This standard textbook of modern graph theory, now in its fifth edition, combines the authority of a classic with the engaging freshness of style that is the hallmark of active mathematics. It covers the core material of the subject with concise yet reliably complete proofs, while offering glimpses of more advanced methods in each field by one or two deeper results, again with proofs given in full detail.
The book can be used as a reliable text for an introductory course, as a graduate text, and for self-study.
Author(s): Reinhard Diestel
Series: Graduate Texts in Mathematics 173
Edition: Professional Edition
Publisher: Springer
Year: 2017
Language: English
Pages: 474
Title Page
Preface
First edition
Second edition
Third edition
Fourth edition
Fifth edition
Contents
1. The Basics
1.1 Graphs
1.2 The degree of a vertex
1.3 Paths and cycles
1.4 Connectivity
1.5 Trees and forests
1.6 Bipartite graphs
1.7 Contraction and minors
1.8 Euler tours
1.9 Some linear algebra
1.10 Other notions of graphs
Exercises
Notes
2. Matching, Covering and Packing
2.1 Matching in bipartite graphs
2.2 Matching in general graphs
2.3 The Erdös-Pósa theorem
2.4 Tree packing and arboricity
2.5 Path covers
Exercises
Notes
3. Connectivity
3.1 2-Connected graphs and subgraphs
3.2 The structure of 3-connected graphs
3.3 Menger’s theorem
3.4 Mader’s theorem
3.5 Linking pairs of vertices
Exercises
Notes
4. Planar Graphs
4.1 Topological prerequisites
4.2 Plane graphs
4.3 Drawings
4.4 Planar graphs: Kuratowski’s theorem
4.5 Algebraic planarity criteria
4.6 Plane duality
Exercises
Notes
5. Colouring
5.1 Colouring maps and planar graphs
5.2 Colouring vertices
5.3 Colouring edges
5.4 List colouring
5.5 Perfect graphs
Exercises
Notes
6. Flows
6.1 Circulations
6.2 Flows in networks
6.3 Group-valued flows
6.4 k-Flows for small k
6.5 Flow-colouring duality
6.6 Tutte’s flow conjectures
Exercises
Notes
7. ExtremalGraph Theory
7.1 Subgraphs
7.2 Minors
7.3 Hadwiger’s conjecture
7.4 Szemerédi’s regularity lemma
7.5 Applying the regularity lemma
Exercises
Notes
8. Infinite Graphs
8.1 Basic notions, facts and techniques
8.2 Paths, trees, and ends
8.3 Homogeneous and universal graphs
8.4 Connectivity and matching
8.5 Recursive structures
8.6 Graphs with ends: the complete picture
8.7 The topological cycle space
8.8 Infinite graphs as limits of finite ones
Exercises
Notes
9. Ramsey Theory for Graphs
9.1 Ramsey’s original theorems
9.2 Ramsey numbers
9.3 Induced Ramsey theorems
9.4 Ramsey properties and connectivity
Exercises
Notes
10. Hamilton Cycles
10.1 Sufficient conditions
10.2 Hamilton cycles and degree sequences
10.3 Hamilton cycles in the square of a graph
Exercises
Notes
11. Random Graphs
11.1 The notion of a random graph
11.2 The probabilistic method
11.3 Properties of almost all graphs
11.4 Threshold functions and second moments
Exercises
Notes
12. Graph Minors
12.1 Well-quasi-ordering
12.2 The graph minor theorem for trees
12.3 Tree-decompositions
12.4 Tree-width
12.5 Tangles
12.6 Tree-decompositions and forbidden minors
12.7 The graph minor theorem
Exercises
Notes
A. Infinite sets
B. Surfaces
Hints for all theExercises
Hints for Chapter 1
Hints for Chapter 2
Hints for Chapter 3
Hints for Chapter 4
Hints for Chapter 5
Hints for Chapter 6
Hints for Chapter 7
Hints for Chapter 8
Hints for Chapter 9
Hints for Chapter 10
Hints for Chapter 11
Hints for Chapter 12
Index
Symbol Index
