Reconfiguration of vertex colouring and forbidden induced subgraphs

TL;DR

This paper studies the reconfiguration graph of vertex colourings, characterizing when it is connected based on forbidden induced subgraphs, and provides bounds on its diameter for certain graph classes.

Contribution

It characterizes the connectivity of the reconfiguration graph for $k$-colourable graphs with forbidden induced subgraphs and explores new classes where the reconfiguration graph is connected.

Findings

Connectedness characterized for $H$-free graphs with $H$ as an induced subgraph of $P_4$ or $P_3+P_1$.

Reconfiguration graph is connected with diameter at most $4n$ for ($2K_2$, $C_4$)-free graphs.

Reconfiguration graph is connected for ($P_5$, $C_4$)-free graphs.

Abstract

The reconfiguration graph of the $k$-colourings, denoted $\mathcal{R}_k(G)$, is the graph whose vertices are the $k$-colourings of $G$ and two colourings are adjacent in $\mathcal{R}_k(G)$ if they differ in colour on exactly one vertex. In this paper, we investigate the connectivity and diameter of $\mathcal{R}_{k+1}(G)$ for a $k$-colourable graph $G$ restricted by forbidden induced subgraphs. We show that $\mathcal{R}_{k+1}(G)$ is connected for every $k$-colourable $H$-free graph $G$ if and only if $H$ is an induced subgraph of $P_4$ or $P_3+P_1$. We also start an investigation into this problem for classes of graphs defined by two forbidden induced subgraphs. We show that if $G$ is a $k$-colourable ($2K_2$, $C_4$)-free graph, then $\mathcal{R}_{k+1}(G)$ is connected with diameter at most $4n$. Furthermore, we show that $\mathcal{R}_{k+1}(G)$ is connected for every $k$-colourable ($P_5$, $C_4$)-free graph $G$.