which were studied in various contexts, e.g. in [11], [13], [9], [16], [12]).A geometric graph is defined as a pair (K,£), where V is a set of points in general positionin the plane and £ is a subset of the set of all line segments connecting points of V.Two geometric graphs (K,£),(V',Ef) are said to be isomorphic if there exists a bijection/ : V —• Vf satisfying the following two conditions:

(i) v\Vi G E if and only if f(v\)f(v2) G £',(ii) two line segments 1^2,^4 G E cross if and only if the corresponding line segments

f(vi)f(v2)J(v3)f(v4)eE' cross.

The following theorem may be proved by the method used in the proof of Theorem 3.

532 J- Nesetfil and P. Valtr

Theorem 11. For every geometric graph (V,E) there exists a geometric graph (W,F) suchthat, for every partition W = C\ U. . . U Q , there exists a set V ^ W with the followingthree properties:

(i) The subgraph of (W,F) induced by V is isomorphic to (V,E) as a geometric graph,(ii) V c Cio for some i0,(Hi) the convex hull of V contains no point of W \ V.

3. The above proof of Theorem 6 does not guarantee that conv(Q') n R = Q'. We do notknow whether Theorem 6 with the extra condition con$Qf)DR = Qf holds. In general wehave the following question: does there exist a planar point set Q for which an analogueof Theorem 6 holds but such an analogue does not hold if we further require that thecorresponding set Qf should satisfy conv(g') Pi R = Q ?

4. The minimal size of the set R in Theorem 3 is bounded by a primitive recursive function(by Shelah's proof of the Hales-Jewett Theorem, [18]). However, the best lower boundwe have is only quadratic (in \P$. The quadratic lower bound holds even if we deletecondition (iii) in Theorem 3.

5. If we delete condition (iii) in Theorem 3, then, for k = 2 and for any set P in generalposition, it is possible to find a set R of size O(n2) satisfying Theorem 3 (without (iii)),where n = \P\. On the other hand, for any positive integer rc, there is a set P of size n ingeneral position for which the size of any set R satisfying Theorem 3 (without (iii)) is atleast Q(n2/ logn).

Acknowledgment

We would like to thank the referee for his very careful work.

References

[1] Barany, I. and Fiiredi, Z. (1987) Empty simplices in Euclidean space. Canadian Math. Bull. 30436-445.

[2] Erdos, P. and Szekeres, G. (1935) A combinatorial problem in geometry. Compositio Math. 2463-470.

[3] Goodman, J. E. and Pollack, R. (1991) The complexity of point configurations. Discrete Appl.Math. 31 167-180.

[4] Goodman, J. E. and Pollack, R. (1993) Allowable sequences and order types in discrete andcomputational geometry, in: Pach, J. (ed.), New trends in discrete and computational geometry,Springer-Verlag.

[5] Graham, R. L., Rothschild, B., and Spencer, J., (1980) Ramsey Theory, J. Wiley & Sons, NewYork.

[6] Hales, A. W., and Jewett, R. I., (1963) Regularity of Positional Games, Trans. Am. Math. Soc.106 222-229.


[7] Harborth, H., (1978) Konvexe Funfecke in ebenen Punktmengen. Elem. Math. 33 116-118.[8] Horton, J. D., (1983) Sets with no empty convex 7-gons. Canadian Math. Bull. 26 482-484.[9] Korte B., and Lovasz, L., (1985) Posets, matroids, and greedoids. in: Lovasz, L. and Recski, A.

(eds.), Matroid theory, North-Holland, 239-265.[10] Korte, B. Lovasz, L. and Schrader, R. (1991) Greedoids, Springer-Verlag, Berlin.[11] Lovasz, L. (1979) Topological and algebraic methods in graph theory. In: Bondy, A. and Murty,

U. S. R. (eds.), Graph Theory and Related Topics, Academic Press 1-14.[12] Moser, W. and Pach, J. (1993) Recent developments in combinatorial geometry. In: Pach, J.

(ed.) New trends in discrete and computational geometry, Springer-Verlag.[13] Nesetril, J. Poljak, S. and Turzik, D. (1981) Amalgamation of matroids and its applications, J.

Comb. Th. B 31 9-22.[14] Nesetfil, J. and Rodl, V. (eds.) (1990) Mathematics of Ramsey Theory, Springer-Verlag (1990).[15] Nesetril, J. and Valtr, P. (preprint) Order types containing approximately an affine transforma-

tion of the grid k x k.[16] Pach, J. (1991) Notes on Geometric Graph Theory. DIM ACS Series in Discrete Mathematics

and Theoretical Computer Science 6 273-285.[17] Ramsey, F. P. (1930) On a problem of formal logic. Proc. Lond. Math. Soc, II. Ser. 30 264-286.[18] Shelah, S. (1988) Primitive recursive bounds for van der Waerden numbers. Journal AMS 1

683-697.[19] Valtr, P. (1992) Convex independent sets and 7-holes in restricted planar point sets. Discrete

Comput. Geom. 7 135-152.

The Enumeration of Self-Avoiding Walks andDomains on a Lattice

H. N. V. TEMPERLEY

Emeritus Professor of Applied Mathematics, Swansea, Wales, U.K.Thorney House, Thorney, Langport, Somerset, U.K.

1. Introduction

The domain problem is of interest in magnetism and the walks problem in polymer scienceand molecular biology. Almost the only analytic information is due to Edwards - see forexample Edwards (1) and many earlier papers. His method is approximate, but is reliableenough to give correct critical exponents.

2. The Cluster Method

We use an approach based on Mayer's classical work on the imperfect gas, namely webegin by counting all chain configurations and then remove those configurations in whichone or more points coincide. Our chains consist of links lying along the lines of a planesquare or simple cubic lattice, the ends of the links being on lattice points. A clusterconsists of two or more links connected up in succession with two or more end-pointscoinciding. Figure 1 shows some of the smallest clusters.

The full lines correspond to links on the chain, the dotted lines join points that areconstrained to be coincident. Cluster (a) corresponds to two links lying together on a lineof the lattice, cluster (b) to three links lying together on a line. Cluster (c) may lie on asquare of lines on the lattice or on two adjacent lines, or on one line. Cluster (d) can onlylie along one line of the lattice. To form the cluster series we weight each cluster with

(a) (b) (c)

Figure 1

536 H.N.V. Temperley

z to the power of the number of links, multiplied by the number of embeddings of thecluster in the lattice. We also introduce a factor —1 for each dotted line, correspondingto the fact that such configurations are eventually to be subtructed out of the "walks"generating function. It turns out that the only types of cluster that we need are whatMayer called "irreducible" but most graph theorists would call them multiply connected.(A graph containing an articulation point can be factored into two smaller graphs andwe do not need such a graph in our cluster generating function.

In Temperley (4) (1988) it is formally proved that the generating function for non-selfintersecting walks with end-to-end displacement specified by selector variables is the sameas the reciprocal of the generating function for chains of clusters and single links.

For example, for the plane square lattice, if z is the variable whose power counts thenumber of steps, and the powers of ew and e1^ count horizontal and vertical steps, thefirst few terms of the generating function for self-avoiding walks on this lattice are easilyseen to be

l+z(2 cos 0+2cos </>)+z2[2cos 20+2cos 2</>+4cos(0+(/>)+4cos(0 -(/>)]+z3 [2 cos z0+2 cos 30+8 cos(20+0)+8 cos(20 - </>)+8 cos(0+20)+8 cos(0 - 20)]+z 4 [2cos40+2cos40+. . .

(1)whereas the crude generating function for walks with self-intersections allowed is simply

[ l - z ( 2 c o s 0 + 2cos(/>)]-1. (2)

The term in z2 in (1) differs from z2(2cos0 + 2cos</>)2 by the term 4z2 which correspondsto four two-step walks involving "immediate reversals" the only two-step walks thatintersect themselves. These immediate reversals are enumerated by the cluster sum (a) inFigure 1. The generating function for walks of two steps without immediate reversals maybe written z2(2cos0 + 2cos0)2 — 4z2 simply subtracting the immediate reversals. It is alsothe coefficient of z2 in the modified generating function [1 — 2(2cos 0 + 2cos 0) + 4z2]~l.However if we want self-avoiding walks involving more steps, we have to introduce furtherconnecting terms, the first of which is z3 times the cluster sum in Figure l(b), and there arefurther terms in z4 involving clusters of four steps etc. In Temperley (4) (1988) it is formallyproved that a generating function such as (1) is obtained from the crude generating func-tion (2) by subtracting out clusters involving self-intersections and that all walks involvingself-intersections can be expressed as products of single links and irreducible clusters.

Explicitly we obtain the cluster generating function simply by taking the reciprocal ofthe walks generating function, e.g.,

expression(l) = [1 — 2z(cos0 + cos 0) + cluster sums]"1 (3)

<- Crude G.F. -> <- Clusters - •

In one dimension the only self-avoiding walks are lines of steps all to the right or allto the left, and we can write all the series down explicitly.

[1] Walks G.F. = 1 + 2z cos 0 + 2z2 cos 20 + 2z3 cos 30 + ...

The Enumeration of Self-Avoiding Walks and Domains on a Lattice 537

Table 1 Terms up to z15 for the plane square lattice

Coo(z) - 4 z 2 -12z4 -60z6 -332z8 -1948z10 -11708z12

d ' i (z) -8z 6 -64z8 -424z10 -2608z12

-4z 8 -36z1 0 -256z12

C2,2(Z)

Ci,o(z)C2,i(z)C3,o(z)C3,2(z)

2z +2z3 +14z5 +78z i +482z9

4z9+2926Z11

H-20Z11

-71789z14

-16184z14

-1268z14

-32z14

+18OO6z13

+120z13

16z15

4zi5

Table 2 Terms up to z14 for the simple cubic lattice

Co,o,o(z)C u,o(z)C2,o,o(z)C2,2,o(z)C3,i,o(z)

Ci,o,o(z)Cixi(z)C m ( z )

-6z 2

2z

-30z4

+2z3

-366z6

-8z6

+26z5

-5022z8

-96z8

-8z8

+394z7

-76062z10

-1032z10

-232z10

+5778z9

-48z9

-14z9

-1230462z12

-9840z12

-3888z12

+90714Z11

-578Z11

+60Z11

-20787102z14

-69512z14

-58176z14

-128z14

-8z1 4

+1490378z13

+5424z13

+1904z13

where the crude generating function is (1 — 2zcos0) 1 and the effect of the cluster termsis to remove walks with self-intersections. In (4) the term 2z2 corresponds to the clusterl(a) in Figure 1, the term — 2z3cos0 to the cluster l(b) in Figure 1, the term 2z4 to thesum of all cluster sums involving four links and so on.

Since generating functions for self-avoiding walks on the plane square lattice, weightedaccording to the end distances of the walks, were available in King's College, London,having been originally obtained by Watson for up to 15 steps, a programme was writtenby G. Evans of Swansea Computing Centre to take the reciprocal of the walks generatingfunction, thus obtaining the cluster series. The results for the plane square lattice forwalks of up to 16 steps are tabulated in Temperley (5) (1989). Guttman meanwhileverified Watson's original work on the plane square lattice and extended the series out to25 steps. He also obtained the generating function for walks on the simple cubic latticefor up to 15 steps. The results of taking the reciprocal of the extended generating functionfor the plane square lattice and of the newly available generating function for the simplecubic lattice are reported in Temperley (6) (1991).

The results (up to z14) for the plane square lattice and simple cubic lattice are repro-duced in Table 1. Here Co,o(z) gives the generating function corresponding to clusterscorresponding to zero end to end spacing, Ci,o(z) to clusters corresponding to end to endspacing of one lattice distance, coefficient (cos 6 + cos (/>), Cii(z) to clusters correspondingto one horizontal and one vertical lattice distance, coefficient (cos0cos(/>) etc. For thesimple cubic lattice (Table 2), we use an obvious extension of the notation for the planesquare lattice, for example C\$${z) is the coefficient of (cos# + cosc/> + cost/;) etc.

538 H.N.V. Temperley

3. Discussion of the results

Two things stand out on looking at these data. First, the coefficients of the correspondingpowers of z in the cluster series are very much smaller than the corresponding terms inthe original "walks" generating function. (The sum of the coefficients of zN in the originalgenerating functions is of the order of 3N for the plane square lattice and 5N for thesimple cubic lattice.) Second, the corresponding coefficients in the cluster series decreasevery rapidly as the end to end distance in lattice spacings increases. It is clear from theseresults that we have obtained a very rapidly converging set of successive approximationsto the generating function series. In fact in the plane square lattice we can replace thegenerating function by

[1 - 2z(cos 9 + cos </>) - Co,o(z) - C

and still have the generating function reproduced up to the term in z6, with only a smallcorrection to this and higher terms arising from the Cij(z) and later series. Similarly forthe simple cubic lattice it will be a good approximation to work with just the cluster seriesQ),o,o(z) a n d Ci?o,o(z) a n d neglect the rest. Recall that the values of z that we are interestedin for series analysis are small fractions, [2.638...]-1 for the plane square lattice and[4.638...]~{ for the simple cubic lattice. (The numbers 2.638 and 4.638 are the expectednumbers of ways of adding another step to a long walk without causing a self-intersection.These are slightly less than the numbers 3 and 5 which would correspond just to avoiding"immediate reversals".)

In statistical mechanics we are mainly interested in obtaining the limiting number ofwalks as (2.638...)N</>(iV) where </>(iV) is a slowly varying function of N, usually a poweror logarithm. Sometimes we are interested in quantities like the mean square end-to-enddisplacement, obtained from the second derivative of the generating function with respectto 6 or (j>.

As in problems in the theory of numbers, our first approximation just determinesthe number 2.638... and gives a preliminary estimate of </>(iV). Higher approximations,introducing Cij(z) and later series should give better estimates of the function of (j)(N).This approach to the generating function seems to give a great deal more insight thandirect analysis of the generating function. As the result of very complicated analysis,Guttman concluded that the generating function for the plane square lattice had aconfluent singularity of logarithmic type. Looking at our series Co,o(z) and Ci5o(z), Guttmanconcluded that they both diverge logarithmically at the critical value z = (2.638...)"*,which seems to confirm Guttman's conclusion.

4. Conclusion

A mathematical physicist would prefer an exact result for the generating function, suchas those obtained by Onsager for the two-dimensional ferromagnetic, and by Lieb andBaxter for the six and eight vertex models and the hard hexagon model. However, suchexact results are still scarce and we usually have to be content with series analysis orrenormalization type approximations.

The Enumetation of Self-Avoiding Walks and Domains on a Lattice 539

Only a few theory of numbers type problems have exact treatments, for example the

Hardy-Ramanujan-Rademacher recipes for calculating exactly the number of partitions

of N. Our work suggests that our self-avoiding walk problem is amenable to successive

improvements of the asymptotic results as in other theory of numbers and lattice-point

type problems.

5. Acknowledgement

The references acknowledge help by AJ. Guttman and others.

References

[1] Edwards, S.F. (1986) Theory of Polymer Dynamics (Doi, M. and Edwards, S. K, eds.), OxfordUniversity Press

[2] Guttmann, A. J. (1987) On the critical behaviour of self-avoiding walks, J. Phys. A: Math. Gen.20 1839-1854.

[3] Temperley, H. N. V. (1957) On the statistical mechanics of non-crossing chains: part 1, Trans.Faraday Soc. 53 1065-1073.

[4] Temperley, H. N. V. (1988) New results on the enumeration of non-intersecting random walks,Discrete Appl. Maths. 19 367-379.

[5] Temperley, H. N. V. (1989) On the statistical mechanics of non-crossing chains: part 2 J. Phys.A: Math. Gen. 22 L843-L847.

[6] Temperley, H. N. V. (1991) On the statistical mechanics of non-crossing chains: part 3 J. Phys.A: Math. Gen. 24 L609-L613.

An Extension of Foster's Network Theorem

PRASAD TETALI

AT & T Bell Labs, Murray Hill, NJ 07974.Email: [email protected]

Consider an electrical network on n nodes with resistors r,7 between nodes i and j . Let R,denote the effective resistance between the nodes. Then Foster's Theorem [5] asserts that

where i ~ j denotes i and j are connected by a finite r,;. In [10] this theorem is provedby making use of random walks. The classical connection between electrical networks andreversible random walks implies a corresponding statement for reversible Markov chains.In this paper we prove an elementary identity for ergodic Markov chains, and show thatthis yields Foster's theorem when the chain is time-reversible.

We also prove a generalization of a resistive inverse identity. This identity was known forresistive networks, but we prove a more general identity for ergodic Markov chains. Weshow that time-reversibility, once again, yields the known identity. Among other results,this identity also yields an alternative characterization of reversibility of Markov chains(see Remarks 1 and 2 below). This characterization, when interpreted in terms of electricalcurrents, implies the reciprocity theorem in single-source resistive networks, thus allowingus to establish the equivalence of reversibility in Markov chains and reciprocity in electricalnetworks.

1. Foster's Theorem

Let P = (Pij) denote the nby n transition probability matrix of an ergodic Markov chainwith stationary distribution n, and let us assume that Pn = 0 for all I Furthermore, letH = (Htj) denote the expected first-passage matrix (also of size nxn) of the above chain.Thus Htj denotes the expected time it takes to reach state j from state i. We call the //, ;

the hitting times. Here then is our result, which will easily imply Foster's Theorem.

Theorem 1. With the notation above,

Uj

542 P. Tetali

We give two elementary proofs of this identity. Having found the first, the author realizedthat the shorter second proof is hidden in a proof of [2, Theorem 1], unbeknown to theauthors of [2].

Proof 1. Let Nlkj denote the expected number of visits to k in a random walk from i to j .

Then, [8, equation (34) (page 221)] implies that, for k =/= j ,

That is

Summing both sides over k (^ j),

which implies

so

i

Finally, summing over j , we obtain

2^ njPjiHij = / J ( l — 7ij) = n— 1.Uj j

Proof 2. Note simply that

J V iE «jpj'H>j = E *J ( E

pj'H*j) = E *JWJJ - !]•ij J V

Since Hjj = 1/UJ, this implies that

/*j - 1] = n - 1. D

It is well known that a Markov chain is time-reversible if, and only if, it can berepresented as a random walk on an undirected weighted graph. Moreover, if the weightsare interpreted to be electrical conductors (inverses of resistors), there is a pleasantcorrespondence between the electrical properties of such resistor networks and reversibleMarkov chains ([4, 1, 10]).

More precisely, given an undirected graph with weight ci; = cp on the edge ij, definea random walk with transition probability matrix P — (Py), where Ptj = Ctj/J2jcij-If ij is not an edge then Ptj = 0; in particular, Pn = 0 for all i. This walk has thestationary measure TT,- = ^2jCtj/C, where C = ^ / / c i7 - Reversibility follows from the fact

An Extension of Foster's Network Theorem 543

that UiPij = UjPjt = Cij/C. Using the classical interpretation of ci;- as the conductancel/rij, Chandra et a\. [1] showed that

Hij + Hji = CRih (1)

where Ry is the effective resistance between i and j .Given all this, it is easy to deduce Foster's Theorem from Theorem 1. Indeed, as P is

reversible and (1) holds, we have

Uj

z_

2. Reciprocity and reversibility

The following resistive inverse identity is well known in electrical network theory. Givenconductances ci; and an all-pairs effective resistance matrix R = {Rtj}, with Ru = 0 for allU define two (n — 1) x (n — 1) matrices c = (ci;) and R = (R,;) by the formulae

Ru = [Rtn + R n j - Rtj]/2, l < U j < n - l .

Then c is the inverse of R:

cR = Rc = In_u

where In-\ is the identity matrix of order n— 1. This identity can be generalized as follows.

Let P = (Ptj) be a probability transition matrix of an ergodic Markov chain on nstates, with Pa = 0 for all i. Define an (n — 1) x (n — 1) matrix P = (P^) by setting, for1 < ij < n- 1,

Pa = *i = 5 ^ nfPy and Pi; = -TCfPy.

Furthermore, for 1 < j,k < n— 1 and j =£ /c, let Hjj = Hjn+Hnj, and #,* = Hjn+Hnk-Hjk.

544 P. Tetali

Theorem 2. With the notation above,

Proof. The basic identity we use is the triangle inequality for hitting times. [9, Proposition9-58] asserts that

NZy — HXy = ,

Hzx =JVJ

(2)

(3)

Recall that N*z denotes the expected number of visits to y in a random walk from x toz. From (2) and (3) we have, for all j and /c,

NHjk = * * - .

J nk

Now consider the summation implicit in the statement of the theorem:

n - l n - l

Nin

nk

n-\0±nk

By taking means conditional on the first outcome, we see that the last expression is equalto

nkn - l

An analogous argument shows that S^PjiHkj = Sik, completing the proof.7=1

•

Remark 1. For reversible chains, we have Hjk = Hkj. This is because, for all i and j ,

Hjn + Hnk + Hkj = Hjk + Hkn + Hnj, (4)

A proof of this can be found in [3], alternatively, (4) can be verified directly by using theformula for the hitting times in terms of either resistances (see [10]) or the fundamentalmatrix (see [8]). Thus the proof of Theorem 2 becomes simpler for the reversible case;in particular, we do not need to use equations (2) and (3). Note that the resistive inverseidentity follows by using the analogs mentioned in the previous section: essentially,TiiPtj = ctj/C and Hu + Hjt = CRih for all ij.

Remark 2. Another interesting consequence of Theorem 2 is that the property in (4) isnot only necessary but also sufficient to imply reversibility. Indeed, (4) implies that H issymmetric, which, in turn, implies that P is symmetric, i.e., ntPtj = TijPjt for all i, j .

We now show that identity (4) has an interesting electrical interpretation. First, recallthe following reciprocity theorem from electrical networks (see [7] for a proof).

Theorem 3. The voltage V across any branch of a network, due to a single current source Ianywhere else in the network, is equal to the voltage across the branch at which the sourcewas originally located if the source is placed at the branch across which the voltage V wasoriginally measured.

Using the techniques from [4], it was shown in [10] that, for any network of unit resistors,

(a) the induced voltage Vzy with a unit current flowing into x and out of y is equal toN*y/d(z), where d(z) is the degree of z, and

(b) the reciprocity theorem is equivalent to the fact that N*y/d(z) = Nxy/d(x) for all x,z

and

Essentially the same proof can be used to show that the reciprocity theorem in generalresistive networks (i.e., not necessarily with unit resistors) is equivalent to the statementthat

N*y/n(z) = Nzxy/n(x)

holds for all x,z, and j/(^= x,z). Using relation (2) above, it is easy to see that this is thesame as identity (4). In view of Remarks 1 and 2 above, we have thus established thefollowing assertion.

Corollary 4. Reversibility in ergodic Markov chains is equivalent to reciprocity in electricalnetworks.

Corollary 5. Given P and n, the hitting times (Htj) can be computed with a single matrixinversion, and conversely, given the hitting times, P and n can be computed with a singlematrix inversion.

Proof. In view of Theorem 2, we only need to show:

(a) how to compute H from H, and(b) to compute P from P.

(a) For 1 < ij < n — 1, we have

Hin = 2^ N™ = 2_^k k

Hni = " » Him

and

Hij = Hin + Hnj —

546 P. Tetali

Thus we can first compute Hin and Hni, for all i < n, and then compute Htj for1 < ij < n— 1.

(b) We need to compute nn and Pni, since the rest of the information is available in P.Since n is stochastic and nP = n, we have

and

Remark 3. We defined P and H by treating n as a special state of the chain. Clearly, wecould have chosen any other state j and carried out a similar analysis.Remark 4. In the reversible case, we can interpret H as R, and using part(a) of theproof of Corollary 5, we can write a formula for the hitting times in terms of effectiveresistances. This gives an alternative proof of the main result in [10]:

where c(k) is the sum ]TW cuw of the conductances at node k.

Remark 5. [8, Theorem 4.4.12] gives an alternative way of computing the chain, given all-pairs hitting times. However, the method outlined above seems simpler, since the solutioncan be written in essentially one equation - Theorem 2.

Finally we comment that these identities are useful in designing randomized on-linealgorithms (essentially extending several results of [2]), and we refer to [11] for this work.The author thought of Theorem 2 while trying to extend the results of [2].

Note added in proof

Thanks to David Aldous, Theorem 1 has further been generalized as follows (see [11]).For any n-state ergodic Markov chain, we have ^ - • UiPtjHij < n — 1, with equality onlyunder reversibility of the chain.

References

[1] Chandra, A. K., Raghavan, P., Ruzzo, W. L., Smolensky, R. and Tiwari, P. (1989) The electricalresistance of a graph captures its commute and cover times. Proc. of the 21st Annual ACMSymp. on Theory of Computing 574-586.

[2] Coppersmith, D., Doyle, P., Raghavan, P. and Snir, M. (to appear) Random Walks on WeightedGraphs, and Applications to On-line Algorithms. Jour, of the ACM.

[3] Coppersmith, D., Tetali, P. and Winkler, P. (1993) Collisions among random walks on a graph.SIAM J. on Discrete Math. 6 363-374.

[4] Doyle, P. G. and Snell, J. L. (1984) Random Walks and Electric Networks, The MathematicalAssociation of America.


[5] Foster, R. M. (1949) The Average impedance of an electrical network. Contributions to Applied-Mechanics (Reissner Anniversary Volume), Edwards Bros., Ann Arbor, Mich. 333-340.

[6] Foster, R. M. (1961) An extension of a network theorem. IRE Trans. Circuit Theory 8 75-76.[7] Hayt Jr., W. H. and Kemmerly, J. E. (1978) Engineering Circuit Analysis, McGraw-Hill, 3rd ed.[8] Kemeny, J. G. and Snell, J. L. (1983) Finite Markov Chains, Springer-Verlag.[9] Kemeny, J. G., Snell, J. L. and Knapp, A. W. (1976) Denumerable Markov Chains, Springer-

Verlag.[10] Tetali, P. (1991) Random Walks and the effective resistance of networks. J. Theoretical Proba-

bility 4 101-109.[11] Tetali, P. (1994) Design of on-line algorithms using hitting times. Proceedings of the 5th annual

ACM-SIAM Symp. on Discrete Algorithms, Virginia 402-411.

Randomised Approximation in the Tutte Plane

D. J. A. WELSH

Mathematical Institute and Merton College, University of Oxford

It is shown that unless NP collapses to random polynomial time RP, there can be nofully polynomial randomised approximation scheme for the antiferromagnetic version ofthe Q-state Potts model.

1. Introduction

Exact evaluation of the Tutte polynomial is a computational problem that contains theproblems of computing the partition function of the Potts model in statistical physics,determining the weight enumerator of a linear code, the Jones polynomial of an alternatingknot and many other well-known combinatorial problems. Unfortunately, even for veryrestricted classes it is known to be computationally infeasible, that is #P-hard, except ata few very special points or along a few special curves. For details see [1].

Nevertheless, the problem does not go away, and in the same way as a randomisedpolynomial time (KP)-algorithm is a very attractive and practical solution in the caseof decision problems such as primality testing, it would be extremely interesting andpractically important if a fully polynomial time randomised approximation scheme (fpras)could be shown to exist for most of the Tutte plane. Here we consider this problem.

2. Statement of results

For a graph G with edge set E the Tutte polynomial can be defined as a 2-variablepolynomial,

(2.1) T(G;x,y) = ^ ( x - l)r{E)-r{A)(y - l)W-'W.

The rank function r is defined by r(A) = \V(G)\ —k(A), where k(A) denotes the number ofconnected components of the subgraph having A as its edge set.

Here we shall be concentrating on just a few of the specialisations of T along particularcurves of the (x, y)-plane.

550 D. J. A. Welsh

First we define the Q-state Potts model on a general graph G. A state a of the modelconsists of a mapping of the vertex set V into the set {1,2, ...,Q} and the energy orHamiltonian of that state is

where the sum is over all (ij) that are joined by an edge, and 3 is the usual delta function.The partition function Z is then given by

G

where the sum is over all possible states a. It is not difficult to show (see [6]), that,

Z = Q(eK - i)l"l-i (

VIts importance is that in the stochastic version of the Potts model, the Gibbs measure \irepresenting the probability that the system finds itself in the state o is given by

Now consider the effect of the parameter K. The 'high probability states' are those forwhich H(a) is low. But

H(a)=

and thus if K is positive, it favours o in which neighbouring spins are the same - that isthe attractive or ferromagnetic case. Conversely, if K is negative, the favoured states arethose in which neighbourhood spins are different, in other words we have the repulsive orantiferromagnetic case. For more details of this, see [6].

We can summarise the above in the following statements.

(2.2) For positive integer g, along the hyperbola

T equals (up to an easily computed constant) the partition function Z of the g-statePotts model. We denote this by

T ~ ZQ along HQ*

Note also that HQ has 2 branches, the positive branch

. H+ = { ( x , ^ ) : ( x - l ) ( y - l ) = e, x > l , y > l }

corresponds to the ferromagnetic Potts model. The negative branch HQ = HQ\HQ.

Many other specialisations of T are given in [6]. However, here we emphasize just two.

(2.3) Along the line y = 0, T specialises to the chromatic polynomial P(G;A). In partic-ular,

P(G; /c ) - 7 (G; l - / c ,0 ) .

Now let F denote a finite Abelian group and let to be any orientation of the edges of the

Randomised Approximation in the Tutte Plane 551

graph G. A nowhere zero T-flow on G is a map </> : E(G) —• F\{0} such that Kirchhoff'slaws (under the group operation) hold at each vertex. It is a remarkable fact that thenumber of such flows depends only on G and the order of the group F.

The simplest proof of this is to show that if |F| denotes the order of F, then the numberof nowhere zero F-flows on G is given by (-l)W-k{G)+lT(G;0,1 - |F|), where k(G) is thenumber of connected components of G. It follows that the number of such flows in agroup of order k is a polynomial in k, which we call the flow polynomial of G, and denoteby F(G\k). In other words:

(2.4) Along the line x = 0, T specialises to the flow polynomial, F(G;A), which forpositive integer k counts the number of nowhere zero /.-flows on G. In particular,

When G is a planar graph there is a well-known exact correspondence between k-colourings of G and /c-flows on any plane dual G* of G. This correspondence does notwork for nonplanar G, since G* does not exist. However, it is just an example of a generalcorrespondence between the Tutte polynomial of a matroid M (defined exactly as in (2.1),with rank interpreted as matroid rank) and its dual matroid M*. This correspondence isthat

Thus, from (2.3) and (2.4) we get the duality between chromatic and flow polynomials.A fully polynomial randomised approximation scheme (fpras) for estimating T(G,x,y) is

a randomised algorithm that takes as input a graph G, a pair of rationals (a, b), and e > 0,and produces as output a random variable Y such that

Pr((l-e)T(G;a,b)< Y < (1 + e)T(G;a,b)) > ,

and, moreover, does so within time that is bounded by a polynomial function of the sizeof input and e~l.

The existence of such a scheme is the exact analogue for counting problems of an RPalgorithm for a decision problem, hence it is a very positive statement.

In an important paper, Jerrum and Sinclair [2], have shown that there exists a fprasfor the ferromagnetic Ising problem. This corresponds to the Q = 2 Potts model and thus,their result can be restated in the terminology of this paper as follows.

(2.5) There exists a fpras for estimating T along the positive branch of the hyperbola H2-

However, it seems to be difficult to extend the argument to prove a similar result for the<2-state Potts model with Q > 2, and this remains one of the outstanding open problemsin this area.

A second result of Jerrum and Sinclair is the following.

(2.6) There is no fpras for estimating the antiferromagnetic Ising partition function unlessNP = RP.

Since it is regarded as highly unlikely that NP = RP, this can be taken as evidence ofthe intractability of the antiferromagnetic problem.

552 D. J. A Welsh

Examination of (2.6) in the context of its Tutte plane representation shows that it canbe restated as follows.

(2.7) Unless NP = RP, there is no fpras for estimating T along the curve

{(x,y) : ( x - l ) ( y - l ) = 2, 0 < y < 1}.

The following is an extension of this result.

Theorem. On the assumption that NP ^ RP, the following statements are true.

(2.8) Even in the planar case, there is no fully polynomial randomised approximation schemefor T along the negative branch of the hyperbola H^.

(2.9) For Q = 2,4,5,..., there is no fully polynomial randomised approximation scheme forT along the curves

HQD{X< 0}.

It is worth emphasising that the above statements do not rule out the possibility ofthere being a fpras at specific points along the negative hyperbolae. For example;

(2.10) T can be evaluated exactly at (-1,0) and (0,-1), which both lie on H^-(2.11) There is no inherent obstacle to there being a fpras for estimating the number of

4-colourings of a planar graph.

I do not believe such a scheme exists, but cannot see how to prove it. It certainly is notruled out by any of our results. I therefore pose the following specific question.

(2.12) Problem. Is there a fully polynomial randomised approximation scheme for count-ing the number of /c-colourings of a planar graph for any fixed k > 4?

I conjecture that the answer to (2.12) is negative.Similarly, since, by Seymour's theorem [4], every bridgeless graph has a nowhere zero

6-flow, there is no obvious obstacle to the existence of a fpras for estimating the numberof /c-flows for k > 6. Thus a natural question, which is in the same spirit as (2.12), is thefollowing.

(2.13) Show that there does not exist a fpras for estimating T at (0,-5). More generally,show that there is no fpras for estimating the number of /c-flows for k > 6.

Again, although, because of (2.9), a large section of the relevant hyperbola has no fpras,there is nothing to stop such a scheme existing at isolated points.

Another point of special interest is (0, —2). Mihail and Winkler [3] have shown, amongother things, that there exists a fpras for counting the number of ice configurations in a4-regular graph.

An ice configuration is an assignment of directions to the edges of G such that at eachvertex, there are exactly the same number of edges pointing in as pointing out. Countingice-configurations is a longstanding problem in statistical physics.

In the special case where G is 4-regular it is not difficult to verify that ice configurationscorrespond to nowhere zero flows in the group Z3, and thus:

(2.14) For 4-regular graphs counting nowhere-zero 3-flows has a fpras.

In other words, because of (2.4):

(2.15) There is a fpras for computing T at (0,-2) for 4-regular graphs.

If we dualise this (and assume planarity) it gives:

(2.16) For planar graphs in which every face is a quadrangle there is a fpras for counting3-colourings.

However, the degree restriction is quite severe. For a start, it demands 2-colourabilityand suggests a question that may be easier to settle.

(2.17) Is there a fpras for counting Q-colourings in bipartite planar graphs?

Again, exact counting is hard (see Vertigan and Welsh (1992)), and I suspect the answerto this too is negative.

3. Proof of the Theorem

We say that a set X of edges of G is a Q-cut if there is a partition of the vertex set V ofG into Q, possibly empty, subsets 7i,..., VQ, such that each member of X joins a pair ofvertices belonging to different members of the partition. We claim that the following istrue.

Lemma. For Q >2, determining the maximum size of a Q-cut in a graph is NP-hard.

When 2 = 2, this is just the problem MAX CUT, which is well known to be NP-hard.For Q > 3, the result may also be well known, but I cannot find it in the literature, sosupply the following proof.

Proof of the Lemma. Let $0 be a polynomial time algorithm for max Q-cut, Q > 3. Thefollowing transformation will give a polynomial time algorithm for max (Q — l)-cut. Givena graph G, on, say, n vertices, form a new graph G by adding a set U of m new vertices

and joining each of them to each vertex of G. For m at least I ], it is easy to prove

that there exists a Q-cut of maximum size in G' that has U as one of the sets in theQ-partition. Thus it gives a polynomial time algorithm for finding the maximum size of a(Q — l)-cut in G. Since max 2-cut is NP-hard, the result follows by induction on Q. •

Returning now to the proof of the main theorem, let RQ = HQ D {X < 0, y > 0}.Suppose that there exists a fpras for T along RQ. For Q = 2, we know from [2, result(2.7)] that this is impossible, unless NP = RP. For integer Q, strictly greater than 2,exactly the same argument as used in [2] for the Q = 2 case will work, the only differencebeing that MAX Q-CUT takes the role that MAX-CUT (= MAX 2-CUT) took in theirargument. In other words, the existence of a fpras in the region RQ would mean a randompolynomial time decision procedure for max Q-cut, and thus would imply NP = RP.

Now let RQ = HQ n {x < 0, y < 0}. I show that if there exists a fpras si for T in thisregion, it will give a fpras for T in the region Rl

Q. But as we have seen, this will implyNP — RP. To prove the assertion, we use the tensor product developed in [1]. Let N beany graph and let e be a special edge of N. The tensor-product G © N is formed by glueing

554 D. J. A. Welsh

a copy of N to each edge / of G by identifying the edges e and / , and then deleting both.The following relationship holds between T(G) and T(G 0 N);

(3.1) T(G 0 N;x;y) = cT(G\X, 7),

where c, X and Y are easily computable functions of Q, x, y and Af. Explicitly

( 2 - 1 ) 7 '(3.2) X =

(3.3) y =

( x - 1)7" - 7 ' '

( 2 - 1)7"• - 1 ) 7 ' - 7 " '

where 7', 7" are respectively given by 7 ' = T(N'e;x,y) and 7" = T(N'J;x,y), and N'e (N'J)denote the deletion (contraction) of e from N, and

c = (XT" - T)r{E)(yT" - 7) |£ |- r (£ )(e - 1)-|£|

where 7 = 7 ' + 7".In the special case where N is the graph on two vertices and three parallel edges, this

gives the 2-thickening of G and (3.2) and (3.3), after some simplification, reduce to

(3.4) X = 1 + - ^ —

(3.5) Y = y2.

Thus, let (a, ft) £ RQ, SO that 0 < b < 1. Form the 2-thickening of G to get G and applythe postulated fpras stf to G at the point (ar,br) £ K , where ft' is the negative squareroot of ft. Using (3.1) with (a, ft) = (X, Y) and (a', ft') = (x,y) gives the fpras in the regionRQ. But this cannot exist unless NP = RP, by our previous argument.

All of the above holds for general positive integer Q. Now we turn to the particularcase 6 = 3, which is where we can say more.

Let RQ = HQ n {0 < x < 1}. Suppose that <$/ is a polynomial time algorithm thatgives a fpras for 7 on RQ. When G is planar we may use srf to obtain a fpras for 7 on#Q by the simple expedient of forming the dual graph G* and using the duality relation

But when Q = 3, we notice that the whole of the previous argument goes through justfor the class of planar graphs. This is because deciding MAX 3-CUT is NP-hard in theplanar case. It contains the problem of deciding whether a planar graph is 3-colourable,and this is a known NP-hard problem. Hence for Q = 3, the duality argument willwork and shows that the whole of the negative hyperbola H$ has no fpras, unlessNP = RP. •

It is frustrating that even for general graphs, I cannot see how to prove non-approximability along the whole negative hyperbola, except when Q = 3. It would behighly surprising if it were not true. However, the technique used to prove the Q = 3 casewill not work. Nor will the obvious idea of trying to combine /c-thickenings and /c-stretchesas in [1]. Thus, if there does exist a proof using 'tensor transformations' it will have to


involve transformations that are very much more complicated. For example, the smallesttensor transformation that cannot be expressed in terms of a composition of thickeningsand stretches is to take N = X4. This leads to the transformation (x,y) >—• (X, Y), where

n 6 , (Q ~ 1)(*3 + 2*2 + 2xy + r + x + y)- 1) + 2x2(y - 1) - 3y(x +1)4- 2xy2 - 2(x + j>)

and 7 is obtained by interchanging x and y.Apart from the questions already raised, we should also emphasise that the above says

nothing about the inability to estimate T along HQ for fractional Q. It would be verysurprising if such a result were not true, but again it is likely to need a new technique,since a tensor transformation will only shift a point along the hyperbola HQ containingit.

Acknowledgement

I am very grateful for very helpful discussions with Martin Grotschel, Mark Jerrum,David Johnson and Paul Seymour, and acknowledge also the support of Esprit WorkingGroup 'RAND\

References

[1] Jaeger, R, Vertigan, D. L. and Welsh, D. J. A. (1990) On the computational complexity of theJones and Tutte polynomials. Math. Proc. Camb. Phil Soc. 108, 35-53.

[2] Jerrum, M. and Sinclair, A. (1990) Polynomial-time approximation algorithms for the Isingmodel (Extended Abstract). Proc. 17th ICALP, Springer-Verlag, 462-475.

[3] Mihail, M. and Winkler, P. (1991) On the number of Eulerian orientations of a graph. BellcoreTechnical Memorandum, TM-ARH-018829.

[4] Seymour, P. D. (1981) Nowhere-zero 6-flows. J. Combinatorial Theory (B) 30, 130-135.[5] Vertigan, D. L. and Welsh, D. J. A. (1992) The computational complexity of the Tutte plane:

the bipartite case. Combinatorics, Probability and Computing 1, 181-187.[6] Welsh, D. J. A. (1993) Complexity: Knots Colourings and Counting. London Mathematical

Society Lecture Note Series 186, Cambridge University Press.

On Crossing Numbers, and Some UnsolvedProblems

HERBERT S. WILFf

University of Pennsylvania, Philadelphia, PA 19104-6395

This paper is about two rather disjoint subjects. First I want to discuss a new result ofEd Scheinerman and myself [4], in which we found a connection between the rectilinearcrossing number problem for graphs and an old question in geometric probability that wasasked by JJ. Sylvester [5]. Second I want to propose a number of unsolved problems incombinatorics and graph theory, some of which I have previously aired, but which havenot appeared in print before.

1. On probability and crossing numbers

Here is the rectilinear crossing number problem. For a given graph G, suppose we drawG in the plane with straight line edges. Among all ways of doing so, we define k(G), therectilinear crossing number of G, to be the smallest number of crossings of edges that canbe achieved.

For the complete graph Kn the values K(KH) are known for 1 ^ n 9, and they are0,0,0,0,1,3,9,19,36, respectively. For n = 10 the number is 61 or 62.

It is also well known that

c, _ >im « £ >exists, and that 0 < c\ < 00, though its exact value is unknown.

Here is Sylvester's question about geometric probability. Let K be a convex set in theplane. Choose four points independently uniformly at random (iuar) in K. Then withprobability 1 there are two possibilities: either the convex hull of the four points is aquadrilateral or it is a triangle. Let q(K) denote the probability that the convex hull isa quadrilateral. Sylvester asked for the minimum and maximum values of q(K) over allconvex sets K in the plane.

That question was answered some time ago [1]. The maximum is

1 - ^ = 0 . 7 0 4 . . . ,

t Supported in part by the United States Office of Naval Research

558 H. S. Wilf

which is achieved on an ellipse, and the minimum is 2/3, attained when K is a triangle.Generalize Sylvester's question as follows. Let K be an open set in the plane, of finite

measure (but not necessarily convex, or even connected). Define q(K) as before. Nowdetermine the m/and the sup of q(K) over all such sets K.

The supremum is certainly 1, for we can take K to be a very thin annulus, in whichcase four points selected iuar in K will almost surely span a quadrilateral.

The infimum is another matter entirely, so define

c2 = infq(K),K

over all open plane sets K of finite measure.

Theorem [4]. c\ = ci

Proof. Indeed, let us show first that c\ ^ ci. Fix some region K, choose n points iuar inK, and join each pair of them by a straight line segment. We call this the sample drawing.

The rectilinear crossing number of Kn cannot exceed the average number of crossingsobserved, over all such sample drawings. The latter average is the expectation of a sumof Q) random variables, one for each 4-subset of vertices in the chosen drawing of Kn.The random variable that is attached to a 4-subset is 1 if they span a quadrilateral and 0otherwise, since each crossing that occurs belongs to one and only one such quadrilateral.These random variables all have expectation q(K), hence K(Kn) ^ (^)q(K). If we divideby Q) and take the limit as n —> 00, we find that c\ ^ q(K). If we now take the infimumover all K, the claimed upper bound follows.

Next we claim that c\ ^ ci. For this, fix an optimum drawing of Kn, i.e., a drawing of Kn

with straight line segment edges, in which the smallest number of edge crossings occur. Ateach vertex of this drawing place an open disk of radius e, where e is small enough so thatfor all choices of n vertices, one from each disk, the resulting drawing will still be optimum.

Let K be the union of these n open disks, and consider the following question. If wechoose four distinct disks of K, and choose iuar a point from each of them, what is theprobability q that the resulting quadrilateral is convex?

We answer this question in two ways, and then compare the answers. First, q is thenumber of convex quadrilaterals in our optimum drawing, divided by Q). Since the for-mer are in 1-1 correspondence with edge crossings, there are exactly k(Kn) of them. Thusq = K(Kn)/(

n4).

Second, q(K) is the probability that four points chosen iuar in K will form a convexquadrilateral. But four points so chosen will lie in four distinct disks of K with probabilityl-O(l/n). Thus

q = q(K) + O(l/n) ^ c2 + O(l/n).

If we compare these two answers to the question we obtain

and if we let n —• 00, the result follows. •

As a corollary we get an estimate for the rectilinear crossing number of any graph.

On Crossing Numbers, and Some Unsolved Problems 559

Corollary. Let G be a graph for which exactly M pairs of edges span four distinct vertices.Then the rectilinear crossing number of G (and therefore its crossing number also) is atmost ciM/3.

To get numerical estimates of the universal constant c\ we proceed as follows. Fora lower bound we can use jc(Kw)/Q) for any particular value of n for which a lowerbound on the crossing number is known. Since it is known [(David Singer, personalcommunication)] that K(K\O) ^ 61, we have c\ ^ 61/210 = 0.290...

In the other direction, we need a clever drawing of Kn, i.e., one with few crossings, fora sequence of n —• oo. The best drawing that is known is due to David Singer, and since itis not available in the open literature, we give here a summary of his idea, which is quitesimple to describe.

Theorem [David Singer]. Wfe have lim k(Kn)/(n4) ^ 5/13.

Proof. We give a recursive description of drawings of Kn when n is a power of 3. Whenn = 3, draw a triangle. Inductively, suppose we have described the drawing Dn of Kn

when n = 3q. Then take the drawing Dn and put it in the upper half plane, with they-coordinates of its vertices all distinct. Next, carry out a mapping of the plane to itself bymeans of (x, y) —• (ex, y\ which will map the drawing Dn into another rectilinear drawingwith the same number of crossings, all of which is contained in a thin vertical strip aboutthe y-axis. Make three copies of this thin drawing, and place one of them on each of therays 6 = 0,27r/3,47r/3, and connect all pairs of points with straight line segments. Theparameter e is small enough so that if two vertices are chosen in the same copy of thedrawing, then the infinite line through them separates the other two copies of the drawing.

In this drawing we count the convex quadrilaterals, i.e., the crossings, and we arethereby led to the recurrent inequality satisfied by f(r) = fc(Ky), which is

/ ( r + 1) O / ( r ) + r2(r - l)(5r - 7)/4.

This implies that if n is a power of 3 then Kn can be drawn with straight line edges andno more than

3l2crossings, which proves Singer's theorem. •

These best known estimates combine to give

0.290... = ^ < C l =c2 ^ A =0.3846.. .

2. Some unsolved problems

2.1. Graceful permutationsA permutation a of n letters is graceful if

560 H. S. Wilf

2

1 . 5

1

0 . 5

• "

5

• *

10 15

. • * "

20

Figure 1

which is to say that no unsigned consecutive difference occurs twice. Such a permutationis, of course, a graceful labelling of a path, in the sense of the well known "all trees aregraceful" problem. The permutation

1 2 3 4 5 6N

6 1 5 2 4 3,is graceful.

The problem is to count these permutations, a question that was first raised in [6].If f(n) is the number of them on n letters, then the following is a table of f(n)/4 forn = 4,5,6,. . . , 23, the values having been computed by Dennis Deturck:

1, 2, 6, 8, 10, 30, 74, 162, 332, 800, 2478, 6398, 13980, 35798, 127674,362824, 874336, 2612956, 9642676, 29728748

Deturck has also shown that liminf/(/t)1/" ^ 23. Does the limit exist, as a finite, nonzeronumber? A plot of /(rc)1/n, for n ^ 22, is shown in Figure 1.

2.2. Balanced binomial coefficientsLet p be a prime. We will say that an integer n is p-balanced if, among the nonzerovalues of the binomial coefficients {(£) }£=0, taken modulo /?, there are equal numbers ofquadratic residues and quadratic nonresidues modulo p.

Next, define the set Tp as the set of integers n, 0 ^n ^ p — 1, that are p-balanced.In [3] it was shown that if Tp is empty, then no integer n is p-balanced, and that if Tp

is nonempty, then n is p-balanced iff its p-ary expansion contains at least one digit d G Tp.The question is: for which primes p is the set Tp empty? What is known (A. M. Odlyzko

(p.c.)) is that among all primes less than 1,000,000, the only empty Tp are those withp = 2,3,11. Are there any others?

2.3. Graphical partitionsIf G is a graph of m edges, then the degrees of the vertices of G constitute a partition of theeven integer 2m. Conversely, among the partitions of the integer 2m, some are graphicaland some are not. Let pg(2m) be the probability that a partition of 2m is graphical.

On Crossing Numbers, and Some Unsolved Problems 561

Table 1 Some values of pg(n)

n

2468101214161820

Pg(n)

0.50.40.4545450.4090910.4047620.4025970.40.389610.3922080.389155

n

22243234363850100200

Pg(n)

0.3862280.3853970.3800460.378310.3782050.3768210.3726460.3586 (est.)0.3354 (est.)

The questions are:

(a) Does limm^oopg(2m) exist?(b) Is it 0?

I asked these questions first in 1980, and two of my graduate students at that timeworked on them, but they seem extremely difficult. Numerically, the values of pg(2m)decrease slowly from 0.5 at 2m = 2, down to 0.3726, at 2m = 50. Beyond that I have donesome random sampling to estimate the values 0.3586, from 8000 trials at 2m = 100, and0.3354, from 1100 random samples with 2m = 200.

Erdos and Richmond [2] have shown that

liminf JlnpJln) ^ —7=.

It is clear that \imsupnpg(2n) ^ 0.5, since not both a partition and its conjugate can begraphical. In [2] the upper bound is reduced to 0.4258.

2.4. Triangular matricesLet A be an m x n matrix of 0's and l's. Consider the computational problem: do thereexist permutations P of the rows of A, and Q, of the columns of A such that after carryingout these permutations, A is triangular?

The question we ask concerns the complexity of the problem. Is this problem NP-complete? Or, does there exist a polynomial time algorithm for doing it? The question isrelated to job scheduling with precedence constraints, a well known problem in theoreticalcomputer science, but it falls, in difficulty, between a known easy case and a known hardcase of the general problem.

2.5. Patterns of permutationsLet k < n be given positive integers and let T and a be given permutations, of l,...,/c,and of l , . . . ,n, respectively. We say that o contains the pattern T if there are integers1 ^ i\ < j'2 < ' ' * < ik ^ n such that for all 1 ^ \i < v ^ k we have cr(i//) < a(iv) if andonly if x(p) < T(V).

For example, a permutation contains the pattern (123) iff it has an ascending subse-quence (i.e., a not necessarily consecutive ascending triple of values) of length 3.

562 H. S. Wilf

Given a pattern x of k letters, we let f(n, x) denote the maximum number of occurrencesof the pattern x that can be packed into a permutation of n letters. If the limit

exists, then we call it the packing density of the permutation x.Does every permutation x have a packing density? That is, does the limit always exist?

If it does exist, what is it, expressed in terms of accessible properties of T?It is easy to see that the lower packing density is always strictly positive. Indeed, for a

fixed pattern x of k letters, we have

_ _kk+l(kk~l -

>

To see why, suppose first that n = /cm, and construct a permutation tr as follows. Dividethe letters l , . . . ,n into /c intervals of consecutive letters. Arrange these intervals in thepattern x. That is, if the largest value of x is first, for instance, then put the interval thathas the largest letters first, etc. With no further effort this already guarantees us that thepattern x will appear in o at least (n/k)k = fefc(m-1) times.

But we can do more. Within each interval we can arrange the letters to have themaximum number of T'S that are possible for permutations with n/k letters.

Example. For the pattern x = (132) we can construct a permutation of 9 letters with manyoccurrences of (132) as follows:

a = (132 798465)

In general, this construction shows that

which leads to the result stated. The construction does not, however, always give the bestresult. The packing density of the pattern (132) has been determined by W. Stromquist(p.c), and it is 2 V3 - 3 = 0.4641... .

References

[1] Blaschke, W. (1917) Leipziger Berichte 69 436-453.[2] Erdos, P. and Richmond, L. B. (1989) On graphical partitions, Combinatorics and optimization,

Research Report CORR 89-42, University of Waterloo, 13pp.[3] Garfield, R. and Wilf, H. S. (1992) The distribution of the binomial coefficients modulo p, J.

Number Theory 41, 1-5.[4] Scheinerman, E. R. and Wilf, H. S. The rectilinear crossing number of a complete graph and

Sylvester's "Four Point Problem" of geometric probability, Amer. Math. Monthly, to appear.[5] Sylvester, J. J. (1908) Mathematical Papers, Vol. II (1854-1873), Cambridge University Press,

Cambridge.[6] Wilf, H. S. and Yoshimura, N. (1986) Ranking rooted trees, and a graceful application, in

Perspectives in Computing, Proc. Japan-US Joint Seminar June 4-6, 1986, in Kyoto, AcademicPress, Boston.

combinatorics ,geometry and probability

Documents

paul erdos xitoast

lattice graphs

hamming graphs

distanceregular graphs

erdosturan law

wheelfree graphs

cambridge cb2

press syndicate