Recoloring some hereditary graph classes

The reconfiguration graph of the $k$-colorings, denoted $R_k\left(G\right)$, is the graph whose vertices are the $k$-colorings of $G$ and two colorings are adjacent in $R_k\left(G\right)$ if they differ in color on exactly one vertex. A graph $G$ is said to be recolorable if $R_{\ell }\left(G\right)$ is connected for all $\ell \ge \chi \left(G\right)+1$. In this paper, we study the recolorability of several graph classes restricted by forbidden induced subgraphs. We prove some properties of a vertex-minimal graph $G$ which is not recolorable. We show that every (triangle, $H$)-free graph is recolorable if and only if every (paw, $H$)-free graph is recolorable. Every graph in the class of $(2K_2,H)$-free graphs, where $H$ is a 4-vertex graph except $P_4$ or $P_3+P_1$, is recolorable if $H$ is either a triangle, paw, claw, or diamond. Furthermore, we prove that every ($P_5$, $C_5$, house, co-banner)-free graph is recolorable.