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arXiv · 1802.09230

The linkedness of cubical polytopes

Hoa Thi Bui·Guillermo Pineda-Villavicencio·Julien Ugon
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Abstract

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 $k$-linked if, for every set of $2k$ distinct vertices organised in arbitrary $k$ pairs of vertices, there are $k$ vertex-disjoint paths joining the vertices in the pairs. Larman and Mani in 1970 proved that simplicial $d$-polytopes, polytopes with all their facets being combinatorially equivalent to simplices, are $\floor{(d+1)/2}$-linked; this is the maximum possible linkedness given the facts that a $\floor{(d+1)/2}$-linked graph is at least $(2\floor{(d+1)/2}-1)$-connected and that some of these graphs are $d$-connected but not $(d+1)$-connected. Here we establish that cubical $d$-polytopes are also $\floor{(d+1)/2}$-linked for every $d\ne 3$; this is again the maximum possible linkedness for such a class of polytopes.

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Hoa Thi Bui, Guillermo Pineda-Villavicencio, Julien Ugon. 2018-02-26. The linkedness of cubical polytopes. https://arxiv.org/abs/1802.09230

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