Rainbow matchings and cycle-free partial transversals of Latin squares

In this paper we show that properly edge-colored graphs GG with |V(G)|≥4δ(G)−3|V(G)|≥4δ(G)−3 have rainbow matchings of size δ(G)δ(G); this gives the best known bound for a recent question of Wang. We also show that properly edge-colored graphs GG with |V(G)|≥2δ(G)|V(G)|≥2δ(G) have rainbow matchings of size at least δ(G)−2δ(G)2/3δ(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,nKn,n has a rainbow matching of size n−o(n)n−o(n), or equivalently that every Latin square of order nn has a partial transversal of size n−o(n)n−o(n) (an asymptotic version of the Ryser–Brualdi conjecture). In this direction we also show that every Latin square of order nn has a cycle-free partial transversal   of size n−o(n)n−o(n).