Graphs of Groups on Surfaces

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