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Kempe-Locking Configurations

61 Meeting House Road, Bedford Corners, NY 10549, USA
Mathematics 2018, 6(12), 309; https://doi.org/10.3390/math6120309
Received: 27 October 2018 / Revised: 28 November 2018 / Accepted: 4 December 2018 / Published: 7 December 2018
(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)
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Abstract

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. View Full-Text
Keywords: graph coloring; Kempe chain; Kempe-locking; Birkhoff diamond graph coloring; Kempe chain; Kempe-locking; Birkhoff diamond
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Tilley, J. Kempe-Locking Configurations. Mathematics 2018, 6, 309.

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