CONTEMPORARY MATHEMATICS

352

Graph Colorings Marek Kubale

Editor

Graph Colorings

http://dx.doi.org/10.1090/conm/352

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

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10 9 8 7 6 5 4 3 2 1 09 08 07 06 05 04

Contents

Preface Graph coloring History of graph coloring Models of graph coloring Preface to the English Edition

Chapter 1. Classical Coloring of Graphs ADRIAN KOSOWSKI, KRZYSZTOF MANUSZEWSKI

1.1. Basic terms and definitions 1.2. Classical vertex-coloring 1.3. Classical edge-coloring

Chapter 2. On-line Coloring of Graphs PIOTR BOROWIECKI

2.1. On-line and off-line coloring 2.2. On-line coloring algorithms 2.3. Worst case effectiveness of on-line coloring 2.4. Expected effectiveness of on-line coloring 2.5. Susceptibility of graphs 2.6. Coloring of intersection graphs 2.7. Applications to resource management

Chapter 3. Equitable Coloring of Graphs HANNA FURMANCZYK

3.1. Equitable vertex-coloring 3.2. Equitable total coloring

Chapter 4. Sum Coloring of Graphs MICHAL MALAFIEJSKI

4.1. Definition and simple properties 4.2. The complexity of the sum coloring problem 4.3. Generalizations of the sum coloring problem 4.4. Some applications of the sum coloring problem

Chapter 5. T-Coloring of Graphs ROBERT JANCZEWSKI

5.1. The spans 5.2. Sets of forbidden distances 5.3. T-colorings of graphs 5.4. T-spans and T-chromatic numbers 5.5. Homomorphisms and T-graphs

v

ix ix X

xi xii

1 1 6

16

21 21 23 24 27 28 29 31

35 35 50

55 55 58 63 64

67 67 69 70 71 73

vi CONTENTS

5.6. Estimates and exact values 5.7. The computational complexity 5.8. Approximation algorithms 5.9. Applications

Chapter 6. Rank Coloring of Graphs

74 75 76 77

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

8.1. Basic properties of the model 8.2. Consecutively colorable bipartite graphs 8.3. The span of interval coloring 8.4. Deficiency of graphs

Chapter 9. Circular Coloring of Graphs ADAM NADOLSKI

9.1. Circular coloring of the vertices of a graph 9.2. Circular coloring of the edges of a graph

Chapter 10. Path Coloring and Routing in Graphs J AKUB BIALOGRODZKI

10.1. Basic definitions 10.2. Known results 10.3. Applications

Chapter 11. List Colorings of Graphs KONRAD PIWAKOWSKI

11.1. Notation and definitions 11.2. Bipartite and 2-choosable graphs 11.3. The Haj6s Construction 11.4. D-choosability and Brooks theorem 11.5. Planar graphs 11.6. Graphs for which X = X< 11.7. (k, r)-choosability 11.8. Edge-list coloring

Chapter 12. Ramsey Colorings of Complete Graphs TOMASZ Dzmo

12.1. Notation and basic definitions 12.2. Ramsey numbers

105 105 109 113 115

123 123 130

139 139 143 151

153 153 154 156 157 158 159 160 161

163 163 164

CONTENTS

12.3. Values and properties of classical Ramsey numbers 12.4. Nonclassical Ramsey numbers 12.5. Applications of Ramsey numbers

Chapter 13. Placing Guards in Art Galleries by Graph Coloring PAWEL ZYLINSKI

13.1. Introduction 13.2. Fisk's proof 13.3. The orthogonal art gallery theorem 13.4. Orthogonal polygons with holes 13.5. Final remarks

Bibliography

Index

Authors' addresses

vii

166 170 173

177 177 180 182 183 188

189

203

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. However, he was able to use Kempe's technique to prove that every map could be 5-colored. Meanwhile, progress on the FCP was slow and painful. In 1913 Birkhoff showed that certain configurations in a map are reducible, in the sense that a 4-coloring of a fragment of a map can be extended to a coloring of the whole map. This idea of reducibility turned out to be crucial in the eventual proof of the theorem. In 1976, after many attempts at solving and partial results on the FCP, Appel and Haken announced a complete proof, later published in [7]. This outstanding achievement was based on the method of "reducible configurations" and a substantial amount of computer time. The argument of Appel and Haken required a massive computation. It was, in fact, the first case where the computer assisted researchers in finding the argument by eliminating a large number of particular cases. This revolutionary incorporation of computations in the combinatorial proof was met with some skepticism. Later, Seymour, Robertson et al. [304] found a proof of FCT that requires much less machine involvement. Likely, other researchers will produce a human-checkable proof soon. Nevertheless it is now clear that combinatorial optimization can use computers not only as the devices to search, but also as assistants in proofs.

As far as the complexity issue of graph coloring is considered, we have already mentioned Karp's NP-hardness proof of vertex-coloring [195]. The next milestone result is due to Garey and Johnson [97] who proved that finding a coloring of value not worse than twice the chromatic number is NP-hard. Bellare et al. (18] strenghtened this result by showing that the problem is not approximable within n 117-e: for any c: > 0. The best currently known polynomial-time approximation algorithm is due to Halld6rsson [145] and has O(n(log log n) 2 / log3 n) performance guarantee.

Obviously, coloring a political map is equivalent to coloring the vertices of its dual planar graph. But a graph also has edges which can be colored. The edge-coloring problem was posed in 1880 in relation to the FCP. The first paper that dealt with this subject was written by Tait [325]. In his paper Tait proved that the FCP is equivalent to the problem of edge-coloring every planar 3-connected cubic graph with 3 colors. In 1916 Konig (214] proved that every bipartite graph can be d-edge-colored, where dis the maximum vertex degree. Later, Vizing (341] proved a

PREFACE xi

very strong result, asserting that every simple graph can be edge-colored with ~ + 1 colors. In other words, this result says that the problem is approximable with an absolute error guarantee of 1 on simple graphs. The first proof of NP-hardness of edge-coloring is due to Holyer [166]. Of course, edge-coloring can be regarded as a special case of vertex-coloring, namely the problem of coloring the vertices of the corresponding line graph.

Models of graph coloring

For many problems which cannot be easily reduced to classical coloring of the vertices or edges of an associated graph we introduce more general, nonclassical models of coloring. In general, the coloring of a graph can consist of assigning colors to vertices, edges, and faces of a plane graph or any combination of the above sets simultaneously. Moreover, for each model of coloring we have various rules on legality or optimality of solutions. Nonclassical models can introduce additional conditions of the usage of colors by making it possible to use more than one color per element, permitting the splitting of colors into fractions or admitting the swathing of colors. In general, there are several dozen graph coloring models described in the literature [183]. However, only a dozen or so are relevant (from a practical point of view). These include list coloring, chromatic sum coloring, equitable coloring, fractional coloring, rank coloring, harmonious coloring, radio-coloring, interval coloring, circular coloring, strong coloring and path coloring. Most of them deal with vertex-coloring and edge-coloring, but some of them are defined only for vertex-coloring (e.g. harmonious) or edge-coloring (e.g. interval). In this book we consider in detail the models of coloring, which are most useful from a practical point of view. Those include:

• classical coloring, • equitable coloring, • chromatic sum coloring, • T-coloring, • rank coloring, • harmonious coloring, • interval coloring, • circular coloring, • path coloring, • list coloring.

The overhelming majority of these models can be regarded as generalizations of classical graph coloring in the sense that they yield legal colorings.

Individual chapters, devoted to various models, are largely independent of each other and can be read separately. In each of them we highlight algorithmic aspects, i.e. the construction of polynomial-time algorithms for graph coloring in the dis-cussed model. These algorithms are exact if a particular subproblem admits such a solution or approximate in the general case. The above selection of models is purposeful. While preparing the selection, potential applications were taken into account. They include such diverse areas of technology as task scheduling, cellular telephony, connection networks, radiocommunication, VLSI design and production systems, etc.

Not only models but also modes are important in algorithmic graph coloring. Therefore one of the first introductory chapters of this book is devoted to on-line

xii PREFACE

coloring. On the other hand, the last two chapters are devoted to interesting varia-tions of coloring (Ramsey-type colorings) and applications in art-gallery problems.

Preface to the English Edition

This book is an extended and updated translation of a book originally written in Polish and published in 2002 as "Optymalizacja dyskretna. Modele i metody kolorowania graf6w". The holder of the copyright, Wydawnictwa Naukowo-Tech-niczne (Polish Scientific and Technical Publishers) graciously provided the rights to the English edition pro bono. The authors express their gratitude to the reviewers of the Polish edition, Professors M. Borowiecki and Z. Lone, for numerous corrections.

The Polish edition has been awarded the prize of the Ministry of National Ed-ucation. This English edition is the result of several years' effort of young scientists of three Polish academic centers: Gdansk University of Technology, the University of Gdansk and the University of Zielona G6ra. All these schools have programs devoted to algorithmic graph coloring. While Professor Marek Kubale of Gdansk University of Technology coordinated the entire project, the authors of the chap-ters were members of his Ph.D. seminar devoted to algorithmic aspects of graph theory. Some of them now conduct their own research and head their own research groups elsewhere. The work on this book was partially supported by a number of grants from the Polish State Committee for Scientific Research. In addition, Robert Janczewski and Michal Malafiejski were supported by FNP.

Several authors received scientific prizes for the work published in this volume. After the publication of the Polish original in October 2002, a group of com-

puter scientists and mathematicians from the University of Kentucky (Drs. Jerzy Jaromczyk, Zbigniew Lone (on leave from Warsaw University of Technology), Vic-tor Marek and Miroslaw Truszczynski) and the Rochester Institute of Technology (Dr. Stanislaw Radziszowski) approached the American Mathematical Society on behalf of the authors of the volume proposing the publication of the English version. From the beginning the project enjoyed the support of AMS staff: Dr. S. Gelfand and Ms. Christine Thivierge of the Acquisitions Department. Thanks to the gen-erosity of the copyright holder (Polish Scientifical and Technical Publishers) this translation can now be presented to the American reader. The text was translated by the authors and then revised by the American team.

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acyclic orientation, 58 admissible colors, 153 algorithm

LST, 24 connected sequential, 12 Depressive, 101 effective, 24 First-Fit (FF), 23 greedy, 23, 62 greedy independent sets, 14 k-absolute approximation, 4, 61 k-approximation, 62 largest-first (LF), 9, 76 on-line, 21 polynomial time approximation, 61 random sequential, 9 recolor, 18 relative approximation, 4 saturation LF (SLF), 13, 77 Scattered Coloring (SCC), 30 sequential coloring, 8 SLI, 12 smallest-last (SL), 10, 49, 76 T-DSATUR, 77 T-LF, 76 T-SL, 76

arc expansion, 145

backtracking method, 58 benchmark, 6 Bk, 61 bull, 19

cactus, 4, 76, 110 cartesian product, 108 channel assignment, 32 chromatic sum, 55, 58, 59 Chvatal's Art Gallery Theorem, 177 clique, 2 Cn, 61 cograph, 63, 82 colliding paths, 139 color interchange, 8 color requirement function, 63

Index

203

coloring best, 55 critical, 165 equitable, 35 greedy, 8 harmonious, 95 k-total, 50 line-distinguishing, 96 of a path, 139 partial harmonious, 100 r-circular, 124, 128 r-fractional, 125 total equitable, 51 vertex, 3

comet, 40, 48, 99 complete network, 163 contiguous multichromatic sum, 64 convex, 173

hull, 173 quadrilateral, 182 quadrilateralization, 182

core of graph, 2 Cr,s, 99 cycle, 3, 76, 98 cyclic production systems, 133 cylinder, 112

D-assignment, 157 ~-Equitable Coloring Conjecture, 38 dart, 64 deficiency, 105 degree of vertex, 1 density, 1 diagonal

internal, 180 digraph

symmetric, 141 dipath, see also path, directed distance

between vertices, 2 distributed resource allocation, 64 division rule, 164 Dynamic Storage Allocation problem (DSA),

31

204

edge chromatic sum, 55 k-assignment, 161 k-coloring, 163 £-coloring, 161 span, 68 sum coloring, 61 T-span, 71

edge-coloring, 16, 106 classical, 16 consecutive, 106 equitable, 38 function, 16 interval, see also edge-coloring, consecu-

tive r-circular, 130, 132

edge-contraction, 187 edge-list assignment, 161 edges

adjacent, 1 nonadjacent, 1

equitable chromatic threshold, 38 Equitable Coloring Conjecture, 37 equitable independence-partition, 36

fan, 18 frequency assignment problem, 77

generalized stars, 150 graph, 1

bipartite, 2, 37, 62, 63, 76 bipartite with ~ ::; 5, 60 biregular, 110 biregular of type (~1, ~2), 110 chordal, 159 circular-arc, 29, 30, 33, 85 Class 0, 106 Class 1, 17, 58 Class 2, 17, 58 Class 2a, 131 Class 2b, 131 claw-free, 64 complement, 41 complete, 36, 57, 76 complete bipartite, 57 complete multipartite, 37 complete split, 42 conflict, 32, 140 connected, 2, 37 cubic, 3, 36, 40 cubic planar, 59, 60 D-choosable, 157 d-degenerate, 33, 41, 160 dense, 62 disconnected, 42 double convex, 111 dual, 180 edge k-choosable, 161 equitably k-colorable, 36

INDEX

G%, 125 hard-to-color, 5, 28 Hertz's, 116 induced by a digraph, 142 interval, 26, 27, 29, 30, 38, 60, 63, 82-84 Johnson's, 25, 27 k-choosable, 153 k-chromatic, 3 k-colorable, 3 k-edge colorable, 17 k-partite, 2 (k,r)-choosable, 160 1-splittable, 41 line, 17, 36, 55, 58, 63 multipartite, 37 outerplanar, 40, 112 perfect, 63, 161 planar, 40, 57, 60, 64 r-regular, 3 reel, 48, 126 regular, 40, 60, 76 regular bipartite, 57, 60 scheduling, 133 slightly hard-to-color, 5 sparse, 62 split, 26, 41 subcubic, 40, 76 subcubic bipartite, 60, 61 threshold, 92 triangulation, 177, 180 tripartite, 2

grid, 112 guard, 177

mobile, 177 vertex, 178

hash function, 102 homomorphism, 73 hypercube, 39, 100 hyperedge, 172 hypergraph, 172

s-uniform, 172

index chromatic, 17, 36, 38 circular chromatic, 130, 132 equitable chromatic, 38 list chromatic, 161

intersection graph, 29 interval, 67, 105

join of graphs, 2, 35

k-assignment, 153 k-coloring, 3 k-core, 100 k-edge coloring, 17 k-tree, 61

partial, 64

(k, d)-coloring, 123 kite, 117 Kn, 61 Kr,s, 61

£-coloring, 153 list assignment, 153 load of a set, 140 load of a set of requests, 141 load of an edge, 140 (L, r)-coloring, 160

Malafiejski's rosette, 107 makespan minimization, 55 Manhattan model, 65

dogleg-free, 65 maximum independent set, 62 mean chromatic number, 22 mean performance function, 27 mean performance ratio, 27 minimum edge ranking spanning tree prob-

lem, 91 multichromatic sum, 63 multicoloring, 63

contiguous, 63 proper, 63

near-triangulation, 158 neighborhood

closed, 37 network graph, 32 number

chromatic, 3, 57 chromatic circular, 123 chromatic equitable, 36 chromatic fractional, 125 chromatic harmonious, 95 chromatic line-distinguishing, 96 chromatic of a path, 140 chromatic of a set of requests, 141 circular chromatic, 128 clique, 2 cost-chromatic, 63 cyclomatic, 76, 110 equitable total chromatic, 51 graph Ramsey, 165 harmonious, see also number, chromatic

harmonious hypergraph Ramsey, 172 independence, 35 list chromatic, 153 matching, 41 multicolor Ramsey, 165 T-chromatic, 71 total chromatic, 50 two color Ramsey, 163

off-line coloring problem, 22 on-line chromatic number, 22

INDEX

on-line coloring problem, 22 on-line path coloring, 32

205

optimum cost chromatic partiton problem, 63

parallel Cholesky factorization of matrices, 86

parallel computations, 65 path, 3, 40, 64, 97

connecting vertices, 2 directed, 139 induced by a dipath, 142 realizing a request, 141 undirected, 139

path networks, 65 performance guarantee, 4 performance guarantee function, 24 performance ratio, 24 permutation graph, 26 Pn, 61 polygon

convex, 174 orthogonal, 178 rectilinear, 177 simple, 177 with holes, 177

polygon tree, 4 prism, 10 prismatoid, 10 problem

routing, 141 product

cartesian, 43 strong tensor, 43 weak tensor, 43

ranking c-edge, 90 edge, 87 vertex, 79

request, 141, 142 resource allocation, 55 routing, 141

scheduling, 64 scheduling theory, 55 session scheduling, 65 set

(r, T)-initial, 75 independent, 2 k multiple of s, 69 of forbidden distances, 69 r-initial, 69

set of paths realizing requests, 141 set of requests

all-to-all, 142 broadcasting, 142 gossiping, 142 one-to-all, 142

206

span, 113 of fan, 18 of a coloring, 67 of a set, 67 of an interval coloring, 113

spider, 150 star, 3, 40 storage allocation problem in warehouse, 65 strength, 56, 57 sum coloring, 55, 61, 62, 64 sum multicoloring, 63 sum multicoloring problem, 55 susceptibility, 28 symmetric cycles, 147 symmetric line, 146 symmetric trees, 148

T-coloring, 70 e-optimal, 72 optimal, 72

T-graph, 73 T-saturation degree, 77 T-span, 71 The Pigeonhole Principle, 164 theorem

Brook's, 57 three dimensional matching problem, 59 torus, 112 · Total Coloring Conjecture, 51 tree, 58, 61, 63

complete binary, 40 elimination, 81 oriented, 151 triangle-free polygon, 41

treewidth, 60 triangulation, 177, 180

even, 187

union of graphs, 43

vertex chromatic sum, see also chromatic sum k-coloring

k-coloring, see also k-coloring multicoloring, 63 universal, 51

vertex £-coloring, see also £-coloring vertex-coloring

equitable, 36 vertices

adjacent, 1 nonadjacent, 1

Vizing, 117 VLSI design, 55, 63-65

Wavelength Division Multiplexing (WDM), 32

wheel, 3 broken spoke, 48, 126

INDEX

multiaxle, 131 Wn, 61

Authors' addresses

(1) Jakub Bialogrodzki (e-mail: hilikus@wp.pl): Department of Algorithms and System Modeling, Faculty of Electron-

ics, Telecommunication and Informatics, Gdansk University of Technol-ogy, 80-952 Gdansk, ul. Narutowicza 11/12, Poland.

(2) Piotr Borowiecki (e-mail: p.borowiecki@im.uz.zgora.pl): Department of Discrete Mathematics and Computer Science, Faculty

of Sciences, University of Zielona G6ra, 65-516 Zielona G6ra, ul. prof Z. Szafrana 4a, Poland.

(3) Dariusz Dereniowski (e-mail: deren@eti.pg.gda.pl): Department of Algorithms and System Modeling, Faculty of Electron-

ics, Telecommunication and Informatics, Gdansk University of Technol-ogy, 80-952 Gdansk, ul. Narutowicza 11/12, Poland.

(4) Tomasz Dzido (e-mail: tdz@math.univ.gda.pl): Institute of Mathematics, Gdansk University, 80-952 Gdansk, ul. Wita

Stwosza 57, Poland. (5) Hanna Furmanczyk (e-mail: hanna@math.univ.gda.pl):

Institute of Mathematics, Gdansk University, 80-952 Gdansk, ul. Wita Stwosza 57, Poland.

(6) Krzysztof Giaro (e-mail: giaro@eti.pg.gda.pl): Department of Algorithms and System Modeling, Faculty of Electron-

ics, Telecommunication and Informatics, Gdansk University of Technol-ogy, 80-952 Gdansk, ul. Narutowicza 11/12, Poland.

(7) Robert Janczewski (e-mail: skalar@eti.pg.gda.pl): Department of Algorithms and System Modeling, Faculty of Electron-

ics, Telecommunication and Informatics, Gdansk University of Technol-ogy, 80-952 Gdansk, ul. Narutowicza 11/12, Poland.

(8) Adrian Kosowski (e-mail: akosowski@o2.pl): Department of Algorithms and System Modeling, Faculty of Electron-

ics, Telecommunication and Informatics, Gdansk University of Technol-ogy, 80-952 Gdansk, ul. Narutowicza 11/12, Poland.

(9) Marek Kubale (e-mail: kubale@pg.gda.pl): Department of Algorithms and System Modeling, Faculty of Electron-

ics, Telecommunication and Informatics, Gdansk University of Technol-ogy, 80-952 Gdansk, ul. Narutowicza 11/12, Poland.

(10) Michal Malafiejski (e-mail: mima@eti.pg.gda.pl): Department of Algorithms and System Modeling, Faculty of Electron-

ics, Telecommunication and Informatics, Gdansk University of Technol-ogy, 80-952 Gdansk, ul. Narutowicza 11/12, Poland.

( 11) Krzysztof Manuszewski (e-mail: manus@eti. pg.gda. pl):

207

208 AUTHORS' ADDRESSES

Department of Algorithms and System Modeling, Faculty of Electron-ics, Telecommunication and Informatics, Gdansk University of Technol-ogy, 80-952 Gdansk, ul. Narutowicza 11/12, Poland.

(12) Adam Nadolski (e-mail: nadolski@eti.pg.gda.pl): Department of Algorithms and System Modeling, Faculty of Electron-

ics, Telecommunication and Informatics, Gdansk University of Technol-ogy, 80-952 Gdansk, ul. Narutowicza 11/12, Poland.

(13) Konrad Piwakowski (e-mail: coni@pg.gda.pl): Department of Algorithms and System Modeling, Faculty of Electron-

ics, Telecommunication and Informatics, Gdansk University of Technol-ogy, 80-952 Gdansk, ul. Narutowicza 11/12, Poland.

(14) Pawel Zylinski (e-mail: impz@univ.gda.pl): Institute of Mathematics, Gdansk University, 80-952 Gdansk, ul. Wit a

Stwosza 57, Poland.

Titles in This Series

352 Marek Kubale, Editor, Graph colorings, 2004 351 George Yin and Qing Zhang, Editors, Mathematics of finance, 2004 350 Abbas Bahri and Sergiu Klainerman, Editors, Noncompact problems at the

intersection of geometry, analysis, and topology, 2004 349 Alexandre V. Borovik and Alexei G. Myasnikov, Editors, Computational and

experimental group theory, 2004 348 Hiroshi Isozaki, Editor, Inverse problems and spectral theory, 2004 347 Motoko Kotani, Tomoyuki Shirai, and Toshikazu Sunada, Editors, Discrete

geometric analysis, 2004 346 Paul Goerss and Stewart Priddy, Editors, Homotopy theory: Relations with algebraic

geometry, group cohomology, and algebraic K-theory, 2004 345 Christopher Heil, Palle E. T. Jorgensen, and David R. Larson, Editors, Wavelets,

frames and operator theory, 2004 344 Ricardo Baeza, John S. Hsia, Bill Jacob, and Alexander Prestel, Editors,

Algebraic and arithmetic theory of quadratic forms, 2004 343 N. Sthanumoorthy and Kailash C. Misra, Editors, Kac-Moody Lie algebras and

related topics, 2004 342 Janos Pach, Editor, Towards a theory of geometric graphs, 2004 341 Hugo Arizmendi, Carlos Bosch, and Lourdes Palacios, Editors, Topological

algebras and their applications, 2004 340 Rafael del Rio and Carlos Villegas-Blas, Editors, Spectral theory of Schriidinger

operators, 2004 339 Peter Kuchment, Editor, Waves in periodic and random media, 2003 338 Pascal Auscher, Thierry Coulhon, and Alexander Grigor'yan, Editors, Heat

kernels and analysis on manifolds, graphs, and metric spaces, 2003 337 Krishan L. Duggal and Ramesh Sharma, Editors, Recent advances in Riemannian

and Lorentzian geometries, 2003 336 Jose Gonzalez-Barrios, Jorge A. Leon, and Ana Meda, Editors, Stochastic models,

2003 335 Geoffrey L. Price, B. Mitchell Baker, Palle E.T. Jorgensen, and PaulS. Muhly,

Editors, Advances in quantum dynamics, 2003 334 Ron Goldman and Rimvydas Krasauskas, Editors, Topics in algebraic geometry and

geometric modeling, 2003 333 Giovanni Alessandrini and Gunther Uhlmann, Editors, Inverse problems: Theory

and applications, 2003 332 John Bland, Kang-Tae Kim, and Steven G. Krantz, Editors, Explorations in

complex and Riemannian geometry, 2003 331 Luchezar L. Avramov, Marc Chardin, Marcel Morales, and Claudia Polini,

Editors, Commutative algebra: Interactions with algebraic geometry, 2003 330 S. Y. Cheng, C.-W. Shu, and T. Tang, Editors, Recent advances in scientific

computing and partial differential equations, 2003 329 Zhangxin Chen, Roland Glowinski, and Kaitai Li, Editors, Current trends in

scientific computing, 2003 328 Krzysztof Jarosz, Editor, Function spaces, 2003 327 Yulia Karpeshina, Giinter Stolz, Rudi Weikard, and Yanni Zeng, Editors,

Advances in differential equations and mathematical physics, 2003 326 Kenneth D. T-R McLaughlin and Xin Zhou, Editors, Recent developments in

integrable systems and Riemann-Hilbert problems, 2003 325 Seok-Jin Kang and Kyu-Hwan Lee, Editors, Combinatorial and geometric

representation theory, 2003

TITLES IN THIS SERIES

324 Caroline Grant Melles, Jean-Paul Brasselet, Gary Kennedy, Kristin Lauter, and Lee McEwan, Editors, Topics in algebraic and noncommutative geometry, 2003

323 Vadim Olshevsky, Editor, Fast algorithms for structured matrices: theory and applications, 2003

322 S. Dale Cutkosky, Dan Edidin, Zhenbo Qin, and Qi Zhang, Editors, Vector bundles and representation theory, 2003

321 Anna Kaminska, Editor, Trends in Banach spaces and operator theory, 2003 320 William Beckner, Alexander Nagel, Andreas Seeger, and Hart F. Smith,

Editors, Harmonic analysis at Mount Holyoke, 2003 319 W. H. Schikhof, C. Perez-Garcia, and A. Escassut, Editors, Ultrametric functional

analysis, 2003 318 David E. Radford, Fernando J. 0. Souza, and David N. Yetter, Editors,

Diagrammatic morphisms and applications, 2003 317 Hui-Hsiung Kuo and Ambar N. Sengupta, Editors, Finite and infinite dimensional

analysis in honor of Leonard Gross, 2003 316 0. Cornea, G. Lupton, J. Oprea, and D. Tanre, Editors, Lusternik-Schnirelmann

category and related topics, 2002 315 Theodore Voronov, Editor, Quantization, Poisson brackets and beyond, 2002 314 A. J. Berrick, Man Chun Leung, and Xingwang Xu, Editors, Topology and

Geometry: Commemorating SISTAG, 2002 313 M. Zuhair Nashed and Otmar Scherzer, Editors, Inverse problems, image analysis,

and medical imaging, 2002 312 Aaron Bertram, James A. Carlson, and Holger Kley, Editors, Symposium in

honor of C. H. Clemens, 2002 311 Clifford J. Earle, William J. Harvey, and Sevin Recillas-Pishmish, Editors,

Complex manifolds and hyperbolic geometry, 2002 310 Alejandro Adem, Jack Morava, and Yongbin Ruan, Editors, Orbifolds in

mathematics and physics, 2002 309 Martin Guest, Reiko Miyaoka, and Yoshihiro Ohnita, Editors, Integrable systems,

topology, and physics, 2002 308 Martin Guest, Reiko Miyaoka, and Yoshihiro Ohnita, Editors, Differentiable

geometry and integrable systems, 2002 307 Ricardo Weder, Pavel Exner, and Benoit Grebert, Editors, Mathematical results in

quantum mechanics, 2002 306 Xiaobing Feng and Tim P. Schulze, Editors, Recent advances in numerical methods

for partial differential equations and applications, 2002 305 Samuel J. Lomonaco, Jr. and Howard E. Brandt, Editors, Quantum computation

and information, 2002 304 Jorge Alberto Calvo, Kenneth C. Millett, and Eric J. Rawdon, Editors, Physical

knots: Knotting, linking, and folding geometric objects in ~ 3 , 2002 303 William Cherry and Chung-Chun Yang, Editors, Value distribution theory and

complex dynamics, 2002 302 Yi Zhang, Editor, Logic and algebra, 2002 301 Jerry Bona, Roy Choudhury, and David Kaup, Editors, The legacy of the inverse

scattering transform in applied mathematics, 2002

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstoref.

This book is devoted to problems in graph coloring, which can be viewed as one area of discrete optimization. Chapters are dedicated to various models and are largely indepen-dent of one another. In each chapter, the author highlights algorithmic aspects of the presented models, i.e., the construction of polynomial-time algorithms for graph coloring.

This is an expanded and updated translation of the prizewinning book originally published in Polish, Optymalizacja dyskretna. Modele i metody kolorowania graf6w. It is suitable for graduate students and researchers interested in graph theory.

ISBN 0- 8218-3458- 4

9 7 80821 8 3 4589