Generalized
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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)
Table of Contents
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
