Rainbow matchings and cycle-free partial transversals of Latin squares

In this paper we show that properly edge-colored graphs Gwith |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 Gwith |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 rainbowmatching 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). © 2014 Elsevier B.V. All rights reserved.