Edge-covers in d-interval hypergraphs

TL;DR

This paper investigates edge covers in d-interval hypergraphs, extending classical results by employing topological methods, and introduces bounds on edge-covering numbers based on independence parameters, providing new insights and proofs.

Contribution

It extends the analysis of d-interval hypergraphs by bounding the edge-covering number using topological tools and offers a new proof of existing bounds.

Findings

Bound on edge-covering number in terms of independence parameters

Extension of KKM theorem to product of simplices

New proof of Tardos-Kaiser result

Abstract

A d-interval hypergraph has d disjoint copies of the unit interval as its vertex set, and each edge is the union of d subintervals, one on each copy. Extending a classical result of Gallai on the case d = 1, Tardos and Kaiser used topological tools to bound the ratio between the transversal number and the matching number in such hypergraphs. We take a dual point of view, and bound the edge-covering number (namely the minimal number of edges covering the entire vertex set) in terms of a parameter expressing independence of systems of partitions of the d unit intervals. The main tool we use is an extension of the KKM theorem to products of simplices, due to Peleg. Our approach also yields a new proof of the Tardos-Kaiser result.