Bounds for the collapsibility number of a simplicial complex and non-cover complexes of hypergraphs

TL;DR

This paper establishes bounds on the collapsibility number of simplicial complexes, especially for non-cover complexes of hypergraphs, extending previous graph-based results to hypergraphs and introducing a sequence of bounds that can be tight.

Contribution

It extends bounds for collapsibility numbers from graphs to hypergraphs and introduces a new sequence of bounds that can be tight for certain complexes.

Findings

Non-cover complex of a hypergraph is (|V(H)| - γ_i(H) - 1)-collapsible.

The new bounds generalize and include previous upper bounds as special cases.

The sequence of bounds al M_k(X) can be tight for k-vertex decomposable complexes.

Abstract

The collapsibility number of simplicial complexes was introduced by Wegner in order to understand the intersection patterns of convex sets. This number also plays an important role in a variety of Helly type results. We show that the non-cover complex of a hypergraph $\mathcal{H}$ is $|V(\mathcal{H)}|- \gamma_i(\mathcal{H})-1$-collapsible, where $\gamma_i(\mathcal{H})$ is the generalization of independence domination number of a graph to hypergraph. This extends the result of Choi, Kim and Park from graphs to hypergraphs. Moreover, the upper bound in terms of strong independence domination number given by Kim and Kim for the Leray number of the non-cover complex of a hypergraph can be obtained as a special case of our result. In general, there can be a large gap between the collapsibility number of a complex and its well-known upper bounds. In this article, we construct a sequence of upper bounds $\mathcal{M}_k(X)$ for the collapsibility number of a simplicial complex $X$, which lie in this gap. We also show that the bound given by $\mathcal{M}_k$ is tight if the underlying complex is $k$-vertex decomposable.