无爪图和多面体映射的圈可穿越性
Cycle Traversability for Claw-free Graphs and Polyhedral Maps
摘要：
设$G$是一个图，$v\in V(G)$和$S\subseteq V(G)\backslash v$的大小至少为$k$。关于图连通性的一个重要结果是，如果$v$和$S$是$k$ -链接的，则顶点$v$和$S$之间的$(k-1)$ -链接可以扩展为$v$和$S$之间的$k$ -链接，使得$(k-1)$ -链接的端点也是$k$ -链接的端点.我们首先证明一个推广完美的结果表明，如果两个不相交的集合$S_1$和$S_2$是$k$ -链接，然后一个$t$链接（$t< k$）在两个不相交的集合$S_1$之间MT和$S_2$之间的$k$链路可以扩展到$S_1$和$S_2$之间的$k$链路，使得端$t$链接的顶点保留在$k$链接中。其次，我们可以利用这些结果证明，一个3连通无爪图总是有一个圈通过任何给定的五个顶点，但避免任何其他一个指定的顶点。我们还表明，这一结果是尖锐的，展示了一个无限的家庭3-连通无爪图，其中没有循环包含一定的一组六个顶点，但避免第七个指定的顶点。我们的主要结果的一个直接推论表明，一个3-连通无爪图有一个拓扑轮子式$W_k$，其中$k\le 5$当且仅当它有一个度至少为$k$的顶点。最后，我们还表明，一个图多面体嵌入在一个表面总是有一个循环通过任何给定的三个顶点，但避免任何其他指定的顶点。结果是最好的可能的意义上说，多面体嵌入假设是必要的，有无限多个图多面体嵌入任何表面没有循环包含一定的一组四个顶点，但避免第五个指定的顶点。
英文摘要：
Let $G$ be a graph, and $v\in V(G)$ and $S\subseteq V(G)\backslash v$ of size at least $k$. An important result on graph connectivity due to Perfect states that, if $v$ and $S$ are $k$-linked, then a $(k-1)$-link between a vertex $v$ and $S$ can be extended to a $k$-link between $v$ and $S$ such that the endvertices of the $(k-1)$-link are also the endvertices of the $k$-link. We begin by proving a generalization of Perfect's result by showing that, if two disjoint sets $S_1$ and $S_2$ are $k$-linked, then a $t$-link ($t< k$) between two disjoint sets $S_1$ and $S_2$ can be extended to a $k$-link between $S_1$ and $S_2$ such that the endvertices of the $t$-link are preserved in the $k$-link. Next, we are able to use these results to show that a 3-connected claw-free graph always has a cycle passing through any given five vertices but avoiding any other one specified vertex. We also show that this result is sharp by exhibiting an infinite family of 3-connected claw-free graphs in which there is no cycle containing a certain set of six vertices but avoiding a seventh specified vertex. A direct corollary of our main result shows that, a 3-connected claw-free graph has a topological wheel minor $W_k$ with $k\le 5$ if and only if it has a vertex of degree at least $k$. Finally, we also show that a graph polyhedrally embedded in a surface always has a cycle passing through any given three vertices but avoiding any other specified vertex. The result is best possible in the sense that the polyhedral embedding assumption is necessary, and there are infinitely many graphs polyhedrally embedded in any surface having no cycle containing a certain set of four vertices but avoiding a fifth specified vertex.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
