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 $\varphi$ of $G$ such that $\varphi (u)=p(u)$ 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. Dvořák, 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 Dvořák, Masařík, Musílek, and Pangrác [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.

arXiv comment23 pages (Lemma 3.7 updated)