Edge-covers in d-interval hypergraphs
Combinatorics
2016-05-09 v1
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.
Keywords
Cite
@article{arxiv.1605.01942,
title = {Edge-covers in d-interval hypergraphs},
author = {Ron Aharoni and Ron Holzman and Shira Zerbib},
journal= {arXiv preprint arXiv:1605.01942},
year = {2016}
}