Linkages in Polytope Graphs

TL;DR

This paper improves bounds on the linkedness of polytopes, providing exact values for certain dimensions and analyzing minimal linkedness for polytopes with specific vertex counts, including constructing examples that meet these bounds.

Contribution

It offers new lower bounds on polytope linkedness, determines exact minimal linkedness for 7-, 10-, and 13-dimensional polytopes, and constructs examples meeting these bounds.

Findings

Exact minimal linkedness for 7-, 10-, and 13-dimensional polytopes.

Sharp lower bounds for polytopes with up to (6d+7)/5 vertices.

Construction of examples meeting the derived bounds.

Abstract

A graph is k-linked if any k disjoint vertex-pairs can be joined by k disjoint paths. We improve a lower bound on the linkedness of polytopes slightly, 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, a sharp lower bound on minimal linkedness is derived, and examples meeting this lower bound are constructed. These examples contain a class of examples due to Gallivan.