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

TL;DR

The paper addresses flexible list coloring and proves that every graph with maximum average degree mad$<3$ is weighted $ε$-flexibly $3$-choosable for some explicit $ε>0$, thereby covering a large class of $2$-degenerate graphs. It introduces a generalized reducible subgraph framework that allows reducible subgraphs of arbitrary size and uses a probabilistic $(k,ε,α)$-distribution approach to establish flexible choosability via a reduction-closure argument. A comprehensive toolkit for identifying reducible subgraphs is developed, including handling of blocks, terminal blocks, and special configurations like diamonds and certain subgraphs, with paths and Gallai-tree considerations. The main result implies, in particular, that planar graphs of girth at least $6$ are $ε$-flexibly $3$-choosable, and the authors also perform a discharging analysis to support the structural claims, leading to a robust framework for future extensions in sparse graph classes.

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 $φ$ of $G$ such that $φ(u) = p(u)$ for at least $ε|U|$ vertices $u \in U$. We say that $G$ is $ε$-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 $ε$ proportion of these coloring preferences. Dvořák, Norin, and Postle [Journal of Graph Theory, 2019] asked whether every $d$-degenerate graph is $ε$-flexibly $(d+1)$-choosable for some constant $ε= ε(d) > 0$. In this paper, we prove that there exists a constant $ε> 0$ such that every graph with maximum average degree less than $3$ is $ε$-flexibly $3$-choosable, which gives a large class of $2$-degenerate graphs which are $ε$-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 $ε$-flexibly $3$-choosable for some constant $ε> 0$. 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.

Figures (1)

• Figure 1: The figure shows the graphs $H_5$ and $H_7$, which are $(3,\varepsilon,\alpha)$-reducible under certain conditions.

Theorems & Definitions (57)

• Theorem 1.3: DMMP6
• Theorem 1.5
• Theorem 1.6
• Lemma 2.1
• Definition 2.2
• Lemma 2.3
• proof
• Definition 2.4
• Lemma 2.5
• proof