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Open AccessArticle

Kempe-Locking Configurations

61 Meeting House Road, Bedford Corners, NY 10549, USA
Mathematics 2018, 6(12), 309;
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)
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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Tilley, J. Kempe-Locking Configurations. Mathematics 2018, 6, 309.

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