CONTEMPORARY MATHEMATICS 352 Graph Colorings Marek Kubale Editor http://dx.doi.org/10.1090/conm/352 Graph Colorings CoNTEMPORARY MATHEMATICS 352 Graph Colorings Marek Kubale Editor American Mathematical Society Providence, Rhode Island Editorial Board Dennis DeTurck, managing editor Andreas Blass Andy R. Magid Michael Vogeli us This work was originally published in Polish by Wydawnictwa Naukowo-Techniczne under the title "Optymalizacja dyskretna. Modele i metody kolorowania graf6w", © 2002 Wydawnictwa N aukowo-Techniczne. The present translation was created under license for the American Mathematical Society and is published by permission. 2000 Mathematics Subject Classification. Primary 05Cl5. Library of Congress Cataloging-in-Publication Data Optymalizacja dyskretna. English. Graph colorings/ Marek Kubale, editor. p. em.- (Contemporary mathematics, ISSN 0271-4132; 352) Includes bibliographical references and index. ISBN 0-8218-3458-4 (acid-free paper) 1. Graph coloring. I. Kubale, Marek, 1946- II. Title. Ill. Contemporary mathematics (American Mathematical Society); v. 352. QA166 .247.06813 2004 5111.5-dc22 2004046151 Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Math- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to reprint-permissien@ams. erg. Excluded from these provisions is material in chapters for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). © 2004 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. § The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http: I /www. ams. erg/ 10 9 8 7 6 5 4 3 2 1 09 08 07 06 05 04 Contents Preface ix Graph coloring ix History of graph coloring X Models of graph coloring xi Preface to the English Edition xii Chapter 1. Classical Coloring of Graphs ADRIAN KOSOWSKI, KRZYSZTOF MANUSZEWSKI 1 1.1. Basic terms and definitions 1 1.2. Classical vertex-coloring 6 1.3. Classical edge-coloring 16 Chapter 2. On-line Coloring of Graphs PIOTR BOROWIECKI 21 2.1. On-line and off-line coloring 21 2.2. On-line coloring algorithms 23 2.3. Worst case effectiveness of on-line coloring 24 2.4. Expected effectiveness of on-line coloring 27 2.5. Susceptibility of graphs 28 2.6. Coloring of intersection graphs 29 2.7. Applications to resource management 31 Chapter 3. Equitable Coloring of Graphs HANNA FURMANCZYK 35 3.1. Equitable vertex-coloring 35 3.2. Equitable total coloring 50 Chapter 4. Sum Coloring of Graphs MICHAL MALAFIEJSKI 55 4.1. Definition and simple properties 55 4.2. The complexity of the sum coloring problem 58 4.3. Generalizations of the sum coloring problem 63 4.4. Some applications of the sum coloring problem 64 Chapter 5. T-Coloring of Graphs ROBERT JANCZEWSKI 67 5.1. The spans 67 5.2. Sets of forbidden distances 69 5.3. T-colorings of graphs 70 5.4. T-spans and T-chromatic numbers 71 5.5. Homomorphisms and T-graphs 73 v vi CONTENTS 5.6. Estimates and exact values 74 5.7. The computational complexity 75 5.8. Approximation algorithms 76 5.9. Applications 77 Chapter 6. Rank Coloring of Graphs DARIUSZ DERENIOWSKI 79 6.1. Vertex ranking 79 6.2. Edge ranking 87 Chapter 7. Harmonious Coloring of Graphs MAREK KUBALE 95 7.1. Introduction 95 7.2. Graphs with known harmonious number 97 7.3. Bounds for the harmonious chromatic number of general graphs 99 7.4. Algorithm Depressive 101 7.5. Applications 102 Chapter 8. Interval Edge-Coloring of Graphs KRZYSZTOF GIARO 105 8.1. Basic properties of the model 105 8.2. Consecutively colorable bipartite graphs 109 8.3. The span of interval coloring 113 8.4. Deficiency of graphs 115 Chapter 9. Circular Coloring of Graphs ADAM NADOLSKI 123 9.1. Circular coloring of the vertices of a graph 123 9.2. Circular coloring of the edges of a graph 130 Chapter 10. Path Coloring and Routing in Graphs J AKUB BIALOGRODZKI 139 10.1. Basic definitions 139 10.2. Known results 143 10.3. Applications 151 Chapter 11. List Colorings of Graphs KONRAD PIWAKOWSKI 153 11.1. Notation and definitions 153 11.2. Bipartite and 2-choosable graphs 154 11.3. The Haj6s Construction 156 11.4. D-choosability and Brooks theorem 157 11.5. Planar graphs 158 11.6. Graphs for which X = X< 159 11.7. (k, r)-choosability 160 11.8. Edge-list coloring 161 Chapter 12. Ramsey Colorings of Complete Graphs TOMASZ Dzmo 163 12.1. Notation and basic definitions 163 12.2. Ramsey numbers 164 CONTENTS vii 12.3. Values and properties of classical Ramsey numbers 166 12.4. Nonclassical Ramsey numbers 170 12.5. Applications of Ramsey numbers 173 Chapter 13. Placing Guards in Art Galleries by Graph Coloring PAWEL ZYLINSKI 177 13.1. Introduction 177 13.2. Fisk's proof 180 13.3. The orthogonal art gallery theorem 182 13.4. Orthogonal polygons with holes 183 13.5. Final remarks 188 Bibliography 189 Index 203 Authors' addresses 207 Preface Optimization problems can be naturally divided into two categories: continuous and discrete. Continuous optimization problems are those that allow continuous variables. Generally speaking the aim of such problems is to find a set of real numbers or a real function, which optimize a certain criterion. On the other hand, discrete optimization problems are those that contain discrete variables. The aim of such problems is to find a combinatorial object, which optimizes a certain criterion function in a finite space of legal solutions. On the whole, each of these two types of problems requires separate solution techniques. This book is devoted to selected problems of combinatorial optimization, otherwise known as discrete optimization. The fact that such combinatorial calculations are becoming increasingly important stimulates research and development of the field. Common programmers' expe- rience shows that the number of combinatorial computations appearing in user programs increases more rapidly than the number of numerical computations. The reason for this is that besides the traditional fields of applications of mathematics in engineering and physics, discrete structures appear now more frequently than continuous ones. Discrete optimization is a branch of applied mathematics and theoretical computer science that includes various topics of graph theory, network design, mathematical programming, sequencing and scheduling, as well as many others. One of its areas is graph coloring, to which this book is entirely devoted. Graph coloring Graph coloring is one of the oldest and best-known problems of graph theory. As people became accustomed to applying the tools of graph theory to the solu- tion of real-world technological and organizational problems, new chromatic models emerged as a natural way of tackling many practical situations. Internet statistics show that graph coloring is one of the central issues in the collection of several hundred classical combinatorial problems [60]. The reason for this is the simplicity of its formulation and seemingly natural solution on the one hand, and numerous potential applications on the other. Unfortunately, high computational complexity prevents the efficient solution of numerous problems by means of graph coloring. For example, the deceptively simple task of deciding if the chromatic number of a graph, i.e. the smallest number of colors that can be assigned to the vertices of a graph so that no pair of adjacent vertices is colored with the same color, is at most 3 remains NP-complete [194]. In practice this means that our task cannot be solved in polynomial time and, consequently, it is impossible to find a chromatic solution to a graph on several dozens of vertices in a reasonable time. Needless to say, graphs of this size are definitely too small to be considered satisfactory in practical applications. ix X PREFACE History of graph coloring The origins of graph coloring may be traced back to 1852 when de Morgan wrote a letter to his friend Hamilton informing him that one of his students had observed that when coloring the counties on an administrative map of England only four colors were necessary in order to ensure that adjacent counties were given different colors. More formally, the problem posed in the letter was as follows: What is the least possible number of colors needed to fill in any map (real or invented) on the plane? The problem was first published in the form of a puzzle for the public by Cayley in 1878. The first "proof" of the Four Color Problem (FCP) was presented by Kempe in [198]. For a decade following the publication of Kempe's paper, the FCP was considered solved. For his accomplishment Kempe was elected a Fellow of the Royal Society and later the President of the London Mathematical Society. He even presented refinements of his proof. The case was not closed, however. Heawood [158] stated that he had discovered an error in Kempe's proof- an error so serious that he was unable to repair it. In his paper, Heawood gave an example of a map which, although easily 4-colorable, showed that Kempe's proof technique did not work in general.