Author | Ringel, Gerhard. author |
---|---|
Title | Map Color Theorem [electronic resource] / by Gerhard Ringel |
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg, 1974 |
Connect to | http://dx.doi.org/10.1007/978-3-642-65759-7 |
Descript | XII, 194 p. online resource |
1. Problems, Illustrations, History -- 1.1.The Four Color Problem -- 1.2. Map Color Theorem -- 1.3. The Thread Problem -- 1.4. Unilateral Surfaces -- 2. Graph Theory -- 2.1. Chromatic Number -- 2.2. Rotations of Graphs -- 2.3. Orientable Cases 7 and 10 -- 3. Classification of Surfaces -- 3.1. The Concept of Topology -- 3.2. Polyhedra -- 3.3. Elementary Operations -- 3.4. Normal Form for Orientable Surfaces -- 3.5. Normal Form for Non-Orientable Surfaces -- 3.6. Standard Models -- 3.7. Partial Polyhedra -- 4. Graphs on Surfaces -- 4.1. Embedding Theorem -- 4.2. Dual Polyhedra -- 4.3. Heawoodโs Inequality -- 4.4. Genus of Graphs -- 4.5. Non-Orientable Genus of Graphs -- 4.6. Kleinโs Bottle -- 5. Combinatorics of Embeddings -- 5.1. Triangular Embeddings -- 5.2. Orientable Special Cases -- 5.3. Outline for General Cases -- 6. Orientable Cases 1, 4, and 9 -- 6.1. Orientable Case 4 -- 6.2. Arithmetic Combs -- 6.3. Orientable Case 1 -- 6.4. Coil Diagrams -- 6.5. Orientable Case 9 -- 7. Orientable Cases 11, 2, and 8 -- 7.1. Example for n=35 -- 7.2. Orientable Case 11 -- 7.3. The Additional Adjacency Problem -- 7.4. Orientable Case 2 -- 7.5. Additional Adjacency Problem -- 7.6. Orientable Case 8 -- 8. Non-Orientable Cases (Index 1) -- 8.1. Method of Doubling -- 8.2. Non-Orientable Cases 0, 3, 7 -- 8.3. Cascades -- 8.4. Orientable Application -- 9. Solutions of Index 2 and 3 -- 9.1. Examples and Method -- 9.2. Orientable Cases 3 and 5 -- 9.3. Orientable Case 6 -- 9.4. Non-Orientable Case 9 -- 10. Construction by Induction -- 10.1. An Index 3 Induction -- 10.2. An Index 2 Induction -- 10.3. Non-Orientable Cases 1, 2, 6, and 10 -- 11. Orientable Case 0 -- 11.1. Currents from Non-Abelian Groups -- 11.2. Examples -- 11.3. General Solution -- 12. Related Problems -- 12.1. Questions about Rotations -- 12.2. Questions about Embeddings -- References