# Graph Edge Coloring

## Vizing's Theorem and Goldberg's Conjecture

1. Edition March 2012

344 Pages, Hardcover*Professional Book*

**978-1-118-09137-1**

### Short Description

Written by world authorities on graph theory, this book features many new advances and applications in graph edge coloring, describes how the results are interconnected, and provides historial context throughout. Chapter coverage includes an introduction to coloring preliminaries and lower and upper bounds; the Vizing fan; the Kierstead path; simple graphs and line graphs of multigraphs; the Tashkinov tree; Goldberg's conjecture; extreme graphs; generalized edge coloring; and open problems. It serves as a reference for researchers interested in discrete mathematics, graph theory, operations research, theoretical computer science, and combinatorial optimization, as well as a graduate-level course book for students of mathematics, optimization, and computer science.

Written by world authorities on graph theory, this book features many new advances and applications in graph edge coloring, describes how the results are interconnected, and provides historial context throughout. Chapter coverage includes an introduction to coloring preliminaries and lower and upper bounds; the Vizing fan; the Kierstead path; simple graphs and line graphs of multigraphs; the Tashkinov tree; Goldberg's conjecture; extreme graphs; generalized edge coloring; and open problems. It serves as a reference for researchers interested in discrete mathematics, graph theory, operations research, theoretical computer science, and combinatorial optimization, as well as a graduate-level course book for students of mathematics, optimization, and computer science.

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

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.