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Graph Edge Coloring: Vizing's Theorem and Goldberg's Conjecture

ISBN: 978-1-118-09137-1
344 pages
February 2012
Graph Edge Coloring: Vizing
Features recent advances and new applications in graph edge coloring

Reviewing recent advances in the Edge Coloring Problem, Graph Edge Coloring: Vizing's Theorem and Goldberg's Conjecture provides an overview of the current state of the science, explaining the interconnections among the results obtained from important graph theory studies. The authors introduce many new improved proofs of known results to identify and point to possible solutions for open problems in edge coloring.

The book begins with an introduction to graph theory and the concept of edge coloring. Subsequent chapters explore important topics such as:

  • Use of Tashkinov trees to obtain an asymptotic positive solution to Goldberg's conjecture

  • Application of Vizing fans to obtain both known and new results

  • Kierstead paths as an alternative to Vizing fans

  • Classification problem of simple graphs

  • Generalized edge coloring in which a color may appear more than once at a vertex

This book also features first-time English translations of two groundbreaking papers written by Vadim Vizing on an estimate of the chromatic class of a p-graph and the critical graphs within a given chromatic class.

Written by leading experts who have reinvigorated research in the field, Graph Edge Coloring is an excellent book for mathematics, optimization, and computer science courses at the graduate level. The book also serves as a valuable reference for researchers interested in discrete mathematics, graph theory, operations research, theoretical computer science, and combinatorial optimization.

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Preface xi

1 Introduction 1

1.1 Graphs 1

1.2 Coloring Preliminaries 2

1.3 Critical Graphs 5

1.4 Lower Bounds and Elementary Graphs 6

1.5 Upper Bounds and Coloring Algorithms 11

1.6 Notes 15

2 Vizing Fans 19

2.1 The Fan Equation and the Classical Bounds 19

2.2 Adjacency Lemmas 24

2.3 The Second Fan Equation 26

2.4 The Double Fan 31

2.5 The Fan Number 32

2.6 Notes 39

3 Kierstead Paths 43

3.1 Kierstead's Method 43

3.2 Short Kierstead's Paths 46

3.3 Notes 49

4 Simple Graphs and Line Graphs 51

4.1 Class One and Class Two Graphs 51

4.2 Graphs whose Core has Maximum Degree Two 54

4.3 Simple Overfull Graphs 63

4.4 Adjacency Lemmas for Critical Class Two Graphs 73

4.5 Average Degree of Critical Class Two Graphs 84

4.6 Independent Vertices in Critical Class Two Graphs 89

4.7 Constructions of Critical Class Two Graphs 93

4.8 Hadwiger's Conjecture for Line Graphs 101

4.9 Simple Graphs on Surfaces 105

4.10 Notes 110

5 Tashkinov Trees 115

5.1 Tashkinov's Method 115

5.2 Extended Tashkinov Trees 127

5.3 Asymptotic Bounds 139

5.4 Tashkinov's Coloring Algorithm 144

5.5 Polynomial Time Algorithms 148

5.6 Notes 152

6 Goldberg's Conjecture 155

6.1 Density and Fractional Chromatic Index 155

6.2 Balanced Tashkinov Trees 160

6.3 Obstructions 162

6.4 Approximation Algorithms 183

6.5 Goldberg's Conjecture for Small Graphs 185

6.6 Another Classification Problem for Graphs 186

6.7 Notes 193

7 Extreme Graphs 197

7.1 Shannon's Bound and Ring Graphs 197

7.2 Vizing's Bound and Extreme Graphs 201

7.3 Extreme Graphs and Elementary Graphs 203

7.4 Upper Bounds for ÷' Depending on Ä and ì 205

7.5 Notes 209

8 Generalized Edge Colorings of Graphs 213

8.1 Equitable and Balanced Edge Colorings 213

8.2 Full Edge Colorings and the Cover Index 222

8.3 Edge Colorings of Weighted Graphs 224

8.4 The Fan Equation for the Chromatic Index X'f  228

8.5 Decomposing Graphs into Simple Graphs 239

8.6 Notes 243

9 Twenty Pretty Edge Coloring Conjectures 245

Appendix A: Vizing's Two Fundamental Papers 269

A. 1 On an Estimate of the Chromatic Class of a p-Graph 269

References 272

A.2 Critical Graphs with a Given Chromatic Class 273

References 278

Appendix B: Fractional Edge Colorings 281

B. 1 The Fractional Chromatic Index 281

B.2 The Matching Polytope 284

B.3 A Formula for X'f  290

References 295

Symbol Index 312

Name Index 314

Subject Index 318

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Michael Stiebitz, PhD, is Professor of Mathematics at the Technical University of Ilmenau, Germany. He is the author of numerous journal articles in his areas of research interest, which include graph theory, combinatorics, cryptology, and linear algebra.

Diego Scheide, PhD, is a Postdoctoral Researcher in the Department of Mathematics at Simon Fraser University, Canada.

Bjarne Toft, PhD, is Associate Professor in the Department of Mathematics and Computer Science at the University of Southern Denmark.

Lene M. Favrholdt, PhD, is Associate Professor in the Department of Mathematics and Computer Science at the University of Southern Denmark.

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“College mathematics collections need just this sort of rarity-accounts of major unsolved problems, elementary but still comprehensive.  Summing Up: Recommended.  Upper-division undergraduates.”  (Choice, 1 September 2012)

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