Rainbow Fractional Matchings

Ron Aharoni, Ron Holzman, Zilin Jiang

Research output: Contribution to journalArticlepeer-review

Abstract

We prove that any family E1,.., E┌rn┐ of (not necessarily distinct) sets of edges in an r-uniform hypergraph, each having a fractional matching of size n, has a rainbow fractional matching of size n (that is, a set of edges from distinct Ei’s which supports such a fractional matching). When the hypergraph is r-partite and n is an integer, the number of sets needed goes down from rn to rn−r+1. The problem solved here is a fractional version of the corresponding problem about rainbow matchings, which was solved by Drisko and by Aharoni and Berger in the case of bipartite graphs, but is open for general graphs as well as for r-partite hypergraphs with r>2. Our topological proof is based on a result of Kalai and Meshulam about a simplicial complex and a matroid on the same vertex set.

Original languageEnglish (US)
Pages (from-to)1191-1202
Number of pages12
JournalCombinatorica
Volume39
Issue number6
DOIs
StatePublished - Dec 1 2019
Externally publishedYes

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Computational Mathematics

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