Introduction to Graph Theory

1 Introduction 1.1 Graphs and Graph Models 1.2 Connected Graphs 1.3 Common Classes of Graphs 1.4 Multigraphs and Digraphs 2 Degrees 2.1 The Degree of a Vertex 2.2 Regular Graphs 2.3 Degree Sequences 2.4 Excursion: Graphs and Matrices 2.5 Exploration: Irregular Graphs 3 Isomorphic Graphs 3.1 The Definition of Isomorphism 3.2 Isomorphism as a Relation 3.3 Excursion: Graphs and Groups 3.4 Excursion: Reconstruction and Solvability 4 Trees 4.1 Bridges 4.2 Trees 4.3 The Minimum Spanning Tree Problem 4.4 Excursion: The Number of Spanning Trees 5 Connectivity 5.1 Cut-Vertices 5.2 Blocks 5.3 Connectivity 5.4 Menger's Theorem 5.5 Exploration: Geodetic Sets 6 Traversability 6.1 Eulerian Graphs 6.2 Hamiltonian Graphs 6.3 Exploration: Hamiltonian Walks and Numbers 6.4 Excursion: The Early Books of Graph Theory 7 Digraphs 7.1 Strong Digraphs 7.2 Tournaments 7.3 Excursion: Decision-Making 7.4 Exploration: Wine Bottle Problems 8 Matchings and Factorization 8.1 Matchings 8.2 Factorization 8.3 Decompositions and Graceful Labelings 8.4 Excursion: Instant Insanity 8.5 Excursion: The Petersen Graph 8.6 Exploration: -Labeling of Graphs 9 Planarity 9.1 Planar Graphs 9.2 Embedding Graphs on Surfaces 9.3 Excursion: Graph Minors 9.4 Exploration: Embedding Graphs in Graphs 10 Coloring 10.1 The Four Color Problem 10.2 Vertex Coloring 10.3 Edge Coloring 10.4 Excursion: The Heawood Map Coloring Theorem 10.5 Exploration: Local Coloring 11 Ramsey Numbers 11.1 The Ramsey Number of Graphs 11.2 Turan's Theorem 11.3 Exploration: Rainbow Ramsey Numbers 11.4 Excursion: Erdos Numbers 12 Distance 12.1 The Center of a Graph 12.2 Distant Vertices 12.3 Excursion: Locating Numbers 12.4 Excursion: Detour and Directed Distance 12.5 Exploration: Channel Assignment 12.6 Exploration: Distance Between Graphs 13 Domination 13.1 The Domination Number of a Graph 13.2 Exploration: Stratification 13.3 Exploration: Lights Out 13.4 Excursion: And Still It Grows More Colorful Appendix 1 Sets and Logic Appendix 2 Equivalence Relations and Functions Appendix 3 Methods of Proof