On the hypergraph connectivity of skeleta of polytopes

TL;DR

This paper proves a strong connectivity property of hypergraphs formed by faces of polytopes, showing that for any dimension, these face-based hypergraphs are highly vertex-connected.

Contribution

It establishes a universal connectivity result for hypergraphs of polytope faces, extending understanding of polytope face structures.

Findings

Hypergraphs of polytope faces are strongly (d-k)-vertex connected.

Connectivity holds for all 0 ≤ k ≤ d-1.

Connectivity depends only on the dimension and face levels.

Abstract

We show that for every $d$-dimensional polytope, the hypergraph whose nodes are $k$-faces and whose hyperedges are $(k+1)$-faces of the polytope is strongly $(d-k)$-vertex connected, for each $0 \leq k \leq d- 1$.