On the graph-connectivity of skeleta of convex polytopes

TL;DR

This paper investigates the connectivity properties of graphs formed from faces of convex polytopes, generalizing Balinski's theorem to higher-dimensional face graphs and determining their vertex connectivity bounds.

Contribution

It extends Balinski's theorem by analyzing the vertex connectivity of face-based graphs of convex polytopes for various dimensions and face levels.

Findings

Determines the maximum vertex connectivity of face graphs for all convex polytopes.

Generalizes Balinski's theorem to higher-dimensional face graphs.

Provides exact bounds for connectivity based on polytope dimension and face level.

Abstract

Given a $d$-dimensional convex polytope $P$ and nonnegative integer $k$ not exceeding $d-1$, let $G_k (P)$ denote the simple graph on the node set of $k$-dimensional faces of $P$ in which two such faces are adjacent if there exists a $(k+1)$-dimensional face of $P$ which contains them both. The graph $G_k (P)$ is isomorphic to the dual graph of the $(d-k)$-dimensional skeleton of the normal fan of $P$. For fixed values of $k$ and $d$, the largest integer $m$ such that $G_k (P)$ is $m$-vertex-connected for all $d$-dimensional polytopes $P$ is determined. This result generalizes Balinski's theorem on the one-dimensional skeleton of a $d$-dimensional convex polytope.