Year: 2026
Type: preprint
Abstract: For a graph $G$, a proper coloring of $G$ is called proper conflict-free if for every non-isolated vertex $u$, there is at least one color appearing exactly once in $N_G(u)$. A graph $G$ is proper conflict-free $f$-choosable if for every list assignment $L$ with $|L(v)|\ge f(v)$ for each vertex $v$, $G$ admits a proper conflict-free $L$-coloring. Recently, Kashima, Škrekovski, and Xu proposed a conjecture on proper conflict-free list coloring. For a graph $G$, let $κ_G:V(G)\to \mathbb{N}$ be defined by \[ κ_G(v)= \begin{cases} 4, & \text{if } d_G(v)=2,\\[4pt] d_G(v)+1, & \text{if } d_G(v)\neq 2. \end{cases} \] They conjectured that every connected graph other than $C_5$ is proper conflict-free $κ_G$-choosable. In this paper, we confirm this conjecture in two classes of graphs with bounded average degree, thereby generalizing results of Kashima, Škrekovski, and Xu and of Wang and Zhang. We prove that every connected graph $G\neq C_5$ with either $\operatorname{mad}(G)<\frac{12}{5}$ or $Δ(G)\le3$ is proper conflict-free $κ_G$-choosable. To prove these results, we introduce a method based on systems of proper conflict-free representatives and develop a construction of auxiliary graphs that preserves the maximum average degree bound.
Source: arXiv (Cornell University)
Authors: Lu, Zhijun, Ying, Qirui, Song, Huimin
Institution: Shandong University
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Awards: G3893774329, G8452174313
OpenAlex ID: W7204940856