发表机构
School of Mathematics and Statistics, Shandong University（山东大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对正常无冲突列表着色猜想，通过引入正常无冲突代表系方法与保最大平均度界的辅助图构造，在两类有界平均度图中证实了该猜想，推广了已有结果。
AI 中文摘要
对于图$G$，若$G$的一个正常着色被称为正常无冲突着色，当且仅当对于每个非孤立顶点$u$，至少存在一种颜色在$N_G(u)$中恰好出现一次。若对于每个顶点$v$满足$|L(v)|\f(v)$的列表指派$L$，$G$都存在一个正常无冲突$L$-着色，则称图$G$是正常无冲突$f$-可选的。\n 近期，Kashima、Škrekovski和Xu提出了一个关于正常无冲突列表着色的猜想。对于图$G$，定义函数$\u03ba_G:V(G)\to \n$为：当$d_G(v)=2$时$\u03ba_G(v)=4$，当$d_G(v)\neq 2$时$\u03ba_G(v)=d_G(v)+1$。他们猜想，除$C_5$外的每个连通图都是正常无冲突$\u03ba_G$-可选的。\n 本文在两类具有有界平均度的图中证实了该猜想，从而推广了Kashima、Škrekovski、Xu以及Wang和Zhang的结果。我们证明，每个满足$\n{mad}(G)<\frac{12}{5}$或$\u0394(G)\f3$且$G\neq C_5$的连通图都是正常无冲突$\u03ba_G$-可选的。为证明这些结果，我们引入了一种基于正常无冲突代表系的方法，并构建了保持最大平均度界的辅助图。
英文摘要
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.
Comments27 pages, 9 figures
