# Graphs of Groups on Surfaces

## Interactions and Models

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.

Hardbound, 378 Pages

Published: April 2001

Imprint: North-holland

ISBN: 978-0-444-50075-5

## Reviews

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

## Contents

• Chapter 1. HISTORICAL SETTING
Chapter 2. A BRIEF INTRODUCTION TO GRAPH THEORY
2-1. Definition of a Graph
2-2. Variations of Graphs
2-4. 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
7-4. The Characteristic of a Group
7-5. Problems
Chapter 8. MAP-COLORING PROBLEMS
8-1. Definitions and the Six-Color Theorem
8-2. The Five-Color Theorem
8-3. The Four-Color Theorem
8-4. Other Map-Coloring Problems: The Heawood Map-Coloring Theorem
8-5. A Related Problem
8-6. A Four-Color Theorem for the Torus
8-7. A Nine-Color Theorem for the Torus and Klein Bottle
8-8. k-degenerate Graphs
8-9. Coloring Graphs on Pseudosurfaces
8-10. The Cochromatic Number of Surfaces
8-11. Problems
Chapter 9. QUOTIENT GRAPHS AND QUOTIENT MANIFOLDS:CURRENT GRAPHS AND THE COMPLETE GRAPH THEOREM
9-1. The Genus of Kn
9-2. The Theory of Current Graphs as Applied to Kn
9-3. A Hint of Things to Come
9-4. Problems
Chapter 10. VOLTAGE GRAPHS
10-1. Covering Spaces
10-2. Voltage Graphs
10-3. Examples
10-4. The Heawood Map-coloring Theorem (again)
10-5. Strong Tensor Products
10-6. Covering Graphs and Graphical Products
10-7. Problems
Chapter 11. NONORIENTABLE GRAPH IMBEDDINGS
11-1. General Theory
11-2. Nonorientable Covering Spaces
11-3. Nonorientable Voltage Graph Imbeddings
11-4. Examples
11-5. The Heawood Map-coloring Theorem, Nonorientable Version
11-6. Other Results
11-7. Problems
Chapter 12. BLOCK DESIGNS
12-1. Balanced Incomplete Block Designs
12-2. BIBDs and Graph Imbeddings
12-3. Examples
12-4. Strongly Regular Graphs
12-5. Partially Balanced Incomplete Block Designs
12-6. PBIBDs and Graph Imbeddings
12-7. Examples
12-8. Doubling a PBIBD
12-9. Problems
Chapter 13. HYPERGRAPH IMBEDDINGS
13-1. Hypergraphs
13-2. Associated Bipartite Graphs
13-3. Imbedding Theory for Hypergraphs
13-4. The Genus of a Hypergraph
13-5. The Heawood Map-Coloring Theorem, for Hypergraphs
13-6. The Genus of a Block Design
13-7. An Example
13-8. Nonorientable Analogs
13-9. Problems
Chapter 14. FINITE FIELDS ON SURFACES
14-1. Graphs Modelling Finite Rings
14-2. Basic Theorems About Finite Fields
14-3. The Genus of Fp
14-4. The Genus of Fpr
14-5. Further Results
14-6. Problems
Chapter 15. FINITE GEOMETRIES ON SURFACES
15-1. Axiom Systems for Geometries
15-2. n-Point Geometry
15-3. The Geometries of Fano, Pappus, and Desargues
15-4. Block Designs as Models for Geometries
15-5. Surface Models for Geometries
15-6. Fano, Pappus, and Desargues Revisited
15-7. 3-Configurations
15-8. Finite Projective Planes
15-9. Finite Affine Planes
15-10. Ten Models for AG(2,3)
15-11. Completing the Euclidean Plane
15-12. Problems
Chapter 16. MAP AUTOMORPHISM GROUPS
16-1. Map Automorphisms
16-2. Symmetrical Maps
16-3. Cayley Maps
16-4. Complete Maps
16-5. Other Symmetrical Maps
16-6. Self -Complementary Graphs
16-7. Self-dual Maps
16-8. Paley Maps
16-9. Problems
Chapter 17. ENUMERATING GRAPH IMBEDDINGS
17-1. Counting Labelled Orientable 2-Cell Imbeddings
17-2. Counting Unlabelled Orientable 2-Cell Imbeddings
17-3. The Average Number of Symmetries
17-4. Problems
Chapter 18. RANDOM TOPOLOGICAL GRAPH THEORY
18-1. Model I
18-2. Model II
18-3. Model III
18-4. Model IV
18-5. Model V
18-6. Model VI- Random Cayley Maps
18-7. Problems
Chapter 19. CHANGE RINGING
19-1. The Setting
19-2. A Mathematical Model
19-3. Minimus
19-4. Doubles
19-5. Minor
19-6. Triples and Fabian Stedman
19-7. Extents on n Bells
19-8. Summary
19-9. Problems
REFERENCES. BIBLIOGRAPHY. INDEX OF SYMBOLS. INDEX OF DEFINITIONS