Multicolored matchings in hypergraphs

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Abstract

For a collection of (not necessarily distinct) matchings M = (M1, M2,:::, Mq) in a hypergraph, where each matching is of size t, a matching M of size t contained in the union [qi=1Mi is called a rainbow matching if there is an injective mapping from M to M assigning to each edge e of M a matching Mi 2 M containing e. Let f(r, t) denote the maximal k for which there exists a collection of k matchings, each of size t, in some r-partite r-uniform hypergraph, such that there is no rainbow matching of size t. Aharoni and Berger showed that f(r, t) 2r–1 (t – 1), proved that the equality holds for r = 2 as well as for t = 2 and conjectured that the equality holds for all r, t. We show that in fact f(r, t) is much bigger for most values of r and t, establish an upper bound and point out a relation between the problem of estimating f(r, t) and several results in additive number theory, which provides new insights into some of those results.

Original languageEnglish (US)
Pages (from-to)3-10
Number of pages8
JournalMoscow Journal of Combinatorics and Number Theory
Volume1
Issue number1
StatePublished - 2011
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory
  • Discrete Mathematics and Combinatorics

Keywords

  • Additive Latin transversals
  • Rainbow matchings

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