Graduate Texts in Mathematics 63 Editorial Board F W Gehring P R Halmos Managing Editor c.e Moore Bela Bollobas Graph Theory An Introductory Course Springer-Verlag New York Heidelberg Berlin Bela Bollobas Department of Pure Mathematics and Mathematical Statistics University of Cambridge 16 Mill Lane Cambridge CB2 ISB ENGLAND Editorial Board P R Halmos Managing Editor Indiana University Department of Mathematics Bloomington, Indiana 47401 USA F W Gehring University of Michigan Department of Mathematics Ann Arbor, Michigan 48104 USA c C Moore University of California at Berkeley Department of Mathematics Berkeley, California 94720 USA AMS Subject Classification: 05Cxx With 80 Figures Library of Congress Cataloging in Publication Data Bollobas, Bela Graph theory (Graduate texts in mathematics: 63) Includes index I Graph theory I Title II Series QA166.B662 511'.5 79-10720 All rights reserved No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag © 1979 by Springer-Verlag New York Inc Softcover reprint of the hardcover 1st edition 1979 876 54 32 ISBN-13: 978-1-4612-9969-1 001: 10.10077978-1-4612-9967-7 e-ISBN-13: 978-1-4612-9967-7 To Gabriella There is no permanent place in the world for ugly mathematics G H Hardy A Mathematician's Apology Preface This book is intended for the young student who is interested in graph theory and wishes to study it as part of his mathematical education Experience at Cambridge shows that none of the currently available texts meet this need Either they are too specialized for their audience or they lack the depth and development needed to reveal the nature of the subject We start from the premise that graph theory is one of several courses which compete for the student's attention and should contribute to his appreciation of mathematics as a whole Therefore, the book does not consist merely of a catalogue of results but also contains extensive descriptive passages designed to convey the flavour of the subject and to arouse the student's interest Those theorems which are vital to the development are stated clearly, together with full and detailed proofs The book thereby offers a leisurely introduction to graph theory which culminates in a thorough grounding in most aspects of the subject Each chapter contains three or four sections, exercises and bibliographical notes Elementary exercises are marked with a - sign, while the difficult ones, marked by + signs, are often accompanied by detailed hints In the opening sections the reader is led gently through the material: the results are rather simple and their easy proofs are presented in detail The later sections are for those whose interest in the topic has been excited: the theorems tend to be deeper and their proofs, which may not be simple, are described more rapidly Throughout this book the reader will discover connections with various other branches of mathematics, including optimization theory, linear algebra, group theory, projective geometry, representation theory, probability theory, analysis, knot theory and ring theory Although most of these connections are nQt essential for an understanding of the book, the reader would benefit greatly from a modest acquaintance with these SUbjects vii viii Preface The bibliographical notes are not intended to be exhaustive but rather to guide the reader to additional material I am grateful to Andrew Thomason for reading the manuscript carefully and making many useful suggestions John Conway has also taught the graph theory course at Cambridge and I am particularly indebted to him for detailed advice and assistance with Chapters II and VIII I would like to thank Springer-Verlag and especially Joyce Schanbacher for their efficiency and great skill in producing this book Cambridge April 1979 Bela Bollobas Contents Chapter I Fundamentals Definitions Paths, Cycles and Trees Hamilton Cycles and Euler Circuits Planar Graphs An Application of Euler Trails to Algebra Exercises Notes 11 16 19 22 25 Chapter II Electrical Networks 26 Graphs and Electrical Networks Squaring the Square Vector Spaces and Matrices Associated with Graphs Exercises Notes 26 33 35 41 43 Chapter III Flows, Connectivity and Matching 44 Flows in Directed Graphs Connectivity and Menger's Theorem Matching Tutte's I-Factor Theorem Exercises Notes 45 50 53 58 61 66 IX Contents X Chapter IV Extremal Problems 67 68 71 75 80 84 87 Paths and Cycles Complete Subgraphs Hamilton Paths and Cycles The Structure of Graphs Exercises Notes Chapter V Colouring 88 I Vertex Colouring 89 93 95 Edge Colouring Graphs on Surfaces fure~ Notes ~ 102 Chapter VI Ramsey Theory 103 I The Fundamental Ramsey Theorems Monochromatic Subgraphs Ramsey Theorems in Algebra and Geometry Subsequences Exercises Notes 103 107 110 liS 119 121 Chapter VII Random Graphs 123 I 124 127 130 133 139 142 144 Complete Subgraphs and Ramsey Numbers-The Use of the Expectation Girth and Chromatic Number-Altering a Random Graph Simple Properties of Almost All Graphs-The Basic Use of Probability Almost Determined Variables-The Use of the Variance Hamilton Cycles-The Use of Graph Theoretic Tools Exercises Notes Chapter VIIl Graphs and Groups 146 I Cayley and Schreier Diagrams Applications of the Adjacency Matrix Enumeration and P6lya's Theorem Exercises Notes 146 155 162 169 173 Subject Index 175 Index of Symbols 179 167 §3 Enumeration and P61ya's Theorem PROOF By Lemma 12 IriS I WI I = ;= L L w(OJ = w(f) aEl JEF(a*) Now clearly F(rx*) = {fERD:f is constant on cycles of rx}, so if ~1' ~2"'" ~m are the cycles of rx and a E ~; means that a is an element of the cycle ~; then and F(rx*)={fERD:r;ER if aE~;,i=1,2, ,m} f(a)=r; Hence I JEF(a*) w(f) = I TI w(rJI~;I = TI I d m (ri)cR ;= k= ( w(r)k )jk(a) = rER TI si da ), d k= giving InS = d L TI slk(a) = z(r; SI' S2"'" aE k = Sd)' o If Ir I has an inverse in the ring A, say if A is a polynomial ring over the rationals, then Theorem 13 can also be written in its more usual form: Let us illustrate now how the theorem can be applied Let us consider again the bracelets made up of five beads, which can be red, blue and green Then D = {1, 2, 3, 4, 5} is the set of places of the beads, R = {r, b, g} is the set of colours (figures) and r is C s , the cyclic group of order generated by the permutation (12345) The cycle sum is Z = a~ + 4a s On choosing A = 7L and w(r) = w(b) = w(g) = 1, we find that Sk = for every k, so 5S = s + 4.3 Since each pattern (bracelet) has weight 1, there are !{3s + 12} = 51 distinct patterns (bracelets) On choosing A = 7L[x, y] and w(r) = 1, w(b) = x, w(g) = y, we find that S = !{(l + x + y)S + 4(1 + X S + yS)} Now it is easy to extract much information from this form of S For example, a bracelet has weight xy2 iff it has red, blue and green beads Thus the number of such bracelets is the coefficient of xy2 in the polynomial S, that is 0/5)(5 !/2!2!) = EXAMPLE What happens if in the previous example we allow reflections? Then r is the dihedral group Ds' the group of symmetries of the regular pentagon, whose cycle sum is a~ + 4a s + 5a l a~ Thus if we take, as before, A = 7L[x, y], w(r) = 1, w(b) = x and w(g) = y, we find that the number of bracelets containing red, blue and green beads is the coefficient of xy2 in lo{(l + x + y)S + 40 + X S + yS) + 5(1 + x + y)(1 + x + y2)2}, that is,3 + = EXAMPLE 168 VIII Graphs and Groups EXAMPLE This is the example P6lya used to illustrate his theorem Place red, blue and yellow balls in the vertices of an octahedron In how many distinct ways can this be done? The group of symmetries of the octahedron has cycle sum a~ + 6ai a4 + 3ai a~ + 6a~ + 8a~: a~ comes from the identity, 6ai a4 from rotations through rc about axes through the vertices, 6a~ from rotations through rc about axes through midpoints of edges, and, finally, a~ is the summand corresponding to a rotation through 2rc/3 about an axis going through the centre of a face On taking A = Z[x, y], w(r) = 1, w(b) = x and w(y) = y, we see that the required number is the coefficient of x y in + x + y)6 + 6(1 + x + y)2(1 + X4 + y4) + 3(1 + x + y)2(1 + x + y2)2 + 6(1 + x + y2)3 + 8(1 + x + y3)2}, l4{(1 that is, It should be clear by now that the theorem loses nothing from its generality if instead of a general commutative ring A we take Z[x,: r E R], the polynomial ring over the integers in variables with the elements of R, and we define the weight function as w(r) = x, Then the pattern sum S contains all the information the theorem can ever give us In particular, if w: R ~ A is an arbitrary weight function then the corresponding pattern sum is obtained by replacing x, by w(r) in S However, if R is large, the calculations may get out of hand if we not choose a "smaller" ring than Z[x,: r E R], which is tailor-made for the problem at hand The choice of a smaller ring is, of course, equivalent to a substitution into S EXAMPLE Place red, blue, green and yellow balls into the vertices of an octahedron Denote by Pi the set of patterns in which the total number of red and blue balls is congruent to i modulo What is 1Pol - 1P 21 ? The cycle sum of the rotation group of the octahedron was calculated in Exercise and was found to be a~ + 6ai a4 + 3ai a~ + 6a~ + 8a~ Let A = C, the field of complex numbers, and put w(r) = w(b) = i, w(g) = w(y) = Then for a pattern/we have Re w(f)= iffEP o , Re w(f)= - if / E P and Re w(f) = if / E PI U P 3· Thus IF 01 - jP 21 is exactly the real part of the pattern sum As s = 2(1 + i), s = 0, S = 2(1 - i) and S4 = 4, we see immediately that after substitution the real part of each term is 0, so IF0 1= 1P I· We were first led to our study of the orbits of a permutation group by our desire to count the number of graphs within isomorphism We realized that this amounted to counting the orbits of the group acting on {O, 1}x, where X = V(2) and rn is the permutation group acting on X which is induced by the symmetric group acting on V So according to P6lya's theorem our problem is solved when we know the cycle sum of the permutation group rn It is now a routine matter to write down an explicit expression for this r: 169 Exercises cycle sum, though we don't display it here since its form is not very inspiring Furthermore, except for small values of n it is too unwieldy for practical calculations and it is much easier to use asymptotic formulae derived by random graph techniques (see Exercises 22 and 23 of Chapter VII) We remark finally that an extension of P6lya's theorem covers the case when there is also a group acting on the range of the functions For instance, if we let S2 act on {a, I} in the example above, we not distinguish between a graph and its complement, and may thereby compute the number of graphs which are isomorphic to their complements EXERCISES Draw the Cayley diagram of the quaternion group