论文标题
立方多面体的连接：超越立方体
The linkedness of cubical polytopes: beyond the cube
论文作者
论文摘要
立方多型是多层室，其所有方面在组合上等效于立方体。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 $链接}。 在上一篇论文\ cite {buipinugo20a}中，我们证明了每个立方$ d $ -polytope is $ \ floor {d/2} $ - 链接。在这里，我们通过建立$ \ floor {（d+1）/2} $ - 立方体$ d $ - polytopes的链接，以加强这种结果。 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。 我们说，如果其图为（强）（强）$ k $ - 链接，则polytope是（强烈）\ textit {$ k $链接}。在本文中，我们还证明，每个$ d \ ne 3 $链接的每个立方体$ d $ - polytope均为$ \ floor {d/2} $ - 链接。 对于这类多型，这些结果最好。
A cubical polytope is a polytope with all its facets being combinatorially equivalent to cubes. 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. In a previous paper \cite{BuiPinUgo20a} we proved that every cubical $d$-polytope is $\floor{d/2}$-linked. Here we strengthen this result by establishing the $\floor{(d+1)/2}$-linkedness of cubical $d$-polytopes, for every $d\ne 3$. 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 say that a polytope is (strongly) \textit{$k$-linked} if its graph is (strongly) $k$-linked. In this paper, we also prove that every cubical $d$-polytope is strongly $\floor{d/2}$-linked, for every $d\ne 3$. These results are best possible for this class of polytopes.
