A fractional version of the Erdős-Faber-Lovász conjecture

Abstract

LetH be any hypergraph in which any two edges have at most one vertex in common. We prove that one can assign non-negative real weights to the matchings ofH summing to at most |V(H)|, such that for every edge the sum of the weights of the matchings containing it is at least 1. This is a fractional form of the Erdős-Faber-Lovász conjecture, which in effect asserts that such weights exist and can be chosen 0,1-valued. We also prove a similar fractional version of a conjecture of Larman, and a common generalization of the two.

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Supported in part by NSF grant MCS 83-01867, AFOSR Grant 0271 and a Sloan Research Fellowship

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Kahn, J., Seymour, P.D. A fractional version of the Erdős-Faber-Lovász conjecture. Combinatorica 12, 155–160 (1992). https://doi.org/10.1007/BF01204719

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AMS subject classification code (1991)

  • Primary: 05 C 65
  • Secondary: 05 B 40
  • 05 C 70