AI 中文总结
研究无标号图的列表染色函数，针对Kaul和Mudrock提出的在无标号情况下Donner结果的类似情况是否成立的问题，证明了\(n\)个顶点无边图的相关猜想，还表明若连通分量满足，则非连通图也满足该结果的无标号类似情况。
AI 中文摘要
给定一个图\(G\)，其色多项式\(P (G, k)\)计算恰当的\(k -\)染色数，而相应的列表染色函数\(P_{\ell} (G, k)\)计算在给每个顶点分配\(k\)种颜色的所有分配方式下恰当染色的最小数量。显然\(P_{\ell} (G, k) \leq P (G, k)\)，1992年Donner表明当\(k\)足够大时\(P_{\ell} (G, k) = P (G, k)\)。1985年Hanlon定义并研究了无标号图的色多项式。2024年Kaul和Mudrock引入了Hanlon概念的列表版本，并提出在无标号情况下Donner结果的类似情况是否成立的问题。他们证明了所有连通点确定图的情况，\(n\)个顶点的无边图情况仍未解决并被作为猜想提出。我们证明了这个猜想，并表明更一般地，如果一个非连通图的所有连通分量都满足，那么该非连通图满足Donner结果的无标号类似情况。
英文摘要
Given a graph $G$, its chromatic polynomial $P (G, k)$ counts proper $k$-colorings, while the corresponding list color function $P_{\ell} (G, k)$ counts the minimum number of proper colorings across all assignments of $k$ colors to each vertex. While it is clear that $P_{\ell} (G, k) \leq P (G, k)$, Donner showed in 1992 that $P_{\ell} (G, k) = P (G, k)$ whenever $k$ is sufficiently large. In 1985, Hanlon defined and studied the chromatic polynomial for an unlabeled graph. A list version of Hanlon's notion was introduced in 2024 by Kaul and Mudrock, who further raised the question of whether the analog of Donner's result holds in the unlabeled case. While they proved this for all connected point-determining graphs, even the case of the edgeless graph on $n$ vertices remained open and was posed as a conjecture. We prove this conjecture and show that it implies that, more generally, a disconnected graph satisfies the unlabeled analog of Donner's result if all of its connected components do.
Comments11 pages
