Preprint Article Version 1 This version is not peer-reviewed

# Kempe-Locking Configurations

Version 1 : Received: 27 October 2018 / Approved: 29 October 2018 / Online: 29 October 2018 (05:08:46 CET)

A peer-reviewed article of this Preprint also exists.

Tilley, J. Kempe-Locking Configurations. Mathematics 2018, 6, 309. Tilley, J. Kempe-Locking Configurations. Mathematics 2018, 6, 309.

Journal reference: Mathematics 2018, 6, 309
DOI: 10.3390/math6120309

## Abstract

Existing proofs of the 4-color theorem succeeded by establishing an unavoidable set of reducible configurations. By this device, their authors showed that a minimum counterexample cannot exist. G.D. Birkhoff proved that a minimum counterexample must satisfy a connectivity property that is referred to in modern parlance as internal 6-connectivity. We show that a minimum counterexample must also satisfy a coloring property, one that we call Kempe-locking. We define the terms Kempe-locking configuration and fundamental Kempe-locking configuration. 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. We describe a methodology for analyzing whether an arbitrary planar triangulation is Kempe-locked; it involves deconstructing the triangulation into a stack of configurations with common endpoints and then creating a bipartite graph of coloring possibilities for each configuration in the stack to assess whether certain 2-color paths can be transmitted from the configuration's top boundary to its bottom boundary. All Kempe-locked triangulations we discovered have two features in common: (1) they are Kempe-locked with respect to only a single edge, say $xy$, and (2) they have a Birkhoff diamond with endpoints $x$ and $y$ as a proper subgraph. On the strength of our various investigations, we are led to 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 to the 4-color theorem are incompatible. It would also point to the singular importance of a particularly elegant 4-connected triangulation of order 9 that consists of a triangle enclosing a pentagon enclosing a single vertex.

## Subject Areas

graph coloring; Kempe chain; Kempe-locking; Birkhoff diamond

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