On a conjecture of Mohar concerning Kempe equivalence of regular graphs

TL;DR

This paper proves Mohar's conjecture that all proper k-colourings of a non-complete k-regular graph are Kempe equivalent for all k ≥ 4, and discusses implications for the antiferromagnetic Potts model.

Contribution

It establishes the validity of Mohar's conjecture for all k ≥ 4, extending previous results for k=3.

Findings

Mohar's conjecture holds for all k ≥ 4.

All proper k-colourings of such graphs are Kempe equivalent.

Implications for the Wang-Swendsen-Kotecký algorithm at zero-temperature.

Abstract

Let $G$ be a graph with a vertex colouring $\alpha$. 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 $\alpha$ 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-Koteck\'{y} algorithm for the antiferromagnetic Potts model at zero-temperature.