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

TL;DR

This work investigates proper conflict-free list colorings in relation to degree-based choosability on sparse graphs. It introduces f-list assignments with $f(v)=d_G(v)+k$ and proves two main results using discharging: (i) if ${\rm mad}(G)<\tfrac{10}{3}$, then $G$ is proper conflict-free $(\deg+3)$-choosable; (ii) if ${\rm mad}(G)<\tfrac{18}{7}$, then $G$ is proper conflict-free $(\deg+2)$-choosable (for connected graphs not equal to $C_5$). The paper further derives planar graph corollaries under girth restrictions and provides a suite of auxiliary results on CF-choosability for small graphs and Theta-graphs to support the discharging arguments. The methods hinge on identifying forbidden configurations in minimal counterexamples and applying careful discharging to contradict the assumed maximum average degree bounds. Overall, the results advance understanding of conflict-free degree-based list coloring in sparse and planar graphs and point to further avenues for tightening constants and broadening classes of graphs covered.

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 $φ$ of $G$ such that for every vertex $v \in V(G)$, $φ(v) \in L(v)$ 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.

Theorems & Definitions (46)

• Conjecture 1.1: KSX2026
• Conjecture 1.2
• Conjecture 1.4: KSX2025-tree
• Theorem 1.5
• Theorem 1.6
• Corollary 1.7
• Claim 2.1
• proof
• Claim 2.2
• proof