This book is for math and computer science majors, for students and representatives of many other disciplines (like bioinformatics, for example) taking courses in graph theory, discrete mathematics, data structures, algorithms. It is also for anyone who wants to understand the basics of graph theory, or just is curious. No previous knowledge in graph theory or any other significant mathematics is required. The very basic facts from set theory, proof techniques and algorithms are sufficient to understand it; but even those are explained in the text. Structurally, the text is divided into two parts where Part II is the generalization of Part I. The first part discusses the key concepts of graph theory with emphasis on trees, bipartite graphs, cycles, chordal graphs, planar graphs and graph coloring. The second part considers generalizations of Part I and discusses hypertrees, bipartite hyper graphs, hyper cycles, chordal hyper graphs, planar hyper graphs and hyper graph coloring. There is an interaction between the parts and within the parts to show how ideas of generalizations work. The main point is to exhibit the ways of generalizations and interactions of mathematical concepts from the very simple to the most advanced. One of the features of this text is the duality of hyper graphs. This fundamental concept is missing in graph theory (and in its introductory teaching) because dual graphs are not properly graphs, they are hyper graphs. However, as Part II shows, the duality is a very powerful tool in understanding, simplifying and unifying many combinatorial relations; it is basically a look at the same structure from the opposite (vertices versus edges) point of view.

Table of Contents

I GRAPHS

1 Basic Definitions and Concepts

2 Trees and Bipartite Graphs

3 Chordal Graphs

4 Planar Graphs

5 Graph Coloring

6 Traversals and Flows

II HYPERGRAPHS

8 Hypertrees and Chordal Hypergraphs

9 Some Other Remarkable Hypergraph Classes

10 Hypergraph Coloring

11 Modeling with Hypergraphs

12 Appendix

About the Author

Vitaly I. Voloshin is a professor at the Department of Mathematics and Statistics, Troy University. He worked at Moldova State University and at the Institute of Mathematics and Informatics of the Moldovan Academy of Sciences (Republic of Moldova, former Soviet Union) for many years, and moved to the United States in 2002. His research areas include graph and hypergraph coloring, theory, algorithms and applications.

In 1993, he originated the direction of mixed hypergraph coloring, and there are now more than 200 scientific publications on the subject, including the author's Fields Institute Monograph. His research has been repeatedly supported by many scientific foundations of Italy, Germany, Canada, China, the United States, and other countries. He was the invited speaker of many major international conferences, held visiting positions at many universities throughout the world, and supervised and was consulted on many MS and PhD theses.