List-Recoloring of Sparse Graphs

Fix a graph $G$, a list-assignment $L$ for $G$, and $L$-colorings $\alpha$ and $\beta$. An $L$-recoloring sequence, starting from $\alpha$, recolors a single vertex at each step, so that each resulting intermediate coloring is a proper $L$-coloring. An $L$-recoloring sequence transforms $\alpha$ to $\beta$ if its initial coloring is $\alpha$ and its final coloring is $\beta$. We prove there exists an $L$-recoloring sequence that transforms $\alpha$ to $\beta$ and recolors each vertex at most a constant number of times if (i) $G$ is triangle-free and planar and $L$ is a 7-assignment, or (ii) $\mathrm{mad}(G)<17/5$ and $L$ is a 6-assignment or (iii) $\mathrm{mad}(G)<22/9$ and $L$ is a 4-assignment. Parts (i) and (ii) confirm conjectures of Dvo\v{r}\'{a}k and Feghali.