Graphs of maximum average degree less than $\frac {11}{3}$ are flexibly $4$-choosable

TL;DR

This paper proves that graphs with maximum average degree less than 11/3 are flexibly 4-choosable, ensuring a significant fraction of vertices with preferred colors can be satisfied under certain list coloring conditions.

Contribution

The authors introduce a new reducible subgraph framework and vertex-partitioning tool to establish flexible list coloring results for graphs with bounded average degree.

Findings

Graphs with max average degree < 11/3 are flexibly 4-choosable.

At least 2^{-145} fraction of preferred colors can be satisfied.

New methods improve understanding of flexible list coloring in sparse graphs.

Abstract

We consider the flexible list coloring problem, in which we have a graph $G$, a color list assignment $L:V(G) \rightarrow 2^{\mathbb N}$, and a set $U \subseteq V(G)$ of vertices such that each $u \in U$ has a preferred color $p(u) \in L(u)$. Given a constant $\varepsilon > 0$, the problem asks for an $L$-coloring of $G$ in which at least $\varepsilon |U|$ vertices in $U$ receive their preferred color. We use a method of reducible subgraphs to approach this problem. We develop a vertex-partitioning tool that, when used with a new reducible subgraph framework, allows us to define large reducible subgraphs. Using this new tool, we show that if $G$ has maximum average degree less than $\frac{11}{3}$, a list $L(v)$ of size $4$ at each $v \in V(G)$, and a set $U \subseteq V(G)$ of vertices with preferred colors, then there exists an $L$-coloring of $G$ for which at least $2^{-145} |U|$ vertices of $U$ receive their preferred color.