Rainbow matchings and cycle-free partial transversals of Latin squares

In this paper we show that properly edge-colored graphs G with | V ( G ) | ≥ 4 δ ( 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 ) | ≥ 2 δ ( G ) have rainbow matchings of size at least δ ( G ) − 2 δ ( G ) 2 / 3 . This result extends (with a weaker error term) the well-known result that a factorization of the complete bipartite graph K n , 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 ) .