围长至少为7的稀疏图的正常无冲突可选择性
Proper conflict-free choosability of sparse graphs with girth at least seven
- 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总结：
本文证明围长至少7且最大平均度小于8/3的图可正常无冲突（度+2）选择着色，并改进平面图围长界至8。
AI中文摘要：
正常无冲突着色是一种正常顶点着色，其中每个非孤立顶点在其开邻域中都有一个颜色恰好出现一次。我们证明了每个围长至少为7且最大平均度小于8/3的有限简单图，从任意顶点列表（列表大小至少为顶点度加2）中允许正常无冲突着色。因此，每个围长至少为8的平面图都是正常无冲突（度+2）可选择的，这改进了由较早的18/7最大平均度定理得到的充分围长界9。证明使用了针对短线程（包括具有公共边界端点的线程）的局部扩展引理。双元素控制集和关联计数产生了一个加权线程不等式，该不等式在放电论证中为每个顶点发送的电荷提供了所需的界。
英文摘要：
A proper conflict-free coloring of a graph is a proper vertex coloring in which every non-isolated vertex has a color appearing exactly once in its neighborhood. A graph $G$ is proper conflict-free $(\mathrm{degree}+2)$-choosable if every list assignment $L$ with $|L(v)|\ge d_G(v)+2$ for each $v\in V(G)$ admits such a coloring from the lists. We prove that every graph with girth at least $7$ and maximum average degree less than $8/3$ is proper conflict-free $(\mathrm{degree}+2)$-choosable. Consequently, every planar graph of girth at least $8$ has this property, improving the previously established sufficient girth bound of $9$.
