Optimal List Recoloring of Subcubic Graphs and Complete Multipartite Graphs

Abstract: For a list-assignment $L$, the reconfiguration graph $C_L(G)$ of a graph $G$ is the graph whose vertices are proper $L$-colorings of $G$ and whose edges link two colorings that differ on only one vertex. If $|L(v)| \ge d(v) + 2$ for every vertex of $G$, it is known that $C_L(G)$ is connected. In this case, Cambie et al. investigated the diameter of $C_L(G)$. They conjectured that $diam(C_L(G)) \le n(G) + \mu(G)$ with $\mu(G)$ the size of a maximum matching of $G$ and proved several results towards this conjecture. We answer to two of their open problems by proving the conjecture for two classes of graphs, namely subcubic graphs and complete multipartite graphs.