Lecture by Prof. Dr. Maria Axenovich
Lecture notes by M´onika Csik´os,Daniel Hoske and Torsten Ueckerdt
1 Contents
1 Introduction 3
2 Notations 3
3 Preliminaries 4
4 Matchings 13
5 Connectivity 17
6 Planar graphs 22
7 Colorings 27
9 Ramsey theory 34
10 Flows 38
11 Random graphs 40
12 Hamiltonian cycles 42
13 Proofs 44 13.1 Kuratowski’s Theorem ...... 61 13.2 Other coloring results ...... 81 13.3 Preparation for Tur´an’stheorem ...... 82 13.4 Induced Ramsey numbers ...... 97 13.5 Flows ...... 99 13.5.1 Group-valued flows ...... 101 13.6 Random graphs ...... 105
References 108
Index 109
2 1 Introduction
These notes include major definitions and theorems of the graph theory lecture held by Prof. Maria Axenovich at KIT in the winter term 2017/18. Most of the content is based on the book “Graph Theory” by Reinhard Diestel [4]. A free version of the book is available at http://diestel-graph-theory.com. The first part includes only formulations and definitions. The second part includes the proofs. Conventions: • G = (V,E) is an arbitrary (undirected, simple) graph • n := |V | is its number of vertices • m := |E| is its number of edges
2 Notations
notation definition meaning