Proving theorems in Tehran

Peter J. CameronG. C. Steward Visiting Fellow

Gonville & Caius CollegeCambridge

March 2008

(after Reading Lolita in Tehran: A Memoir in Booksby Azar Nafisi)

International conference on Combinatorics, Lin-ear Algebra and Graph Colouring, at the Institutefor Studies in Theoretical Physics and Mathematics(IPM) in Tehran, Iran.

IPM grounds

The mountains from IPM

1

Golestan Palace

Mandana

Daily News

• Details of the museums of Tehran (to which wewere taken on excursions)

• The invited speakers’ mathematical genealogy

• Menus, e.g. for Bagali Polo, which we had forlunch

• Summary of Persian music

• Competitions for students, e.g. “Discover themiddle names of the invited speakers”

The winner . . .

A surpriseMy textbook on Algebra has been translated into

Farsi. The translator gave me a copy. (That’s all I get;Iran does not subscribe to the International Copy-right Convention.)

2

Discrete MathematicsVolume 306, Issue 23, 6 December 2006

Special Issue: International Workshop on Combinatorics,Linear Algebra, and Graph Coloring

Guest Editors: R.A. Brualdi, S. Hedayat, H. Kharaghani,G.B. Khosrovshahi, S. Shahriari

Contents

Preface 2989

Guest Editors 2991

G.B. Khosrovshahi and B. Tayfeh-Rezaie

Large sets of t-designs through partitionable sets: A survey 2993

S. Akbari, H. Bidkhori and N. Nosrati

r-Strong edge colorings of graphs 3005

S. Akbari, O. Etesami, H. Mahini, M. Mahmoody and A. Sharifi

Transversals in long rectangular arrays 3011

R.A. Bailey, P.J. Cameron, P. Dobcsanyi, J.P. Morgan and L.H. Soicher

Designs on the web 3014

R. Bean

Latin trades on three or four rows 3028

M. Behbahani and H. Kharaghani

On a new class of productive regular Hadamard matrices 3042

A.E. Brouwer, P.J. Cameron, W.H. Haemers and D.A. Preece

Self-dual, not self-polar 3051

R.A. Brualdi

Algorithms for constructing ð0;1Þ-matrices with prescribed row and column sum vectors 3054

P.J. Cameron, H.R. Maimani, G.R. Omidi and B. Tayfeh-Rezaie

3-Designs from PSLð2;qÞ 3063

P.J. Cameron and C.R. Johnson

The number of equivalence classes of symmetric sign patterns 3074

S. Akbari and H.-R. Fanaı

Some relations among term rank, clique number and list chromatic number of a graph 3078

doi:10.1016/S0012-365X(06)00789-8

N.C. Fiala and W.H. Haemers

5-chromatic strongly regular graphs 3083

Z. Furedi, R.H. Sloan, K. Takata and G. Turan

On set systems with a threshold property 3097

A.S. Hedayat and M. Yang

Efficient crossover designs for comparing test treatments with a control treatment 3112

C.R. Johnson and A.L. Duarte

Converse to the Parter–Wiener theorem: The case of non-trees 3125

C.R. Johnson and C.M. Saiago

The trees for which maximum multiplicity implies the simplicity of other eigenvalues 3130

R. Naserasr and Y. Nigussie

On a new reformulation of Hadwiger’s conjecture 3136

T. Hsu, M.J. Logan and S. Shahriari

The generalized Furedi conjecture holds for finite linear lattices 3140

C. Thomassen

The number of k-colorings of a graph on a fixed surface 3145

R.M. Wilson

A lemma on polynomials modulo pm and applications to coding theory 3154

M. Zaker

Results on the Grundy chromatic number of graphs 3166

P.J. Cameron

Research problems from the 2003 IPM Workshop 3174

The first theoremAn incidence structure consists of a set of “points”

and a set of “blocks”, with a relation of “incidence”between points and blocks.

Let (P, B, I) be an incidence structure. A dualityconsists of a pair of bijective functions f : P → Band g : B → P which “reverse incidence”, in thesense that if point p and block b are incident, then soare the point g(b) and the block f (p). An incidencestructure is self-dual if a duality exists; this meansthat the structure is isomorphic to its dual (with thelabels “point” and “block” reversed).

A duality is a polarity if g is the inverse of f .

Willem Haemers’ question

What is the smallest incidence structure whichis self-dual but not self-polar?

The answerA construction from algebraic geometry shows

that there is an example with 85 points and 85 blocks(a “generalized quadrangle”). Could we do better?

In Tehran we found an example with 8 points and8 blocks. My colleague Donald Preece reduced 8 to7. This is best possible; there is no smaller example.Moreover, there are just four different examples with7 points.

The second theoremA symmetric sign pattern is a symmetric matrix

with entries + and −. Charles Johnson was inter-ested in which properties of a symmetric matrix de-pend only on the signs of its entries. Assuming thereare no zero entries, these form a symmetric sign pat-tern.

Charles Johnson’s questionTwo symmetric sign patterns are equivalent if we

can get from one to the other by a combinationof permuting rows and columns, switching (chang-ing signs of some rows and columns), and possiblychanging all the signs.

How many inequivalent n× n symmetric signpatterns are there?

For n = 1, 2, 3, 4, he knew the numbers: 1, 2, 4, 11respectively.

3

The answerI recognised these numbers: they are the numbers

of graphs on 1, 2, 3, 4 vertices.

u u u uu u u uu u u u

���

���

TT

T

TT

T

But how to prove that the numbers necessarily co-incide?

History to the rescueThirty years earlier, Mallows and Sloane had

shown that the number of graphs, counted up topermutation and switching, is equal to the number ofeven graphs up to permutation. They proved this byfinding formulae for both counting problems, andobserving that they were the same. (Sloane wasgathering data for the first version of his celebratedEncyclopedia of Integer Sequences.)

I found a more “conceptual” proof.It turned out that combining the two proofs led to

the proof of the conjecture about Johnson’s question.

The third theoremThis was work with three postdocs at IPM:

Maimani, Omidi and Tayfeh-Reziae, in connectionwith a problem in design theory.

We have a particular permutation group acting ona set of n elements. (Actually the group PSL(2, q),where n = q + 1). We want to find, for each valueof k, all possible sizes of sets of k-element subsetswhich admit the action of this group, and how manyof each size there are.

Data on the groupTo solve this problem, whe need to know three

things about the group:

• All of its subgroups (these were determined byDickson in the early 20th century).

• Their orbit lengths (these are relatively easy andwere worked out before).

• The so-called “Mobius function” of each possi-ble subgroup. This turns out also to be knownbut is more obscure.

A trailerThere are three “exceptional” subgroups of our

group, which don’t fit into a regular pattern. Theseare the rotation groups of the regular polyhedra:tetrahedron, cube, and dodecahedron.

The values of the Mobius function of these threegroups led to a curious puzzle, which I will touchon in the next lecture.

4