a firs look at graph theory

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John Clark and Derek Allan HoltonDepartment of Mathematics and Statics

University of OtagoNeu/ Zealand

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Published by Sunil Sachdev and printed by Ravi Sachdev atAllied Publishers Limited.A-I04 Mayapuri, Phase n,New Delhi 110064.

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This edition of" A First Look At Graph Theory"© World Scientific Publishing Co.Pte. Ltd. is published by Allied Publishers Limited, 1995 by arrangement withWorld Scientific Publishing Co. Pte. Ltd., Farrer Road, P.O. Box 128, Singapore9128.

First Indian Reprint 1995

Copyright © 1991 by World Scientific Publishing Co. Pte. Ltd.

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To our long-suffering wives,Austina and Marilyn,

with sincere thanks for their patience,and to

John and Alan Clark,who should have seen much more of their father

during their summer holidays.

vii

With the increasing use of computers in society there has been a dramatic growthin all aspects of computer education. At university level, computer science studentsquickly learn that their subject has many facets, some of which are better appreciatedwhen the student has an appropriate mathematical background. Unfortunately, muchof this mathematical background is not the sort found easily in the mathematicaleducation of earlier generations.

Much of the theory of computer science uses an area of mathematics looselydescribed as "discrete mathematics", this term chosen to emphasise its contrastwith the "continuous mathematics" of the more traditional calculus courses. Discretemathematics covers many topics and this book takes a first look at one of these- Graph Theory. This topic has a surprising number of applications, not justto computer science but to many other sciences (physical, biological and social),engineering and commerce.

From what we have said so far, the reader may have got the impression that thisis a book mainly for computer scientists. Not so. Graph Theory is at last beingacknowledged as an important subject in the undergraduate mathematics curriculum.Perhaps one should expect a more theoretical treatment here than in the computerscience setting. However we feel that a blend of the theory with some of its many variedapplications is highly desirable for both disciplines - for those-mainly concerned withthe applications of graphs, the theory helps to strengthen the ideas and point the wayto independent applications; conversely, the applications of graph theory to "realworld" situations reinforces the theoretical aspects and illustrates one of the manyways in which mathematics is applied. As a result this text is a mixture of boththeory and applications and can be used by both the serious mathematics student,her computer science cousin or any other relation keen to learn about one of the mostrapidly growing areas of modern mathematics.

The book began in 1985 as a set of notes for a second year course of 40 one-hourlectures in the Department of Mathematics at the University of Otago. The studentsattending the course, then and in subsequent years, were mainly a mixture of computerscience majors and mathematics majors. Not all topics covered in the book were dealtwith in these lectures and, indeed, some theoretical aspects may be quite difficult forthe average second y~r university student. However, Graph Theory is particularlysuited to selective study and hopefully our treatment here provides material for theindividual teacher to tailor to their course's requirements.

We have also provided a plentiful supply of exercises. These are of varying difficulty.Some deal with the algorithmic aspects of the text, some on the theoretical while

Preface

others introduce some new but related ideas. We encourage the reader to do as manyas possible - mathematics is not a spectator sport and is best appreciated with activeparticipation!We wish to express our sincere thanks to several people involved in the preparation

of this book. Jane Hill helped enormously with the typing and typesetting, JohnMarshall proofread an early version of the text and also made numerous suggestionson style, Gordon Yau prepared some of the diagrams, Maree Watson typed portions ofthe manuscript, Gra.emeMcKinstry provided very useful L\TEXexpertise while MarkBorrie gave frequently-needed computer assistance.However our main thanks go to our families for their patience during the last few

months of the book's preparation.

Prefaceviii

ix

One of the beauties of Graph Theory is that it depends very little on other branchesof mathematics. However in our text we do occasionally rely on the reader havingwhat is often called "mathematical maturity". This means an ability on behalf ofthe reader to understand and appreciate a mathematical argument or proof. Thisability is something that usually is not acquired overnight but is the outcome of anongoing exposure to mathematics and its accompanying logic. Hopefully, the reader'smathematical maturity will grow as he progresses through the text. If so, then we willhave achieved one of the goals of the book.

On a more concrete level, we will assume that the reader knows about the principleof mathematical induction. This is dealt with in many undergraduate textbooks. Inparticular, the text by Mott, Kandel and Baker, mentioned in the section on FurtherReading, has a nice treatment of this.We also assume that the reader is familiar with the notion of a set. We use 0

to denote the empty set and, for two sets A and B, we denote the set difference,consisting of all elements which belong to A but not B, by A-B.

Each of the ten chapters of the book is split up into numbered sections, with, forexample, Section 2.5 denoting the fifth section of Chapter 2. At the end of mostsections there is a collection of numbered exercises. For example, Exercise 2.5.3 refersto the third exercise accompanying Section 2.5. Within each chapter, results such astheorems and corollaries are also numbered consecutively. For example, Theorem 4.3is the third result of Chapter 4 and it is followed by Corollary 4.4.The end of a proof of a theorem or 'corollary is shown by the symbol D.We have referred to several books and articles throughout the text and details of

these are given in the bibliography at the end of the book. Such a reference is givenby a number in square brackets with, for example, [8] referring to the eighth item inthe bibliography.

A Note to the Reader