List-recoloring of two classes of planar graphs

Abstract

For a graph $G$ with a list assignment $L$ and two $L$-colorings $\alpha$ and $\beta$, an $L$-recoloring sequence from $\alpha$ to $\beta$ is a sequence of proper $L$-colorings where consecutive colorings differ at exactly one vertex. We prove the existence of such a recoloring sequence in which every vertex is recolored at most a constant number of times under two conditions: (i) $G$ is planar, contains no $3$-cycles or intersecting $4$-cycles, and $L$ is a $6$-assignment; or (ii) the maximum average degree of $G$ satisfies $\mathrm{mad}(G) < \frac{5}{2}$ and $L$ is a $4$-assignment. These results strengthen two theorems previously established by Cranston.