  The reconfiguration graph of the $k$-colorings, denoted $R_k(G)$, is the
graph whose vertices are the $k$-colorings of $G$ and two colorings are
adjacent in $R_k(G)$ if they differ in color on exactly one vertex. A graph $G$
is said to be recolorable if $R_{\ell}(G)$ is connected for all $\ell\geq
\chi(G)$+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.
