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9781118091371

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

by ; ; ;
  • ISBN13:

    9781118091371

  • ISBN10:

    111809137X

  • Edition: 1st
  • Format: Hardcover
  • Copyright: 2012-02-14
  • Publisher: Wiley
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Summary

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.

Author Biography

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.

Table of Contents

Prefacep. xi
Introductionp. 1
Graphsp. 1
Coloring Preliminariesp. 2
Critical Graphsp. 5
Lower Bounds and Elementary Graphsp. 6
Upper Bounds and Coloring Algorithmsp. 11
Notesp. 15
Vizing Fansp. 19
The Fan Equation and the Classical Boundsp. 19
Adjacency Lemmasp. 24
The Second Fan Equationp. 26
The Double Fanp. 31
The Fan Numberp. 32
Notesp. 39
Kierstead Pathsp. 43
Kierstead's Methodp. 43
Short Kierstead's Pathsp. 46
Notesp. 49
Simple Graphs and Line Graphsp. 51
Class One and Class Two Graphsp. 51
Graphs whose Core has Maximum Degree Twop. 54
Simple Overfull Graphsp. 63
Adjacency Lemmas for Critical Class Two Graphsp. 73
Average Degree of Critical Class Two Graphsp. 84
Independent Vertices in Critical Class Two Graphsp. 89
Constructions of Critical Class Two Graphsp. 93
Hadwiger's Conjecture for Line Graphsp. 101
Simple Graphs on Surfacesp. 105
Notesp. 110
Tashkinov Treesp. 115
Tashkinov's Methodp. 115
Extended Tashkinov Treesp. 127
Asymptotic Boundsp. 139
Tashkinov's Coloring Algorithmp. 144
Polynomial Time Algorithmsp. 148
Notesp. 152
Goldberg's Conjecturep. 155
Density and Fractional Chromatic Indexp. 155
Balanced Tashkinov Treesp. 160
Obstructionsp. 162
Approximation Algorithmsp. 183
Goldberg's Conjecture for Small Graphsp. 185
Another Classification Problem for Graphsp. 186
Notesp. 193
Extreme Graphsp. 197
Shannon's Bound and Ring Graphsp. 197
Vizing's Bound and Extreme Graphsp. 201
Extreme Graphs and Elementary Graphsp. 203
Upper Bounds for ¿' Depending on ¿ and ¿p. 205
Notesp. 209
Generalized Edge Colorings of Graphsp. 213
Equitable and Balanced Edge Coloringsp. 213
Full Edge Colorings and the Cover Indexp. 222
Edge Colorings of Weighted Graphsp. 224
The Fan Equation for the Chromatic Index ¿fp. 228
Decomposing Graphs into Simple Graphsp. 239
Notesp. 243
Twenty Pretty Edge Coloring Conjecturesp. 245
Vizing's Two Fundamental Papersp. 269
On an Estimate of the Chromatic Class of a p-Graphp. 269
Referencesp. 272
Critical Graphs with a Given Chromatic Classp. 273
Referencesp. 278
Fractional Edge Coloringsp. 281
The Fractional Chromatic Indexp. 281
The Matching Polytopep. 284
A Formula for ¿′*fp. 290
Referencesp. 295
Symbol Indexp. 312
Name Indexp. 314
Subject Indexp. 318
Table of Contents provided by Ingram. All Rights Reserved.

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