Rainbow Fractional Matchings

Abstract

We prove that any family E1,..., Ern 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 rnr+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.

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Acknowledgments

We are grateful to Dani Kotlar, Roy Meshulam and Ran Ziv for helpful discussions.

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Correspondence to Ron Aharoni or Ron Holzman or Zilin Jiang.

Additional information

Supported in part by the United States–Israel Binational Science Foundation (BSF) grant no. 2006099, the Israel Science Foundation (ISF) grant no. 2023464 and the Discount Bank Chair at the Technion.

Supported in part by ISF grant no. 409/16.

The work was done when Z. Jiang was a postdoctoral fellow at Technion–Israel Institute of Technology, and was supported in part by ISF grant nos. 409/16, 936/16.

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Aharoni, R., Holzman, R. & Jiang, Z. Rainbow Fractional Matchings. Combinatorica 39, 1191–1202 (2019). https://doi.org/10.1007/s00493-019-4019-y

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Mathematics Subject Classification (2010)

  • 05D15
  • 55U10