简介
Bollob谩s (mathematics, University of Memphis) offers a concise treatment of extremal graph theory. Unlike most graph theory treatises, this book features complete proofs for almost all of its results. Further insights into theory are provided by numerous exercises of varying degrees of difficulty. Although geared toward mathematicians and research students, much of the book is accessible to undergraduate students. This is an unabridged republication of the edition published by Academic Press, London, 1978. Annotation 漏2004 Book News, Inc., Portland, OR (booknews.com)
目录
Preface p. vii
Basic Definitions p. xiii
Connectivity p. 1
Elementary Properties p. 2
Menger's Theorem and its Consequences p. 7
The Structure of 2- and 3-Connected Graphs p. 11
Minimally k-Connected Graphs p. 17
Graphs with Given Maximal Local Connectivity p. 29
Exercises, Problems and Conjectures p. 45
Matching p. 50
Fundamental Matching Theorems p. 52
The Number of 1-Factors p. 59
f-Factors p. 67
Matching in Graphs with Restrictions on the Degrees p. 79
Coverings p. 90
Exercises, Problems and Conjectures p. 95
Cycles p. 102
Graphs with Large Minimal Degree and Large Girth p. 103
Vertex Disjoint Cycles p. 110
Edge Disjoint Cycles p. 119
The Circumference p. 131
Graphs with Cycles of Given Lengths p. 147
Exercises, Problems and Conjectures p. 161
The Diameter p. 169
Diameter, Maximal Degree and Size p. 170
Diameter and Connectivity p. 181
Graphs with Large Subgraphs of Small Diameter p. 194
Factors of Small Diameter p. 206
Exercises, Problems and Conjectures p. 213
Colourings p. 218
General Colouring Theorems p. 221
Critical k-Chromatic Graphs p. 234
Colouring Graphs on Surfaces p. 243
Sparse Graphs of Large Chromatic Number p. 254
Perfect Graphs p. 263
Ramsey Type Theorems p. 270
Exercises, Problems and Conjectures p. 280
Complete Subgraphs p. 292
The Number of Complete Subgraphs p. 293
Complete Subgraphs of r-Partite Graphs p. 309
The Structure of Graphs p. 327
The Structure of Extremal Graphs without Forbidden Subgraphs p. 339
Independent Complete Subgraphs p. 351
Exercises, Problems and Conjectures p. 359
Topological Subgraphs p. 368
Contractions p. 369
Topological Complete Subgraphs p. 378
Semi-Topological Subgraphs p. 386
Exercises, Problems and Conjectures p. 397
Complexity and Packing p. 401
The Complexity of Graph Properties p. 402
Monotone Properties p. 411
The Main Packing Theorem p. 418
Packing Graphs of Small Size p. 425
Applications of Packing Results to Complexity p. 429
Exercises, Problems and Conjectures p. 434
References p. 438
Index of Symbols p. 481
Index of Definitions p. 485
Basic Definitions p. xiii
Connectivity p. 1
Elementary Properties p. 2
Menger's Theorem and its Consequences p. 7
The Structure of 2- and 3-Connected Graphs p. 11
Minimally k-Connected Graphs p. 17
Graphs with Given Maximal Local Connectivity p. 29
Exercises, Problems and Conjectures p. 45
Matching p. 50
Fundamental Matching Theorems p. 52
The Number of 1-Factors p. 59
f-Factors p. 67
Matching in Graphs with Restrictions on the Degrees p. 79
Coverings p. 90
Exercises, Problems and Conjectures p. 95
Cycles p. 102
Graphs with Large Minimal Degree and Large Girth p. 103
Vertex Disjoint Cycles p. 110
Edge Disjoint Cycles p. 119
The Circumference p. 131
Graphs with Cycles of Given Lengths p. 147
Exercises, Problems and Conjectures p. 161
The Diameter p. 169
Diameter, Maximal Degree and Size p. 170
Diameter and Connectivity p. 181
Graphs with Large Subgraphs of Small Diameter p. 194
Factors of Small Diameter p. 206
Exercises, Problems and Conjectures p. 213
Colourings p. 218
General Colouring Theorems p. 221
Critical k-Chromatic Graphs p. 234
Colouring Graphs on Surfaces p. 243
Sparse Graphs of Large Chromatic Number p. 254
Perfect Graphs p. 263
Ramsey Type Theorems p. 270
Exercises, Problems and Conjectures p. 280
Complete Subgraphs p. 292
The Number of Complete Subgraphs p. 293
Complete Subgraphs of r-Partite Graphs p. 309
The Structure of Graphs p. 327
The Structure of Extremal Graphs without Forbidden Subgraphs p. 339
Independent Complete Subgraphs p. 351
Exercises, Problems and Conjectures p. 359
Topological Subgraphs p. 368
Contractions p. 369
Topological Complete Subgraphs p. 378
Semi-Topological Subgraphs p. 386
Exercises, Problems and Conjectures p. 397
Complexity and Packing p. 401
The Complexity of Graph Properties p. 402
Monotone Properties p. 411
The Main Packing Theorem p. 418
Packing Graphs of Small Size p. 425
Applications of Packing Results to Complexity p. 429
Exercises, Problems and Conjectures p. 434
References p. 438
Index of Symbols p. 481
Index of Definitions p. 485
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