论文标题
立方多面体的连接：立方体
The linkedness of cubical polytopes: The cube
论文作者
论文摘要
该论文涉及立方多面体图的链接。如果对于每组$ k $ diScoint对顶点，则具有至少$ 2K $顶点的图形为\ textIt {$ k $链接}，则有$ k $ dertex-disexchaint Paths在Pairs中加入顶点。我们说，如果其图为$ k $ - 链接，则polytope是\ textit {$ k $链接}。我们确定$ d $二维数据集为$ \ lfloor（d+1）/2 \ rfloor $ - 链接，每$ d \ ne 3 $；这是$ d $ - polytope的最大链接。该结果意味着，对于每$ d \ ge 1 $，一个立方$ d $ -polytope为$ \ lfloor {d/2} \ rfloor $ - 链接，它回答了wotzlaw \ cite {ron09}的问题。最后，我们介绍了牢固的联系的概念，该概念比连接性的概念稍强。如果它具有至少$ 2K+1 $ dertices，并且对于每个顶点$ v $ $ g $，则图$ g $是{\ us of $ k $链接}，则该子图$ g-v $ as $ k $ ins $ k $ - 链接。我们表明，立方4个聚件的$ 2 $连接，对于每个$ d \ ge 1 $，$ d $二维立方体都强烈$ \ lfloor {d/2} \ rfloor $ - 链接。
The paper is concerned with the linkedness of the graphs of cubical polytopes. A graph with at least $2k$ vertices is \textit{$k$-linked} if, for every set of $k$ disjoint pairs of vertices, there are $k$ vertex-disjoint paths joining the vertices in the pairs. We say that a polytope is \textit{$k$-linked} if its graph is $k$-linked. We establish that the $d$-dimensional cube is $\lfloor(d+1)/2\rfloor$-linked, for every $d\ne 3$; this is the maximum possible linkedness of a $d$-polytope. This result implies that, for every $d\ge 1$, a cubical $d$-polytope is $\lfloor{d/2}\rfloor$-linked, which answers a question of Wotzlaw \cite{Ron09}. Finally, we introduce the notion of strong linkedness, which is slightly stronger than that of linkedness. A graph $G$ is {\it strongly $k$-linked} if it has at least $2k+1$ vertices and, for every vertex $v$ of $G$, the subgraph $G-v$ is $k$-linked. We show that cubical 4-polytopes are strongly $2$-linked and that, for each $d\ge 1$, $d$-dimensional cubes are strongly $\lfloor{d/2}\rfloor$-linked.
