The Four-Color Problem
The Four-Color Problem
Published on 29. August 2011
258 pages
978-0-08-087339-8 (ISBN)
Description
The Four-Color Problem
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Content
- Front Cover
- The Problem Four-Color, Volume 27
- Copyright Page
- Contents
- Preface
- Introduction
- Chapter 1. Planar Graphs
- 1.1. Planar Representations
- 1.2. The Faces
- 1.3. Maximal Planar Graphs. Straight Line Representations
- Chapter 2. Bridges and Circuits
- 2.1. Bridges in General Graphs
- 2.2. Circuit Bridges
- 2.3. Equivalence of Planar Representations
- 2.4. Uniqueness of Representations
- 2.5. Transfer of Bridges
- 2.6. Characterization of Planar Graphs
- 2.7. Further Observations on Maximal Graphs
- Chapter 3. Dual Graphs
- 3.1. Geometric Definition of Duality
- 3.2. Observations on Graphs and Their Duals
- 3.3. Relational Definition of Duality
- 3.4. Characterization of Planar Graphs
- 3.5. Maximal Bipartite Graphs. Self-Dual Graphs
- Chapter 4. Euler's Formula and Its Consequences
- 4.1. Euler's Formula
- 4.2. Regular Graphs
- 4.3. The Euler Contributions
- Chapter 5. Large Circuits
- 5.1. Circuit Arcs
- 5.2. Hamilton Circuits
- Chapter 6. Colorations
- 6.1. Types of Coloration
- 6.2. Two Colors
- 6.3. Reductions
- 6.4. The Five-Color Theorem
- 6.5. TheTheorem of Brooks
- Chapter 7. Color Functions
- 7.1. Vertex Coloration
- 7.2. The Dual Theory
- 7.3. Color Directed Graphs
- 7.4. Some Special Applications
- Chapter 8. Formulations of the Four-Color Problem
- 8.1. Decomposition into Three Subgraphs
- 8.2. Bipartite Dichotomy
- 8.3. Even Subgraphs
- 8.4. Graphs with Small Face Boundaries
- 8.5. Angle Characters and Congruence Conditions (mod 3)
- Chapter 9. Cubic Graphs
- 9.1. Color Reduction to Cubic Graphs
- 9.2. Configurations in Cubic Graphs
- 9.3. Four-Color Conditions in Cubic Graphs
- 9.4. The Interchange Graph and the Color Problems
- 9.5. Planar Interchange Graphs
- 9.6. Construction of Cubic Graphs
- Chapter 10. Hadwiger's Conjecture
- 10.1. Contractions and Subcontractions
- 10.2. Maximal Graphs and Simplex Decompositions
- 10.3. lndecomposable Graphs
- 10.4. Hadwiger's Conjecture
- 10.5. Wagner's Equivalence Theorem
- 10.6. Contractions to S4
- 10.7. Multiply-Connected Graphs
- Chapter 11. Critical Graphs
- 11.1. Types of Critical Graphs
- 11.2. Contraction Critical Graphs
- 11.3. Edge-Critical Graphs
- 11. 4. Construction of e-Critical Graphs
- 11.5. Conjunctions and Mergers
- 11.6. Amalgamations
- 11. 7. Mergers of Simplexes
- Chapter 12. Planar 5-Chromatic Graphs
- 12.1. Separations
- 12.2. Irreducible Graphs
- 12.3. Reductions for Minor Vertices
- 12.4. Errera Circuits and 5-Components
- 12.5. Lower Bounds for Irreducible Graphs
- Chapter 13. Three Colors
- 13.1. Formulations of the Three-Color Problem
- 13.2. The Theorem of Grötzsch
- Chapter 14. Edge Coloration
- 14.1. General Observations
- 14.2. Coloration of an Augmented Graph
- 14.3. The Theorem ofShannon
- 14.4. The Theorem of Vizing
- Bibliography
- Author Index
- Subject Index
