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Rainbow Fractional Matchings

  • Ron AharoniEmail author
  • Ron HolzmanEmail author
  • Zilin JiangEmail author
Article
  • 26 Downloads

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.

Mathematics Subject Classification (2010)

05D15 55U10 

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Notes

Acknowledgments

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

References

  1. [1]
    R. Aharoni and E. Berger: Rainbow matchings in r-partite r-graphs, Electron. J. Combin. 16, Research Paper 119, (2009).Google Scholar
  2. [2]
    R. Aharoni, E. Berger, M. Chudnovsky, D. Howard and P. Seymour: Large rainbow matchings in general graphs, European J. Combin. 79 (2019), 222–227.MathSciNetCrossRefGoogle Scholar
  3. [3]
    R. Aharoni, D. Kotlar and R. Ziv: Uniqueness of the extreme cases in theorems of Drisko and Erdős-Ginzburg-Ziv, European J. Combin. 67 (2018), 222–229.MathSciNetCrossRefGoogle Scholar
  4. [4]
    N. Alon: Multicolored matchings in hypergraphs, Mosc. J. Comb. Number Theory 1 (2011), 3–10.MathSciNetzbMATHGoogle Scholar
  5. [5]
    N. Alon, G. Kalai, J. Matoušsek and R. Meshulam: Transversal numbers for hypergraphs arising in geometry, Adv. in Appl. Math. 29 (2002), 79–101.MathSciNetCrossRefGoogle Scholar
  6. [6]
    I. Bárány: A generalization of Carathéodory’s theorem, Discrete Math. 40 (1982), 141–152.MathSciNetCrossRefGoogle Scholar
  7. [7]
    J. Barát, A. Gyárfás and G. N. Sárközy: Rainbow matchings in bipartite multigraphs, Period. Math. Hungar. 74 (2017), 108–111.MathSciNetCrossRefGoogle Scholar
  8. [8]
    A. A. Drisko: Transversals in row-Latin rectangles, J. Combin. Theory Ser. A 84 (1998), 181–195.MathSciNetCrossRefGoogle Scholar
  9. [9]
    P. Erdős, A. Ginzburg and A. Ziv: Theorem in the additive number theory, Bull. Res. Counc. Israel Sect. F Math. Phys. 10 (1961), 41–43.MathSciNetzbMATHGoogle Scholar
  10. [10]
    Z. Füuredi: Covering the complete graph by partitions, Discrete Math. 75 (1989), 217–226, Graph theory and combinatorics (Cambridge, 1988).MathSciNetCrossRefGoogle Scholar
  11. [11]
    G. Kalai and R. Meshulam: A topological colorful Helly theorem, Adv. Math. 191 (2005), 305–311.MathSciNetCrossRefGoogle Scholar
  12. [12]
    G. Wegner: d-collapsing and nerves of families of convex sets, Arch. Math. (Basel) 26 (1975), 317–321.MathSciNetCrossRefGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag 2019

Authors and Affiliations

  1. 1.Department of Mathematics TechnionIsrael Institute of TechnologyTechnion City, HaifaIsrael
  2. 2.Department of MathematicsMassachusetts Institute of TechnologyCambridgeMAUSA

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