Mathematics

For junior- to senior-level courses in Graph Theory taken by majors in Mathematics, Computer Science, or Engineering or for beginning-level graduate courses.

Once considered an “unimportant” branch of topology, graph theory has come into its own through many important contributions to a wide range of fields — and is now one of the fastest-growing areas in discrete mathematics and computer science. This new text introduces basic concepts, definitions, theorems, and examples from graph theory. The authors present a collection of interesting results from mathematics that involve key concepts and proof techniques; cover design and analysis of computer algorithms for solving problems in graph theory; and discuss applications of graph theory to the sciences. It is mathematically rigorous, but also practical, intuitive, and algorithmic.

• Self-contained format:

–       Almost all proofs of some exceptionally technical theorems (such as the Four Color Theorem and a Minor Theorem) are contained in the text.

–       Does not assume any special mathematical background beyond the standard undergraduate mathematics courses.

• Explanatory notes – Includesnumerous notes and remarks to explain the “commonsense” point of view, the motivation, and many “hand-waving” arguments, so that the reader gets the best of both worlds–the rigor and the intuition.

• Wealth of examples – Explain both the idea of technical definitions and theorems and the applications in graph theory itself, computer science, and other sciences.

• Numerous end-of-chapter exercises:

–       All solvable with the material presented in the text

–       Vary greatly in difficulty (but should all be attempted by the diligent and alert reader)

–       Omits hard research problems or unsolved problems, in the interest of stimulating and encouraging students to in master the fundamental concepts that are treated in the text itself.

• Hints and clues – Providesnumerous suggestions for many of the more involved exercises, especially those that are important to the development of the field of graph theory itself, keeping gaps in the overall treatment to an absolute minimum.

• Figures:

–       Explains and illuminates graph-theory concepts with approximately 250 figures and diagrams

–       Figures are designed to be simple, uncluttered, and “to the point,” rather than to impress.

–       They are all drawn in the xfig program, which allows the use of exactly the same fonts within the figures as within the text itself.

–       Formulas can be displayed around the figures.

• Algorithms in pseudocode – Presentsa wide range of graph algorithms in a precise pseudocode for easy implementation in any programming language.

• Java programs – Includes a collection of graph algorithms, written in Java, that are ready for compiling and running.

• Unique coverage – Presents manytopics not always covered in modern textbooks including:

– An early treatment of trees, binary trees, and rooted trees, both ordered and unordered

– A fairly comprehensive topological discussion regarding surface graphs

– Numerous types of graph colorings and related algorithms

– A brief introduction to chordal graphs and related algorithms

– A full chapter devoted to certain basic graph algorithms

– Both theoretical and algorithmic treatment of graph matchings

– Numerous enumerative aspects, such as the Matrix Tree Theorem, chromatic polynomials, rook polynomials, and matching polynomials

– An introduction to generating functions, exponential generating functions, and their applications in graph countings.

• Historical remarks – Presentsnumerous classical results, their brief history, and historical remarks

about the authors of various concepts in graph theory.

Preface                                                                                                     

1     Introduction to Graph Theory                                               

2     Basic Concepts in Graph Theory                                      

3     TreesandForests

4     Spanning Trees

5     Fundamental Properties of Graphs and Digraphs

6     Connectivity and Flow

7     Planar Graphs

8     Graph Coloring

9     Coloring Enumerations and Chordal Graphs

10    Independence,Dominance, and Matchings  

11    Cover Parameters and MatchingPolynomials

12    GraphCounting

13    Graph Algorithms

APPENDICES

A Greek Alphabet

B Notation

C Top Ten Online References

Index ix

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