Rainbow matchings in Υ-partite Υ-graphs

Ron Aharoni, Eli Berger

Research output: Contribution to journalArticlepeer-review


Given a collection of matchings μ = (M1,M2,..., Mq) (repetitions allowed), a matching M contained in ∪μ is said to be s-rainbow for μ if it contains representatives from s matchings Mi (where each edge is allowed to represent just one Mi). Formally, this means that there is a function ø : M -> [q] such that e ε Mø(e) for all e ε M, and |Im(ø)| ≥ s. Let f(r, s, t) be the maximal k for which there exists a set of k matchings of size t in some r-partite hypergraph, such that there is no s-rainbow matching of size t. We prove that f(r,s,t) ≥ 2r-1(s - 1), make the conjecture that equality holds for all values of r, s and t and prove the conjecture when r = 2 or s = t = 2. In the case r = 3, a stronger conjecture is that in a 3-partite 3-graph if all vertex degrees in one side (say V1) are strictly larger than all vertex degrees in the other two sides, then there exists a matching of V1. This conjecture is at the same time also a strengthening of a famous conjecture, described below, of Ryser, Brualdi and Stein. We prove a weaker version, in which the degrees in V1 are at least twice as large as the degrees in the other sides. We also formulate a related conjecture on edge colorings of 3-partite 3-graphs and prove a similarly weakened version.

Original languageEnglish
Article numberR119
JournalElectronic Journal of Combinatorics
Issue number1
StatePublished - 25 Sep 2009

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics
  • Applied Mathematics


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