Reconfiguration Graphs for Vertex Colorings of $P_5$-Free Graphs

Abstract

For any positive integer $k,$ the reconfiguration graph for all $k$-colorings of a graph $G,$ denoted by $\mathcal{R}_k(G),$ is the graph where vertices represent the $k$-colorings of $G,$ and two $k$-colorings are joined by an edge if they differ in color on exactly one vertex. Bonamy et al. established that for any 2-chromatic $P_5$-free graph $G,$ $\mathcal{R}_k(G)$ is connected for each $k≥3.$ On the other hand, Feghali and Merkel proved the existence of a $7p$-chromatic $P_5$-free graph $G$ for every positive integer $p,$ such that $\mathcal{R}_{8p}(G)$ is disconnected.
In this paper, we offer a detailed classification of the connectivity of $\mathcal{R}_k(G)$ concerning $t$-chromatic $P_5$-free graphs $G$ for cases $t = 3,$ and $t ≥ 4$ with $t+1 ≤ k ≤(\begin{matrix} t \\ 2 \end{matrix}).$ We demonstrate that $\mathcal{R}_k(G)$ remains connected for each 3-chromatic $P_5$-free graph $G$ and each $k ≥4.$ Furthermore, for each $t≥4$ and $t+1\le k\le (\begin{matrix} t \\ 2 \end{matrix}),$ we provide a construction of a $t$-chromatic $P_5$-free graph $G$ with $\mathcal{R}_k(G)$ being disconnected. This resolves a question posed by Feghali and Merkel.