Graphs of Groups on Surfaces: Interactions and Models

by A.T. White

Overview

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.

Product Details

ISBN-13:	9780444500755
Publisher:	Elsevier Science
Publication date:	04/27/2001
Series:	North-Holland Mathematics Studies , #188
Pages:	378
Product dimensions:	7.17(w) x 10.12(h) x (d)

Table of Contents

Chapter 1. HISTORICAL SETTINGChapter 2. A BRIEF INTRODUCTION TO GRAPH THEORY 2-1. Definition of a Graph 2-2. Variations of Graphs2-3. Additional Definitions 2-4. Operations on Graphs 2-5. ProblemsChapter 3. THE AUTOMORPHISM GROUP OF A GRAPH 3-1. Definitions3-2. Operations on Permutations Groups3-3. Computing Automorphism Groups of Graphs 3-4. Graphs with a Given Automorphism Group3-5. 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. ProblemsChapter 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

Editorial 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

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