### Abstract

In this paper we show that properly edge-colored graphs G with |V(G)|<4δ(G)-3 have rainbow matchings of size δ(G); this gives the best known bound for a recent question of Wang. We also show that properly edge-colored graphs G with |V(G)|<2δ(G) have rainbow matchings of size at least δ(G)-2δ(^{G)2/3}. This result extends (with a weaker error term) the well-known result that a factorization of the complete bipartite graph Kn,_{n} has a rainbow matching of size n-o(n), or equivalently that every Latin square of order n has a partial transversal of size n-o(n) (an asymptotic version of the Ryser-Brualdi conjecture). In this direction we also show that every Latin square of order n has a cycle-free partial transversal of size n-o(n).

Original language | English |
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Pages (from-to) | 96-102 |

Number of pages | 7 |

Journal | Discrete Mathematics |

Volume | 327 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jul 28 2014 |

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### Keywords

- Partial transversals in Latin squares
- Proper edge colorings
- Rainbow matchings

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

### Cite this

*Discrete Mathematics*,

*327*(1), 96-102. https://doi.org/10.1016/j.disc.2014.03.010