This book fills a need for a thorough introduction to graph theory that features both the understanding and writing of proofs about graphs. Verification that algorithms work is emphasized more than their complexity.

An effective use of examples, and huge number of interesting exercises, demonstrate the topics of trees and distance, matchings and factors, connectivity and paths, graph coloring, edges and cycles, and planar graphs. For those who need to learn to make coherent arguments in the fields of mathematics and computer science.

The revision for the second edition emphasizes making the text easier for the students to learn from and easier for the instructor to teach from. There have not been great changes in the overall content of the book, but the presentation has been modified to make the material more accessible, especially in the early parts of the book. Some of the changes are discussed in more detail later in this preface.

**1. Fundamental Concepts.**

What Is a Graph? Paths, Cycles, and Trails. Vertex Degrees and Counting. Directed Graphs.

2. Trees and Distance.

2. Trees and Distance.

Basic Properties. Spanning Trees and Enumeration. Optimization and Trees.

3. Matchings and Factors.

3. Matchings and Factors.

Matchings and Covers. Algorithms and Applications. Matchings in General Graphs.

4. Connectivity and Paths.

4. Connectivity and Paths.

Cuts and Connectivity. k-connected Graphs. Network Flow Problems.

5. Coloring of Graphs.

5. Coloring of Graphs.

Vertex Colorings and Upper Bounds. Structure of k-chromatic Graphs. Enumerative Aspects.

6. Planar Graphs.

6. Planar Graphs.

Embeddings and Euler’s Formula. Characterization of Planar Graphs. Parameters of Planarity.

7. Edges and Cycles.

7. Edges and Cycles.

Line Graphs and Edge-Coloring. Hamiltonian Cycles. Planarity, Coloring, and Cycles.

8. Additional Topics (Optional).

8. Additional Topics (Optional).

Perfect Graphs. Matroids. Ramsey Theory. More Extremal Problems. Random Graphs. Eigenvalues of Graphs.

**Mathematical Background.**

Appendix A:

Appendix A:

**Optimization and Complexity.**

Appendix B:

Appendix B:

**Hints for Selected Exercises.**

Appendix C:

Appendix C:

**Glossary of Terms.**

Appendix D:

Appendix D:

**Supplemental Reading.**

Appendix E:

Appendix E:

**Appendix F:**References.

REVIEW