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Oxford Lecture Series in Mathematics and its Applications 28 Series Editors John Ball Dominic Welsh OXFORD LECTURE SERIES IN MATHEMATICS AND ITS APPLICATIONS 1. J. C. Baez (ed.): Knots and quantum gravity 2. I. Fonseca and W. Gangbo: Degree theory in analysis and applications 3. P. L. Lions: Mathematical topics in fluid mechanics, Vol. 1: Incompressible models 4. J. E. Beasley (ed.): Advances in linear and integer programming 5. L. W. Beineke and R. J. Wilson (eds): Graph connections: Relationships between graph theory and other areas of mathematics 6. I. Anderson: Combinatorial designs and tournaments 7. G. David and S. W. Semmes: Fractured fractals and broken dreams 8. Oliver Pretzel: Codes and algebraic curves 9. M. Karpinski and W. Rytter: Fast parallel algorithms for graph matching problems 10. P. L. Lions: Mathematical topics in fluid mechanics, Vol. 2: Compressible models 11. W. T. Tutte: Graph theory as I have known it 12. Andrea Braides and Anneliese Defranceschi: Homogenization of multiple integrals 13. Thierry Cazenave and Alain Haraux: An introduction to semilinear evolution equations 14. J. Y. Chemin: Perfect incompressible fluids 15. Giuseppe Buttazzo, Mariano Giaquinta and Stefan Hildebrandt: One-dimensional variational problems: an introduction 16. Alexander I. Bobenko and Ruedi Seller: Discrete integrable geometry and physics 17. Doina Cioranescu and Patrizia Donate: An introduction to homogenization 18. E. J. Janse van Rensburg: The statistical mechanics of interacting walks, polygons, animals and vesicles 19. S. Kuksin: Hamiltonian partial differential equations 20. Alberto Bressan: Hyperbolic systems of conservation laws: the one-dimensional Cauchy problem 21. B. Perthame: Kinetic formulation of conservation laws 22. Andrea Braides: T-convergence for beginners 23. Robert Leese and Stephen Hurley (eds): Methods and algorithms for radio channel assignment 24. Charles Sernple and Mike Steel: Phylogenetics 25. Luigi Ambrosio and Paolo Tilli: Topics on analysis in metric spaces 26. Eduard Feireisl: Dynamics of viscous compressible fluids 27. Anotnm Novotny and Ivan Straskraba: Introduction to mathematical theory of compressible flow 28. Pavol Hell and Jaroslav Nesetfil: Graphs and homomorphisms Graphs and Homomorphisms Pavol Hell Simon Fraser University, Burnaby, B.C., Canada and Jaroslav Nesetfil Charles University, Prague, The Czech Republic OXPORD UNIVERSITY PRESS OXPORD UNIVERSITY PRESS Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford. It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Bangkok Buenos Aires Cape Town Chennai Dares Salaam Delhi Hong Kong Istanbul Karachi Kolkata Kuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi Sao Paulo Shanghai Taipei Tokyo Toronto Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York © Oxford University Press 2004 The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2004 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose this same condition on any acquirer A catalogue record for this title is available from the British Library Library of Congress Cataloging in Publication Data (Data available) ISBN 0 19 852817 5 (Hbk) 10 9 8 7 6 5 4 3 21 Typeset by the authors using lATfrjX Printed in Great Britain on acid-free paper by Biddies Ltd, King's Lynn PREFACE This is a book about graph homomorphisms. While graph theory is now an established discipline (within the field of combinatorics), the study of graph ho- momorphismsis onlybeginning to gainwide acceptance.As little asafew years ago, most graph theorists, while passively aware of a few classical results on graph homomorphisms, would not include homomorphisms among the topics of central interest in graph theory. We believe that this perception is changing, principally because of the usefulness of the homomorphism perspective in areas such as graph reconstruction, products, and fractional and circular colourings, andapplications incomplexity theory,artificialintelligence, telecommunication, and, most recently, statistical physics. At the same time, the homomorphism framework strengthens the link between graph theory and other parts of mathematics, making graph theory more attractive, and understandable, to other mathematicians. We feel that the time is ripe to introduce this exciting topic to a wider audience. It was our intention to bring together what we see as the highlights of the theory and its many applications. We hope that our book will be seen as a sampler of this rich theory, of its most interesting results, techniques, and applications. Sample additional results have been included in the Exercises and referredtointheRemarks.Wehopethatthereaderwillbemotivatedtofurther explorethe richliterature.We havetriednotto setourfocus too narrowly;thus thetechniquesandpointsofviewvary,fromalgebraicandalgorithmictoapplied, extremal,andrandomized.Theresultinglackofhomogeneitymeansthatwehave hadto occasionallymake certaincompromisesoncontinuity, levelofexposition, terminology, or organization.We hope the reader will be understanding. One challenge we faced was the intermingling of the various versions of graphs, digraphs, and more general systems. It is typical of the area to freely jumpfromgraphstodigraphs,allowingordisallowingloops,asisdictatedbythe context.We havetriedto be clearateachpointwhat is the correctcontext,but the reader may find it useful to keep in mind the main possibilities, illustrated in Fig. 1.1. In general, our most basic context is that of digraphs, i.e., sets with one binary relation; graphs are viewed as a subclass of digraphs. Occasionally, we shall consider more general relational systems, i.e., sets with several relations of various arities. Homomorphisms are defined the same way in all these contexts—they simply have to preserve all the relations. Most homomorphism- related concepts transfer between these contexts without difficulty—this is pre- cisely what makes jumping between the contexts possible. Many of the results weshalldiscussapplyinthemostgeneralcontextofrelationalsystems;however, if the generalization does not bring a new perspective, we usually just stick to v vi PREFACE the context of digraphs, or even just graphs. One generality we have, for the most part, completely avoided is infinite graphs. By definition, all our graphs, digraphs, and relational systems are finite. This includes, in the case of general relational systems, the number of relations and their arities. However, it does not include sets of graphs and associated notions such as category and partial order; these sets (categories, orders) can be infinite. If the book is to be a sampler,we havewritten Chapter 1 as a mini-sampler. In it, we introduce motivational examples and applications, which are usually taken up in more detail in later chapters. It gives the flavour of algorithmic aspects, to be taken up again in Chapter 5, retractions, to be further discussed in Chapter 2, duality, investigated in Chapters 3 and 5, constraint satisfaction problems, discussed in Chapter 5, as well as structural properties of homomorphism composition, to which we devote Chapter 4. The highlights of Chapter 1 include a simple proof of the Colouring Interpolation Theorem, a generalization of the No-Homomorphism Lemma, the construction of a triangle-free graph to which all cubic triangle-free graphs are homomorphic, a case of the Edge Re- construction Conjecture, and a generalization of a theorem of Frucht on graphs with prescribed automorphism groups. Chapter 2 focuses oncertainbasicconstructions thatoccurfrequently in the rest of the book, emphasizing the product and the retract, but also consider- ing other constructs. These include the shift graph, the exponential graph, and the Lova´sz vector—each of which plays an auxiliary role in Chapter 2, and is a useful concept in its own right. Other basic constructions are discussed in later chapters; we mention here the replacement operation alluded to in Chapter 1 and explored in greater detail in Chapter 4, the indicator and subindicator constructions described in Chapter 5, the Kneser graphs and the rational complete graphs studied in Chapter 6, and so on. Taken together, such constructions are the commonthreadandthe leitmotiv ofthis book.The highlightofthe sections on products include a linear algebra based lower bound on the dimension of a graph, a stronger version of the edge reconstruction result from Chapter 1, a discussion of cancellation and unique factorization properties, a construction of graphs with arbitrarily high odd girth and chromatic number, an exploration of the Product Conjecture, and an elementary proof of the multiplicativity of orientedcycles of prime power length. In the sections on retracts,we prove that an isometric tree, and a shortest cycle, is always a retract of a reflexive graph; we prove a similar result for shortest odd cycles in irreflexive nearly bipartite graphs. We characterize absolute reflexive retracts in several different ways, in- cluding characterizations in terms of majority functions, in terms of the variety of paths, and in terms of the Helly property (or the absence of holes). We prove thatareflexivegraphadmitsawinningstrategyforthecop,inthegameofcops and robbers, if and only if it is dismantlable, and relate dismantlable graphs to absolute reflexive retracts.Finally, we introduce mediangraphs andrelate them to retracts of hypercubes; we also discuss an application of median graphs and retractions in a resource location problem. PREFACE vii In Chapter 3 we consider the order homomorphisms induce on the set of all cores. This order is rich enough to represent all countable partial orders. We considerantichainsinthe homomorphismorder,i.e.,collectionsofincomparable graphs (graphs without homomorphisms between any two of them). Of partic- ular interest are finite maximal antichains, and their structure turns out to be surprisingly revealing. Graphs only have trivial finite maximal antichains, while digraphs have many such antichains, of all possible sizes, arising from duality relationships. This chapter also contains the (probabilistic) proof of the Sparse Incomparability Lemma, of the fact that asymptotically almost all graphs on n vertices are cores,and of the fact that the number of incomparable graphs on n vertices differs little (asymptotically) from the total number of non-isomorphic graphsonn vertices.The densityofthe homomorphismorderisrelatedto duality, revealing an unexpected connection between these two seemingly unrelated concepts.Finally,weshowthatonecangaininterestinginsightsintomanytradi- tionalgraphtopics,suchas,say,Hadwiger’sconjecture,when interpreting them as statements about the homomorphism order. In Chapter 4, we explore the structure, as opposed to just the existence, of the family of homomorphisms among a set of graphs. The difference is notice- able with even just one graph—consider, for instance, a graph having only the identity homomorphisms to itself. Such graphs are called rigid and they are the building blocks of many constructions. We construct many useful examples of rigidgraphs,provethatasymptoticallyalmostallgraphsarerigid,andconstruct infiniterigidgraphswitharbitrarycardinality.Thehomomorphismsamongaset of graphs impose the algebraic structure of a category. We show that every finite category is represented by a set of graphs. This is the generalization of the theorem of Frucht alluded to above. Also, as in the case studied by Frucht, we show that the representing graphs can be required to have any of a number of graph theoretic properties. However, we prove that these properties cannot in- cludehavingboundeddegrees—somewhatsurprisingly,sinceFruchtprovedthat cubic graphs represent all finite groups. In Chapter 5, we explore algorithmic aspects of graph homomorphisms and ofsimilarpartitionproblems.Thehighlightsincludethedichotomyclassification of graph homomorphisms to a fixed target graph H, a proof of the fact that dichotomy for digraph homomorphisms would imply dichotomy for all constraint satisfaction problems, a presentation of consistency-basedalgorithms, and asso- ciateddualities, thatseem to be applicable to allknownpolynomialcasesof the digraph homomorphism problem. We also discuss the use of polymorphisms for the design of polynomial algorithms, and prove that graphs with the same set of polymorphisms define polynomially equivalent problems.We explain how the polymorphism known as the majority function can be used to construct a polynomial time algorithm. We prove the dichotomy classification of list homomorphism problems for reflexive graphs. We present list matrix partition problems in the language of trigraph homomorphisms, and illustrate the richness of the associated algorithms on the case of clique cutsets and generalized split graphs. viii PREFACE Finally,Chapter6setsoutcertainparticularclassesofhomomorphismprob- lems that have been investigated as variants of colourings. The homomorphism perspectiveunifiesthese conceptsandoffersnewquestions.We include adiscus- sion of the circular chromatic number, the fractional chromatic number, the T-span, and the oriented chromatic number. The highlights include a number of equivalent definitions of the circular chromatic number, in terms of H- colourability, in terms of a geometric representation, in terms of orientations, implying, say, Minty’s result on chromatic numbers, and in terms of schedule concurrency. For fractional chromatic numbers we also give equivalent formula- tions, in terms of Kneser graphs, integer linear programs, and zero-sum games, and relate them in several ways to the circular chromatic numbers. We discuss homomorphismsamongstKneser graphsanda proofofKneser’s conjecture.We prove that the span, for any set T, of the cliques K has a limit, closely related n to the fractionalchromatic number of an associatedgraph.We also give bounds onthe orientedchromaticnumbersofplanarandouterplanargraphs,andrelate the oriented chromatic numbers to acyclic chromatic numbers. Our book can be used as a textbook for a second course in graph theory, at the level of a beginning graduate student. (In fact, we have used it for just such a course at Simon Fraser University, Vancouver, Charles University, Prague, Eidgeno¨ssische Technische Hochschule Zu¨rich, Universidade Federal do Rio de Janeiro,andUniversitatPolitecnicadeCatalunya.)Becauseoftherelativeinde- pendence ofthe chapters,the book canalsobe usedasa supplementarytext for a more varied course (at the same graduate or even undergraduate level). One can,forinstance,justpresentChapter1,ourmini-sampler.Inaddition,Chapter 1 can be supplemented by a sequence of combinatorial topics from Chapters 2 and 6. If time permits, a more intensive sequence could complement Chapter 1 with a selection of algebraic topics from Chapters 2, 3, and 4, or of algorithmic topics from Chapters 2 and 5. The exercisesvary indifficulty. The firstfew areusually intendedto give the reader an opportunity to practice the concepts introduced in the chapter; the later ones explore related concepts or even introduce new ones. For the harder exercises we usually give a hint or a reference. We thank our students, friends, and collaborators for checking some of the details in this book. Special thanks to M. Ba´lek, T. Feder, J. Foniok, J. Fiala, J. Huang, A. Kazda, J. Kratochv´ıl, L. Lova´sz, J. Matouˇsek, R. Naserasr, A. Raspaud, V. Rödl, R. Sˇa´mal, I. Sˇvejdarova´, and U. Wagner, who have made numerous suggestions. We are particularly grateful to C. Tardif and X. Zhu for their valuable input; with their participation, we are currently writing a more comprehensivefollow-upbook.Lastbutnotleast,weexpressourdeepgratitude to our teachers, and pioneers of the area, Zdenˇek Hedrl´ın, Aleˇs Pultr, and Gert Sabidussi. We are the only ones to blame for any remaining errors and inconsistencies. The book was written over an extended period of time, and we can only hope that we have managed to make all the parts fit together. PREFACE ix We will maintain a webpage at www.cs.sfu.ca/∼pavol/hombook.html torecordanycorrectionsfoundafterprinting,andprovideotherusefulinforma- tion and links. The reader’s input would be appreciated. Finally, we dedicate this book to all who have encouragedand inspired us in this endeavor, especially Marion Hellova´, Helena Neˇsetˇrilova´,Heather Mitchell, Catherine and Julia Taylor-Hell, Jakub Neˇsetˇril, Sam and Erin Hogg, and Jan and Lenka Hˇrebejk. Pavol Hell, JaroslavNeˇsetˇril Vancouver, Prague,Christmas 2003.