The linkedness of cubical polytopes: beyond the cube

TL;DR

This paper improves the understanding of the linkedness properties of cubical polytopes, showing they are more highly linked than previously known, with results that are optimal for this class.

Contribution

It establishes that cubical d-polytopes are loor{(d+1)/2}-linked and strongly loor{d/2}-linked, strengthening prior results and proving optimal bounds for these properties.

Findings

Cubical d-polytopes are loor{(d+1)/2}-linked for all d ≠ 3.

Cubical d-polytopes are strongly loor{d/2}-linked for all d ≠ 3.

Results are proven to be optimal for cubical polytopes.

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 \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.