The colorful Helly theorem and general hypergraphs

Authors:
Rommel M. Barbosa;Erika M. M. Coelho;Mitre C. Dourado;Jayme L. Szwarcfiter
Affiliations:
INF - Universidade Federal de Goiás, GO, Brazil;COPPE - Universidade Federal do Rio de Janeiro, RJ, Brazil;IM - Universidade Federal do Rio de Janeiro, RJ, Brazil;IM, NCE and COPPE - Universidade Federal do Rio de Janeiro, RJ, Brazil
Venue:
European Journal of Combinatorics
Year:
2012

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Abstract

The definition of the Helly property for hypergraphs was motivated by the Helly theorem for convex sets. Similarly, we define the colorful Helly property for a family of hypergraphs, motivated by the colorful Helly theorem for collections of convex sets, by Lovasz. We describe some general facts about the colorful Helly property and prove complexity results. In particular, we show that it is Co-NP-complete to decide if a family of p hypergraphs is colorful Helly, even if p=2. However, for any fixed p, we describe a polynomial time algorithm to decide if such family is colorful Helly, provided at least p-1 of the hypergraphs are p-Helly.