Reconfiguration graphs for vertex colorings of $P_5$-free graphs

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 $k\geq 3$. 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 $t\geq4$ with $t+1\leq k \leq {t\choose2}$. We demonstrate that $\mathcal{R}_k(G)$ remains connected for each $3$-chromatic $P_5$-free graph $G$ and each $k \geq 4$. Furthermore, for each $t\geq4$ and $t+1 \leq k \leq {t\choose2}$, 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.