GRAPH THEORY WITH APPLICATIONS J. A. Bondy and U. S. R. Murty of Department Combinatorics and Optimization, of University Waterloo, Ontario, Canada NORfH-HOLLAND New York • Amsterdam • Oxford @J.A. Bondy and U.S.R. Muny 1976 First published in Great Britain 1976 by The Macmillan PressLtd. First published in the U.S.A. 1976 by Elsevier Science Publishing Co., Inc. 52 Vanderbilt Avenue. New York. N.V. 10017 Fifth Printing, 1982. Sole Distributor in the U.S.A: Elsevier Science Publishing Co .• Inc. Library of Congress Cataloging in Publication Data Bondy, John Adrian. Graph theory with.applications. Bibliography: p. Includes index. 1. Graph theory. I. Murty, U.S.R.,joint author. II. Title. QA166.B67 1979 511 '.5 75-29826 ISBN 0.;444-19451-7 All rights reserv~d. No part ofthis publication may be reproduced or transmitted. in any form or by any means, without permission. Printed in the United States of America · To our parents Preface This book is intended as an introduction to graph theory. Our aim has been to present what we consider to be the basic material, together with a wide variety of applications,both to other branches of mathematics and to real-world problems. Included are simple new proofs of theorems of Brooks, Chvatal, Tutte and Vizing. The applications have been carefully selected, and are treated in some depth. We have chosen to omit all so-called 'applications' that employ just the language of graphs and no theory. The applications appearing at the end of each chapter actually make use of theory developed earlier in the same chapter. We have also stressed the importance of efficient methods of solving problems. Several good al gorithms are included and their efficiencies are analysed. We do not, however, go into the computer iinplementation of these algorithms. The exercises at the end of each section are of varying difficulty. The harder ones are starred (*) and, for these, hints are provided in appendix I. In some exercises, new. definitions .a re introduced. The reader is recom mended to acquaint himself with these definitions. Other exercises, whose numbers are indicated by bold type, are used in subsequent sections; these should all be attempted. Appendix II consists of a table in which basic properties of four graphs are listed. When new definitions are introduced,· the reader may find it helpful to check his understanding by referring to this table. Appendix III includes a selection of interesting graphs with special properties. These may prove to be useful in testing new conjectures. In appendix IV, we collect together a number of unsolved problems, some known to be very difficult, and others more hopeful. Suggestions for further reading are given in appendix V. Many people have contributed, either directly or indirectly, to this book. We are particularly indebted to C. Berge and D. J. ~. Welsh for introducing us to graph theory, to G. A. Dirac, J. Edmonds, L. Lovasz and W. T. Tutte, whose works have influenced oUf treatment of the subject, to V. Chungphaisan and C. St. J. A. Nash-Williams for their careful reading of the Preface vii manuscript and valuable suggestions, and to the ubiquitous G. O. M. for his kindness and constant encouragement. o We also wish to thank S. B. Maurer, P. J. 'Halloran, C. Thomassen, B. Toft and our colleagues at the University of Waterloo for many helpful comments, and the National Research Council of Canada for its financial support. Finally, we would like to express our appreciation to Joan Selwood for her excellent typing and Diana Rajnovich for her beautiful artwork. J. A. Bondy U. S. R. Murty Contents Preface vi 1 GRAPHS AND SUBGRAPHS 1.1 Graphs and Simple Graphs . 1 1.2 Graph Isomorphism 4 1.3 The Incidence and Adjacency Matrices 7 1.4 Subgraphs . 8 1.5 Vertex Degrees . 10 1.6 Paths and Connection 12 1.7 Cycles . 14 Applications 1.8 The Shortest Path Problem. 15 1.9 Sperner's Lemma. 21 2 TREES 2.1 Trees. 25 2.2 Cut Edges and Bonds .. 27 2.3 Cut Vertices. 31 2.4 Cayley's Formula . 32 Applications 2.5 The Connector Problem .36 3 CONNECTIVITY 3.1 Connectivity. 42 3.2 Blocks . 44 Applications 3.3 Construction of Reliable Communication Networks 47 4 EULER TOURS AND HAMILTON CYCLES 4.1 Euler Tours . 51 4~2 Hamilton Cycles . 53 Applications 4.3 The. Chinese Postman Problem 62 4.4 The Travelling Salesman Problem 65 Contents ix 5 MATCHINGS 5.1 Matchings 70 5.2 Matchings and Coverings in Bipartite Graphs 72 5.3 Perfect Matchings . 76 Applications 5.4 The Personnel Assignment Problem· 80 5.5 The Optimal Assignment Problem 86 . 6 EDGE COLOURINGS 6.1 Edge Chromatic Number 91 6.2 Vizing's Theorem. 93 Applications 63 The Timetabling Problem 96 7 INDEPENDENT SETS AND CLIQUES 7.1 Independent Sets. . 101 7.2 Ramsey's Theorem 103 7.3 Turin's Theorem . . 109 Applications 7.4 Schur's Theorem. 112 7.5 A Geometry Problem . 113 8 VERTEX COLOURINGS 8.1 Chroniatic Number . .117 8.2 Brooks' Theorem . . 122 8.3 Haj6s'· Conjecture. 123 8.4 Chromatic 125 Polynomial~. 8.5 Girth and Chromatic Number 129 Applications 8.6 A Storage Problem 131 9 PLANAR GRAPHS 9.1 Plane and Planar Graphs 135 9.2 Dual Graphs. 139 9.3 Euler's Formula . . 143 9.4 Bridges. . 145 9.5 Kuratowski's Theorem . . 151 9.6 The Five-Colour Theorem and the Four-Colour Conjecture 156 9.7 Nonhamiltonian Planar Graphs . 160 Applications 9.8 A Planarity Algorithm . . 163 x Contents 10 DIRECTED GRAPHS 10.1 Directed Graphs . 171 10.2 Directed Paths 173 10.3 Directed Cycles . 176 Applications 10.4 A Job Sequencing Pr<?blem. 179 10.5 Designing an Efficient Computer Drum 181 10.6 Making a Road System One-Way 182 10.7 Ranking the Participants in a Tournament. 185 11 NETWORKS 11.1 Flows . 191 11.2 Cuts 194 11.3 The Max-Flow Min-Cut Theorem 196 Applications 11.4 Menger's Theorems 203 11.5 Feasible Flows 206 12 THE CYCLE SPACE AND BOND SPACE t 2.1 Circulations and Potential Differences. 212 12.2 The Number of Spanning Trees. 218 Applications 12.3 Perfect Squares . 220 Appendix I Hints to Starred Exercises 227 of Appendix II Four Graphs and a Table their Properties. 232 Appendix III Some Interesting Graphs. 234 Appendix IV Unsolved Problems. 246 Appendix V Suggestions for Further Reading 254 of Glossary Symbols 257 Index 261 1 Graphs and Subgraphs 1.1 GRAPHS AND SIMPLE GRAPHS Many real-world situations can conveniently be described by means of a diagram consisting of a set of points together with lines joining certain pairs of these points. For example, the points could represent people, with lines joining pairs of friends; or the points might be communication centres, with lines representing communication links. Notice that in such diagrams one is mainly interested in whether or not two given points are joined by a line; the manner in which they are joined is immaterial. A mathematical abstrac tion of situations of this type gives rise to the concept of a graph. A graph G is an ordered triple (V(G), E(G), t/lG) consisting of a nonempty set V( G) of vert~ces, a set E( G), disjoint from V( G), of edges, and an incidence function tPG that associates with each edge of G an unordered pair of (not necessarily distinct) vertices of G. If e is an edge and u and v are vertices such that t/lG(e) = UV, then e is said to join u and v; the vertices u and v are called the ends of e. Two examples of graphs should serve to clarify the definition. Example 1 G = (V(G), E(G), t/lG) where V(G)= {VI, V2, V3, V4, vs} E(G) = {el, e2, e3, e4, es, e6, e" es} and tPCi is defined by t/lG(el) = VI V2, t/lG(e2) = V2V3, tPG(e3) = V3V3, t/lG(e4) = V3V4 t/lG(es) = V2V4, t/lG(e6) = V4VS, t/lG(e,) = V2VS, t/lG(es) = V2VS Example 2 H = (V(H), E(H), t/lH) where V(H) = {u, v, W, x, y} E(H) = {a, b, c, d, e, f, g, h} and tPH is defined by tPH(a) = UV, t/lH(b) = uu, t/lH(C) = VW, t/lH(d) = wx t/lH( e) = vx, t/lH(f) = wx, t/lH(g) = ux, t/lH(h) = xy