Flexible list coloring of graphs with maximum average degree less than $3$

Abstract

In the flexible list coloring problem, we consider a graph $G$ and a color list assignment $L$ on $G$, as well as a subset $U \subseteq V(G)$ for which each $u \in U$ has a preferred color $p(u) \in L(u)$. Our goal is to find a proper $L$-coloring $\phi$ of $G$ such that $\phi(u) = p(u)$ for at least $\epsilon|U|$ vertices $u \in U$. We say that $G$ is $\epsilon$-flexibly $k$-choosable if for every $k$-size list assignment $L$ on $G$ and every subset of vertices with coloring preferences, $G$ has a proper $L$-coloring that satisfies an $\epsilon$ proportion of these coloring preferences. Dvo\v{r}\'ak, Norin, and Postle [Journal of Graph Theory, 2019] asked whether every $d$-degenerate graph is $\epsilon$-flexibly $(d+1)$-choosable for some constant $\epsilon = \epsilon(d) > 0$. In this paper, we prove that there exists a constant $\epsilon > 0$ such that every graph with maximum average degree less than $3$ is $\epsilon$-flexibly $3$-choosable, which gives a large class of $2$-degenerate graphs which are $\epsilon$-flexibly $(d+1)$-choosable. In particular, our results imply a theorem of Dvo\v{r}\'ak, Masa\v{r}\'ik, Mus\'ilek, and Pangr\'ac [Journal of Graph Theory, 2020] stating that every planar graph of girth $6$ is $\epsilon$-flexibly $3$-choosable for some constant $\epsilon > 0$. To prove our result, we generalize the existing reducible subgraph framework traditionally used for flexible list coloring to allow reducible subgraphs of arbitrarily large order.