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Graph Theory

10 years 8 months ago
Graph Theory
A well-written book about graph theory.
Reinhard Diestel
Added 16 Feb 2009
Updated 16 Feb 2009
Authors Reinhard Diestel

Table of Content

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
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
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
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
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
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
7. Extremal Graph Theory
7.1 Subgraphs
7.2 Minors
7.3 Hadwiger’s conjecture
7.4 Szemer´edi’s regularity lemma
7.5 Applying the regularity lemma
Exercises
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 The topological end space
Exercises
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
10. Hamilton Cycles
10.1 Simple sufficient conditions
10.2 Hamilton cycles and degree sequences
10.3 Hamilton cycles in the square of a graph
Exercises
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
12. Minors, Trees and WQO
12.1 Well-quasi-ordering
12.2 The graph minor theorem for trees
12.3 Tree-decompositions
12.4 Tree-width and forbidden minors
12.5 The graph minor theorem
Exercises
A. Infinite sets
B. Surfaces
Hints for all the exercises.
Index
Symbol index
 
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