Description

The book, suitable as both an introductory reference and as a text book in the rapidly growing field of topological graph theory, models both maps (as in map-coloring problems) and groups by means of graph imbeddings on sufaces. Automorphism groups of both graphs and maps are studied. In addition connections are made to other areas of mathematics, such as hypergraphs, block designs, finite geometries, and finite fields. There are chapters on the emerging subfields of enumerative topological graph theory and random topological graph theory, as well as a chapter on the composition of English church-bell music. The latter is facilitated by imbedding the right graph of the right group on an appropriate surface, with suitable symmetries. Throughout the emphasis is on Cayley maps: imbeddings of Cayley graphs for finite groups as (possibly branched) covering projections of surface imbeddings of loop graphs with one vertex. This is not as restrictive as it might sound; many developments in topological graph theory involve such imbeddings.

The approach aims to make all this interconnected material readily accessible to a beginning graduate (or an advanced undergraduate) student, while at the same time providing the research mathematician with a useful reference book in topological graph theory. The focus will be on beautiful connections, both elementary and deep, within mathematics that can best be described by the intuitively pleasing device of imbedding graphs of groups on surfaces.

Table of Contents

Chapter 1. HISTORICAL SETTING
Chapter 2. A BRIEF INTRODUCTION TO GRAPH THEORY
2-1. Definition of a Graph
2-2. Variations of Graphs
2-3. Additional Definitions
2-4. Operations on Graphs
2-5. Problems
Chapter 3. THE AUTOMORPHISM GROUP OF A GRAPH
3-1. Definitions
3-2. Operations on Permutations Groups
3-3. Computing Automorphism Groups of Graphs
3-4. Graphs with a Given Automorphism Group
3-5. Problems
Chapter 4. THE CAYLEY COLOR GRAPH OF A GROUP PRESENTATION
4-1. Definitions
4-2. Automorphisms
4-3. Properties
4-4. Products
4-5. Cayley Graphs
4-6. Problems
Chapter 5. AN INTRODUCTION TO SURFACE TOPOLOGY
5-1. Definitions
5-2. Surfaces and Other 2-manifolds
5-3. The Characteristic of a Surface
5-4. Three Applications
5-5. Pseudosurfaces
5-6. Problems
Chapter 6. IMBEDDING PROBLEMS IN GRAPH THEORY
6-1. Answers to Some Imbedding Questions
6-2. Definition of "Imbedding"
6-3. The Genus of a Graph
6-4. The Maximum Genus of a Graph
6-5. Genus Formulae for Graphs
6-6. Rotation Schemes
6-7. Imbedding Graphs on Pseudosurfaces
6-8. Other Topological Parameters for Graphs
6-9. Applications
6-10. Problems
Chapter 7. THE GENUS OF A GROUP
7-1. Imbeddings of Cayley Color graphs
7-2. Genus Formulae for Groups
7-3. Related Results

Details

No. of pages:
378
Language:
English
Copyright:
© 2001
Published:
Imprint:
North Holland
eBook ISBN:
9780080507583
Print ISBN:
9780444500755
Print ISBN:
9780444546692

About the author

A.T. White

Affiliations and Expertise

Western Michigan University, Kalamazoo, MI 49008, USA

Reviews

@qu:...this is a very well-written and readable book, which I recommend on anyone wanting to learn this particular approach to the subject. @source:Bulletin of the London Mathematical Society