9780521802307

Topics in Topological Graph Theory

by
  • ISBN13:

    9780521802307

  • ISBN10:

    052180230X

  • Edition: 1st
  • Format: Hardcover
  • Copyright: 2009-08-10
  • Publisher: Cambridge University Press
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Summary

The use of topological ideas to explore various aspects of graph theory, and vice versa, is a fruitful area of research. There are links with other areas of mathematics, such as design theory and geometry, and increasingly with such areas as computer networks where symmetry is an important feature. Other books cover portions of the material here, but there are no other books with such a wide scope. This book contains fifteen expository chapters written by acknowledged international experts in the field. Their well-written contributions have been carefully edited to enhance readability and to standardize the chapter structure, terminology and notation throughout the book. To help the reader, there is an extensive introductory chapter that covers the basic background material in graph theory and the topology of surfaces. Each chapter concludes with an extensive list of references.

Table of Contents

Forewordp. xv
Prefacep. xvii
Introductionp. 1
Graph theoryp. 1
Graphs in the planep. 10
Surfacesp. 12
Graphs on surfacesp. 14
Embedding graphs on surfacesp. 18
Introductionp. 18
Graphs and surfacesp. 19
Embeddingsp. 20
Rotation systemsp. 23
Covering spaces and voltage graphsp. 26
Enumerationp. 29
Algorithmsp. 30
Graph minorsp. 31
Maximum genusp. 34
Introductionp. 34
Characterizations and complexityp. 36
Kuratowski-type theoremsp. 38
Upper-embeddabilityp. 39
Lower boundsp. 40
Distribution of embeddingsp. 45
Introductionp. 45
Enumerating embeddings by surface typep. 48
Total embedding distributionsp. 51
Congruence classesp. 53
The unimodality problemp. 55
Average genusp. 56
Stratification of embeddingsp. 59
Algorithms and obstructions for embeddingsp. 62
Introductionp. 62
Planarityp. 64
Outerplanarity and face coversp. 66
Disc embeddings and the 2-path problemp. 68
Graph minors and obstructionsp. 69
Algorithms for embeddability in general surfacesp. 73
Computing the genusp. 75
Graph minors: generalizing Kuratowski's theoremp. 81
Introductionp. 81
Graph decompositionsp. 84
Linked decompositionsp. 88
Graphs with bounded tree-widthp. 94
Finding large gridsp. 99
Embedding large gridsp. 107
Colouring graphs on surfacesp. 111
Introductionp. 111
High-end colouringp. 113
A transition from high-end to low-end colouringp. 116
Colouring graphs with few coloursp. 119
Girth and chromatic numberp. 124
List-colouring graphsp. 125
More colouring extensionsp. 127
An open problemp. 129
Crossing numbersp. 133
Introductionp. 133
What is the crossing number?p. 135
General boundsp. 137
Applications to geometryp. 139
Crossing-critical graphsp. 139
Other families of graphsp. 143
Algorithmic questionsp. 144
Drawings in other surfacesp. 146
Conclusionp. 147
Representing graphs and mapsp. 151
Introductionp. 151
Representations of graphsp. 152
Energy and optimal representationsp. 155
Representations of mapsp. 163
Representations of maps in the planep. 170
Representations of incidence geometries and related topicsp. 174
Enumerating coveringsp. 181
Introductionp. 181
Graph coveringsp. 183
Regular coveringsp. 185
Surface branched coveringsp. 190
Regular surface branched coveringsp. 193
Distribution of surface branched coveringsp. 195
Further remarksp. 196
Symmetric mapsp. 199
Introductionp. 199
Representing maps algebraicallyp. 200
Regular mapsp. 205
Cayley mapsp. 210
Regular Cayley mapsp. 212
Edge-transitive mapsp. 218
Maps and mathematicsp. 221
The genus of a groupp. 225
Introductionp. 225
Symmetric embeddings and groups acting on surfacesp. 226
Quotient embeddings and voltage graphsp. 228
Inequalitiesp. 232
Groups of low genusp. 235
Genera of families of groupsp. 239
Embeddings and geometriesp. 245
Introductionp. 245
Surface modelsp. 248
Projective geometriesp. 250
Affine geometriesp. 253
3-configurationsp. 256
Partial geometriesp. 260
Regular embeddings for PG(2,n)p. 264
Problemsp. 265
Embeddings and designsp. 268
Introductionp. 268
Steiner triple systems and triangulationsp. 270
Recursive constructionsp. 273
Small systemsp. 278
Cyclic embeddingsp. 280
Concluding remarksp. 284
Infinite graphs and planar mapsp. 289
Introductionp. 289
Endsp. 290
Automorphismsp. 293
Connectivitiesp. 295
Growthp. 300
Infinite planar graphs and mapsp. 303
Open problemsp. 313
Introductionp. 313
Drawings and crossingsp. 314
Genus and obstructionsp. 317
Cycles and factorsp. 320
Colourings and flowsp. 322
Local planarityp. 324
Thickness, book embeddings and-covering graphsp. 325
Geometrical topicsp. 328
Algorithmsp. 330
Infinite graphsp. 332
Notes on contributorsp. 337
Indexp. 341
Table of Contents provided by Ingram. All Rights Reserved.

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