Extremal Graph Theory

Extremal Graph Theory

by Bela Bollobas

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Product Details

ISBN-13: 9780486435961
Publisher: Dover Publications
Publication date: 06/04/2004
Series: Dover Books on Mathematics
Pages: 512
Product dimensions: 5.50(w) x 8.50(h) x (d)

Table of Contents

Prefacevii
Basic Definitionsxiii
Chapter 1Connectivity1
1.Elementary Properties2
2.Menger's Theorem and its Consequences7
3.The Structure of 2- and 3-Connected Graphs11
4.Minimally k-Connected Graphs17
5.Graphs with Given Maximal Local Connectivity29
6.Exercises, Problems and Conjectures45
Chapter 2Matching50
1.Fundamental Matching Theorems52
2.The Number of 1-Factors59
3.f-Factors67
4.Matching in Graphs with Restrictions on the Degrees79
5.Coverings90
6.Exercises, Problems and Conjectures95
Chapter 3Cycles102
1.Graphs with Large Minimal Degree and Large Girth103
2.Vertex Disjoint Cycles110
3.Edge Disjoint Cycles119
4.The Circumference131
5.Graphs with Cycles of Given Lengths147
6.Exercises, Problems and Conjectures161
Chapter 4The Diameter169
1.Diameter, Maximal Degree and Size170
2.Diameter and Connectivity181
3.Graphs with Large Subgraphs of Small Diameter194
4.Factors of Small Diameter206
5.Exercises, Problems and Conjectures213
Chapter 5Colourings218
1.General Colouring Theorems221
2.Critical k-Chromatic Graphs234
3.Colouring Graphs on Surfaces243
4.Sparse Graphs of Large Chromatic Number254
5.Perfect Graphs263
6.Ramsey Type Theorems270
7.Exercises, Problems and Conjectures280
Chapter 6Complete Subgraphs292
1.The Number of Complete Subgraphs293
2.Complete Subgraphs of r-Partite Graphs309
3.The Structure of Graphs327
4.The Structure of Extremal Graphs without Forbidden Subgraphs339
5.Independent Complete Subgraphs351
6.Exercises, Problems and Conjectures359
Chapter 7Topological Subgraphs368
1.Contractions369
2.Topological Complete Subgraphs378
3.Semi-Topological Subgraphs386
4.Exercises, Problems and Conjectures397
Chapter 8Complexity and Packing401
1.The Complexity of Graph Properties402
2.Monotone Properties411
3.The Main Packing Theorem418
4.Packing Graphs of Small Size425
5.Applications of Packing Results to Complexity429
6.Exercises, Problems and Conjectures434
References438
Index of Symbols481
Index of Definitions485

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