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arXiv · 1510.06964

On a conjecture of Mohar concerning Kempe equivalence of regular graphs

Marthe Bonamy·Nicolas Bousquet·Carl Feghali·Matthew Johnson
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Abstract

Let $G$ be a graph with a vertex colouring $α$. Let $a$ and $b$ be two colours. Then a connected component of the subgraph induced by those vertices coloured either $a$ or $b$ is known as a Kempe chain. A colouring of $G$ obtained from $α$ by swapping the colours on the vertices of a Kempe chain is said to have been obtained by a Kempe change. Two colourings of $G$ are Kempe equivalent if one can be obtained from the other by a sequence of Kempe changes. A conjecture of Mohar (2007) asserts that, for $k \geq 3$, all $k$-colourings of a $k$-regular graph that is not complete are Kempe equivalent. It was later shown that all $3$-colourings of a cubic graph that is neither $K_4$ nor the triangular prism are Kempe equivalent. In this paper, we prove that the conjecture holds for each $k\geq 4$. We also report the implications of this result on the validity of the Wang-Swendsen-Kotecký algorithm for the antiferromagnetic Potts model at zero-temperature.

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Marthe Bonamy, Nicolas Bousquet, Carl Feghali, Matthew Johnson. 2016-09-22. On a conjecture of Mohar concerning Kempe equivalence of regular graphs. https://arxiv.org/abs/1510.06964

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