Kempe changes in $H$-free graphs

Abstract

Given a $k$-colouring of a graph $G$ and two of the colours, a $Kempe$ $chain$ is a connected component of the subgraph of $G$ induced by the vertices coloured with one of these two colours. A $Kempe$ $swap$ changes one colouring into another by interchanging the colours of the vertices in a Kempe chain. Two colourings are $Kempe$ $equivalent$ if each can be obtained from the other by a series of Kempe swaps; the set of Kempe equivalent colourings is called a $Kempe$ $class$. For a graph $G$, let $\chi(G)$ denote its chromatic number and let $\mathcal{C}_{k}(G)$ denote the set of all $k$-colourings of $G$. We say $G$ is $Kempe$ $connected$ if for all $k\ge \chi(G)$, $\mathcal{C}_{k}(G)$ forms a Kempe class. For a graph $H$, graph $G$ is called $H$-$free$ if no induced subgraph of $G$ is isomorphic to $H$. We prove that every $H$-free graph is Kempe connected if and only if $H$ is an induced subgraph of the path on four vertices, $P_4$. The graph 2$K_2$ consists of four vertices and two edges which are not adjacent. We prove that for all $p\ge 0$, there is a $k$-colourable 2$K_2$-free graph $G$ such that $\mathcal{C}_{k+p}(G)$ does not form a Kempe class.