The Linkedness of Cubical Polytopes: The Cube

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 $k$ disjoint pairs of vertices, there are $k$ vertex-disjoint paths joining the vertices in the pairs. We say that a polytope is $k$-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 $d\geqslant 1$, a cubical $d$-polytope is $\lfloor d/2\rfloor$-linked, which answers a question of Wotzlaw (Incidence graphs and unneighborly polytopes, Ph.D. thesis, 2009). Finally, we introduce the notion of strong linkedness, which is slightly stronger than that of linkedness. A graph $G$ is 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 $d\geqslant 1$, $d$-dimensional cubes are strongly $\lfloor d/2\rfloor$-linked.