| Autor: |
James Tilley |
| Jazyk: |
angličtina |
| Rok vydání: |
2018 |
| Předmět: |
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| Zdroj: |
Mathematics, Vol 6, Iss 12, p 309 (2018) |
| Druh dokumentu: |
article |
| ISSN: |
2227-7390 |
| DOI: |
10.3390/math6120309 |
| Popis: |
The 4-color theorem was proved by showing that a minimum counterexample cannot exist. Birkhoff demonstrated that a minimum counterexample must be internally 6-connected. We show that a minimum counterexample must also satisfy a coloring property that we call Kempe-locking. The novel idea explored in this article is that the connectivity and coloring properties are incompatible. We describe a methodology for analyzing whether an arbitrary planar triangulation is Kempe-locked. We provide a heuristic argument that a fundamental Kempe-locking configuration must be of low order and then perform a systematic search through isomorphism classes for such configurations. All Kempe-locked triangulations that we discovered have two features in common: (1) they are Kempe-locked with respect to only a single edge, say x y , and (2) they have a Birkhoff diamond with endpoints x and y as a subgraph. On the strength of our investigations, we formulate a plausible conjecture that the Birkhoff diamond is the only fundamental Kempe-locking configuration. If true, this would establish that the connectivity and coloring properties of a minimum counterexample are indeed incompatible. It would also imply the appealing conclusion that the Birkhoff diamond configuration alone is responsible for the 4-colorability of planar triangulations. |
| Databáze: |
Directory of Open Access Journals |
| Externí odkaz: |
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