Mixing colourings in $2K_2$-free graphs

TL;DR

This paper characterizes the connectedness of the reconfiguration graph of colourings in certain graph classes, constructing a specific example of a 2K2-free graph with a frozen 8-colouring.

Contribution

It completes the classification of when the reconfiguration graph is connected for graphs excluding a fixed path, by providing a counterexample in the case of 2K2-free graphs.

Findings

Constructed a 7-chromatic 2K2-free graph with a frozen 8-colouring.

Shows the reconfiguration graph can be disconnected in certain graph classes.

Settles an open question about colourings in $2K_2$-free graphs.

Abstract

The reconfiguration graph for the $k$-colourings of a graph $G$, denoted $R_{k}(G)$, is the graph whose vertices are the $k$-colourings of $G$ and two colourings are joined by an edge if they differ in colour on exactly one vertex. For any $k$-colourable $P_4$-free graph $G$, Bonamy and Bousquet proved that $R_{k+1}(G)$ is connected. In this short note, we complete the classification of the connectedness of $R_{k+1}(G)$ for a $k$-colourable graph $G$ excluding a fixed path, by constructing a $7$-chromatic $2K_2$-free (and hence $P_5$-free) graph admitting a frozen $8$-colouring. This settles a question of the second author.