Hypergraph Versals

TL;DR

This paper investigates the properties of hypergraph versals, establishing lower bounds on their number under certain conditions and identifying specific exceptions where the bounds differ.

Contribution

It proves that hypergraphs with no edge containing another have at least n+1 versals, except for specific small hypergraphs, extending the understanding of hypergraph structures.

Findings

Hypergraphs without nested edges have at least n+1 versals.

Exceptions occur for the set of all singletons, their complements, and the 4-cycle graph.

The results relate to a special case of the Isolation Lemma with two-value labels.

Abstract

Let H be a hypergraph on n vertices with the property that no edge contains another. We prove some results for a special case of the Isolation Lemma when the label set for the edges of H can only take two values. Given any set of vertices S and an edge e, the weight of S in e is the size of e plus the size of the intersection of S and e. A versal S for an edge e is a set of vertices with weight in e smaller than the weight in any other edge. We show that H always has at least n + 1 versals except if H is either the set of all singletons T_n or the complement of T_n or the 4-cycle graph. In those exceptional cases there are only n versals.