Graphs of Groups on Surfaces

Chapter 1. HISTORICAL SETTINGChapter 2. A BRIEF INTRODUCTION TO GRAPH THEORY 2-1. Definition of a Graph2-2. Variations of Graphs2-3. Additional Definitions2-4. Operations on Graphs2-5. ProblemsChapter 3. THE AUTOMORPHISM GROUP OF A GRAPH3-1. Definitions3-2. Operations on Permutations Groups3-3. Computing Automorphism Groups of Graphs3-4. Graphs with a Given Automorphism Group3-5. ProblemsChapter 4. THE CAYLEY COLOR GRAPH OF A GROUP PRESENTATION4-1. Definitions4-2. Automorphisms4-3. Properties4-4. Products4-5. Cayley Graphs4-6. ProblemsChapter 5. AN INTRODUCTION TO SURFACE TOPOLOGY5-1. Definitions5-2. Surfaces and Other 2-manifolds5-3. The Characteristic of a Surface5-4. Three Applications5-5. Pseudosurfaces5-6. ProblemsChapter 6. IMBEDDING PROBLEMS IN GRAPH THEORY6-1. Answers to Some Imbedding Questions6-2. Definition of "Imbedding"6-3. The Genus of a Graph6-4. The Maximum Genus of a Graph6-5. Genus Formulae for Graphs6-6. Rotation Schemes6-7. Imbedding Graphs on Pseudosurfaces6-8. Other Topological Parameters for Graphs6-9. Applications6-10. ProblemsChapter 7. THE GENUS OF A GROUP7-1. Imbeddings of Cayley Color graphs7-2. Genus Formulae for Groups7-3. Related Results7-4. The Characteristic of a Group7-5. ProblemsChapter 8. MAP-COLORING PROBLEMS8-1. Definitions and the Six-Color Theorem8-2. The Five-Color Theorem8-3. The Four-Color Theorem8-4. Other Map-Coloring Problems:The Heawood Map-Coloring Theorem8-5. A Related Problem8-6. A Four-Color Theorem for the Torus8-7. A Nine-Color Theorem for the Torus and Klein Bottle8-8. k-degenerate Graphs8-9. Coloring Graphs on Pseudosurfaces8-10. The Cochromatic Number of Surfaces8-11. ProblemsChapter 9. QUOTIENT GRAPHS AND QUOTIENT MANIFOLDS:CURRENT GRAPHS AND THE COMPLETE GRAPH THEOREM9-1. The Genus of Kn9-2. The Theory of Current Graphs as Applied to Kn9-3. A Hint of Things to Come9-4. ProblemsChapter 10. VOLTAGE GRAPHS10-1. Covering Spaces10-2. Voltage Graphs10-3. Examples10-4. The Heawood Map-coloring Theorem (again)10-5. Strong Tensor Products10-6. Covering Graphs and Graphical Products10-7. ProblemsChapter 11. NONORIENTABLE GRAPH IMBEDDINGS11-1. General Theory11-2. Nonorientable Covering Spaces11-3. Nonorientable Voltage Graph Imbeddings11-4. Examples11-5. The Heawood Map-coloring Theorem, Nonorientable Version11-6. Other Results11-7. ProblemsChapter 12. BLOCK DESIGNS12-1. Balanced Incomplete Block Designs12-2. BIBDs and Graph Imbeddings12-3. Examples12-4. Strongly Regular Graphs12-5. Partially Balanced Incomplete Block Designs12-6. PBIBDs and Graph Imbeddings12-7. Examples12-8. Doubling a PBIBD12-9. ProblemsChapter 13. HYPERGRAPH IMBEDDINGS13-1. Hypergraphs13-2. Associated Bipartite Graphs13-3. Imbedding Theory for Hypergraphs13-4. The Genus of a Hypergraph13-5. The Heawood Map-Coloring Theorem, for Hypergraphs 13-6. The Genus of a Block Design13-7. An Example13-8. Nonorientable Analogs13-9. ProblemsChapter 14. FINITE FIELDS ON SURFACES14-1. Graphs Modelling Finite Rings14-2. Basic Theorems About Finite Fields14-3. The Genus of Fp14-4. The Genus of Fpr14-5. Further Results14-6. ProblemsChapter 15. FINITE GEOMETRIES ON SURFACES15-1. Axiom Systems for Geometries15-2. n-Point Geometry15-3. The Geometries of Fano, Pappus, and Desargues15-4. Block Designs as Models for Geometries15-5. Surface Models for Geometries15-6. Fano, Pappus, and Desargues Revisited15-7. 3-Configurations15-8. Finite Projective Planes15-9. Finite Affine Planes15-10. Ten Models for AG(2,3)15-11. Completing the Euclidean Plane15-12. ProblemsChapter 16. MAP AUTOMORPHISM GROUPS16-1. Map Automorphisms16-2. Symmetrical Maps16-3. Cayley Maps16-4. Complete Maps16-5. Other Symmetrical Maps16-6. Self -Complementary Graphs16-7. Self-dual Maps16-8. Paley Maps16-9. ProblemsChapter 17.