Linkages in Polytope Graphs

A graph is k-linked if any k disjoint vertex-pairs can be joined by k disjoint paths. We slightly improve a lower bound on the linkedness of polytopes, which results in exact values for the minimal linkedness of 7-, 10- and 13-dimensional polytopes.We analyze in detail linkedness of polytopes on at most (6d+7)/5 vertices. In that case, we derive a sharp lower bound on minimal linkedness, and construct examples meeting this lower bound. These examples contain a class of examples due to Gallivan.