图列表着色的霍尔条件的充分性
Sufficiency of Hall's Condition for Graphic List Coloring
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中文总结 AI 辅助
本文研究图列表着色中Hilton和Johnson提出的霍尔型条件的充分性，刻画了满足该条件的图$H$与$G$的类型，解答了Johnson的问题并获若干图族的完整结果。
中文摘要 AI 辅助
对于公共顶点集$V$上的有限简单图$G$和$H$，若$H$在对所有$v\in V$的列表分配$L(v)=N_G(v)$下存在正常列表着色，则称$H$是$G$-可着色的。这种利用同一顶点集上另一图的邻域对图进行着色的概念，我们称之为图列表着色，它与多个经典主题存在关联，包括相异代表系和图因子分解。本文中，我们研究由Hilton和Johnson于1990年提出的必要霍尔型条件何时对$H$是$G$-可着色也充分。我们刻画了所有满足：只要对$(H,G)$满足霍尔条件，$H$就是$G$-可着色的图$H$，解答了Johnson提出的问题。随后我们考虑对偶问题，即刻画满足：只要$(H,G)$满足霍尔条件，$H$就是$G$-可着色的图$G$。在这一思路下，我们获得了若干图族的完整结果，比如森林、完全多部图和网格图。
英文摘要
For finite simple graphs $G,H$ on a common vertex set $V$, we say that $H$ is $G$-colorable if $H$ admits a proper list coloring with list assignment $L(v)=N_G(v)$ for all $v\in V$. This notion of coloring a graph using the neighborhood of another graph on the same vertex set, which we call \emph{graphic list coloring}, has connections to several classical topics, including systems of distinct representatives and graph factorizations. In this paper, we investigate when a necessary Hall-type condition, introduced by Hilton and Johnson in 1990, is also sufficient for $H$ to be $G$-colorable. We characterize all graphs $H$ that are $G$-colorable whenever the pair $(H,G)$ satisfies Hall's condition, answering a question raised by Johnson. We then consider the dual problem of characterizing graphs $G$ such that, whenever $(H,G)$ satisfies Hall's condition, $H$ is $G$-colorable. In this vein, we obtain complete results for several families of graphs, such as forests, complete multipartite graphs, and grid graphs.
