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商品簡介
目次
商品簡介
Almost two decades have passed since the appearance of those graph theory texts that still set the agenda for most introductory courses taught today. The canon created by those books has helped to identify some main fields of study and research, and will doubtless continue to influence the development of the discipline for some time to come.
Yet much has happened in those 20 years, in graph theory no less than elsewhere: deep new theorems have been found, seemingly disparate methods and results have become interrelated, entire new branches have arisen. To name just a few such developments, one may think of how the new notion of list colouring has bridged the gulf between invuriants such as average degree and chromatic number, how probabilistic methods and the regularity lemma have pervaded extremai graph theory and Ramsey theory, or how the entirely new field of graph minors and tree-decompositions has brought standard methods of surface topology to bear on long-standing algorithmic graph problems.
Yet much has happened in those 20 years, in graph theory no less than elsewhere: deep new theorems have been found, seemingly disparate methods and results have become interrelated, entire new branches have arisen. To name just a few such developments, one may think of how the new notion of list colouring has bridged the gulf between invuriants such as average degree and chromatic number, how probabilistic methods and the regularity lemma have pervaded extremai graph theory and Ramsey theory, or how the entirely new field of graph minors and tree-decompositions has brought standard methods of surface topology to bear on long-standing algorithmic graph problems.
目次
Preface
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 Packing and covering
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 Mengers theorem
3.4 Maders 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: Kuratowskis 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 Tuttes flow conjectures
Exercises
Notes
7 Extremal Graph Theory
8 Infinite Graphs
9 Ramsey Theory for Graphs
10 Hamilton Cycles
11 Random Grapnhs
12 Mionors Trees and WQO
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 Packing and covering
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 Mengers theorem
3.4 Maders 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: Kuratowskis 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 Tuttes flow conjectures
Exercises
Notes
7 Extremal Graph Theory
8 Infinite Graphs
9 Ramsey Theory for Graphs
10 Hamilton Cycles
11 Random Grapnhs
12 Mionors Trees and WQO
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