The linkedness of cubical polytopes: beyond the cube

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 k vertices is k/2 -linked if, for every set of 2 k/2 distinct vertices organised in arbitrary k/2 unordered pairs of vertices, there are k/2 vertex-disjoint paths joining the vertices in the pairs. In a previous paper [2] we proved that every cubical d-polytope is d/2 -linked. Here we strengthen this result by establishing the (d + 1)/2 -linkedness of cubical d-polytopes, for every d = 3.A graph is strongly k/2 -linked if it has at least k vertices and, for every set X of exactly k vertices organised in arbitrary k/2 unordered pairs of vertices, there are k/2 vertex-disjoint paths joining the vertices in the pairs and avoiding the vertices in X not being paired. We say that a polytope is (strongly) k/2 -linked if its graph is (strongly) k/2 -linked. In this paper, we also prove that every cubical d-polytope is strongly (d + 1)/2 -linked, for every d = 3. These results are best possible for such a class of polytopes.