1.Introduction 1
1.1.Graphs and Graph Models 1
1.2.Connected Graphs 9
1.3.Common Classes of Graphs 19
1.4.Multigraphs and Digraphs 26
2.Degrees 31
2.1.The Degree of a Vertex 31
2.2.Regular Graphs 38
2.3.Degree Sequences 43
2.4.Excursion：Graphs and Matrices 48
2.5.Exploration：Irregular Graphs 50
3.Isomorphic Graphs 55
3.1.The Definition of Isomorphism 55
3.2.Isomorphism as a Relation 63
3.3.Excursion：Graphs and Groups 66
3.4.Excursion：Reconstruction and Solvability 76
4.Trees 85
4.1.Bridges 85
4.2.Trees 87
4.3.The Minimum Spanning Tree Problem 94
4.4.Excursion：The Number of Spanning Trees 101
5.Connectivity 107
5.1.Cut-Vertices 107
5.2.Blocks 111
5.3.Connectivity 115
5.4.Menger’s Theorem 124
5.5.Exploration：Geodetic Sets 130
6.Traversability 133
6.1.Eulerian Graphs 133
6.2.Hamiltonian Graphs 140
6.3.Exploration：Hamiltonian Walks and Numbers 152
6.4.Excursion：The Early Books of Graph Theory 156
7.Digraphs 161
7.1.Strong Digraphs 161
7.2.Tournaments 169
7.3.Excursion：Decision-Making 176
7.4.Exploration：Wine Bottle Problems 180
8.Matchings and Factorization 183
8.1.Matchings 183
8.2.Factorization 194
8.3.Decompositions and Graceful Labelings 209
8.4.Excursion：Instant Insanity 214
8.5.Excursion：The Petersen Graph 219
8.6.Exploration：γ-Labelings of Graphs 224
9.Planarity 227
9.1.Planar Graphs 227
9.2.Embedding Graphs on Surfaces 241
9.3.Excursion：Graph Minors 249
9.4.Exploration：Embedding Graphs in Graphs 253
10.Coloring 259
10.1.The Four Color Problem 259
10.2.Vertex Coloring 267
10.3.Edge Coloring 280
10.4.Excursion：The Heawood Map Coloring Theorem 288
10.5.Exploration：Local Coloring 293
11.Ramsey Numbers 297
11.1.The Ramsey Number of Graphs 297
11.2.Turan’s Theorem 307
11.3.Exploration：Rainbow Ramsey Numbers 314
11.4.Excursion：Erdos Numbers 321
12.Distance 327
12.1.The Center of a Graph 327
12.2.Distant Vertices 333
12.3.Excursion：Locating Numbers 341
12.4.Excursion：Detour and Directed Distance 346
12.5.Exploration：Channel Assignment 351
12.6.Exploration：Distance Between Graphs 357