## Abstract

Given a collection of matchings μ = (M_{1},M_{2},..., M_{q}) (repetitions allowed), a matching M contained in ∪μ is said to be s-rainbow for μ if it contains representatives from s matchings M_{i} (where each edge is allowed to represent just one M_{i}). 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) ≥ 2^{r-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 V_{1}) are strictly larger than all vertex degrees in the other two sides, then there exists a matching of V_{1}. 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 V_{1} 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 language | English |
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Article number | R119 |

Journal | Electronic Journal of Combinatorics |

Volume | 16 |

Issue number | 1 |

State | Published - 25 Sep 2009 |

## ASJC Scopus subject areas

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