Degree-choosability of proper conflict-free list coloring of sparse graphs

TL;DR

This paper investigates the degree-choosability of proper conflict-free list coloring in sparse graphs, establishing new bounds for graphs with bounded maximum average degree and planar graphs with large girth.

Contribution

It introduces new results showing that graphs with certain maximum average degrees are proper conflict-free (degree + k)-choosable, extending understanding of coloring in sparse graphs.

Findings

Graphs with max average degree less than 10/3 are proper conflict-free (degree + 3)-choosable.

Graphs with max average degree less than 18/7 are proper conflict-free (degree + 2)-choosable.

Planar graphs with girth at least 5 and 9 are proper conflict-free (degree + 3) and (degree + 2)-choosable, respectively.

Abstract

Given a graph $G$ and a mapping $f:V(G) \to \mathbb{N}$, an $f$-list assignment of $G$ is a function that maps each $v \in V(G)$ to a set of at least $f(v)$ colors. For an $f$-list assignment $L$ of a graph $G$, a proper conflict-free $L$-coloring of $G$ is a proper coloring $\phi$ of $G$ such that for every vertex $v \in V(G)$, $\phi(v) \in L(v)$ and some appears precisely once in the neighborhood of $v$. We say that $G$ is proper conflict-free $f$-choosable if for every $f$-list assignment $L$ of $G$, there exists a proper conflict-free $L$-coloring of $G$. If $G$ is proper conflict-free $f$-choosable and there is a constant $k$ such that $f(v)= d_G(v)+k$ for every vertex $v$ of $G$, then we say $G$ is proper conflict-free $({\rm degree}+k)$-choosable. In this paper, we consider graphs with a bounded maximum average degree. We show that every graph with the maximum average degree less than $\frac{10}{3}$ is proper conflict-free $({\rm degree}+3)$-choosable, and that every graph with the maximum average degree less than $\frac{18}{7}$ is proper conflict-free $({\rm degree}+2)$-choosable. As a result, every planar graph with girth at least $5$ is proper conflict-free $({\rm degree}+3)$-choosable, and every planar graph with girth at least $9$ is proper conflict-free $({\rm degree}+2)$-choosable.