Kempe Equivalence of Colourings of Cubic Graphs

TL;DR

This paper proves that for most cubic graphs, all 3-colourings are Kempe equivalent, except for the complete graph K4 and the triangular prism, confirming a special case of Mohar's conjecture.

Contribution

It establishes that all 3-colourings of cubic graphs are Kempe equivalent except for two specific graphs, advancing understanding of Kempe equivalence in regular graphs.

Findings

All 3-colourings of cubic graphs are Kempe equivalent.

The exceptions are the complete graph K4 and the triangular prism.

This confirms a case of Mohar's conjecture for k=3.

Abstract

Given a graph $G=(V,E)$ and a proper vertex colouring of $G$, a Kempe chain is a subset of $V$ that induces a maximal connected subgraph of $G$ in which every vertex has one of two colours. To make a Kempe change is to obtain one colouring from another by exchanging the colours of vertices in a Kempe chain. Two colourings are Kempe equivalent if each can be obtained from the other by a series of Kempe changes. A conjecture of Mohar asserts that, for $k \geq 3$, all $k$-colourings of $k$-regular graphs that are not complete are Kempe equivalent. We address the case $k=3$ by showing that all $3$-colourings of a cubic graph $G$ are Kempe equivalent unless $G$ is the complete graph $K_4$ or the triangular prism.