Rainbow matchings and partial transversals of Latin squares

In this paper we consider properly edge-colored graphs, i.e. two edges with the same color cannot share an endpoint, so each color class is a matching. A matching is called rainbow if its edges have different colors. The minimum degree of a graph is denoted by δ(G). 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 estimate to a recent question of Wang. Since one obviously needs at least 2δ(G) vertices to guarantee a rainbow matching of size δ(G), we investigate what happens when |V (G)| ≥ 2δ(G). We show that any properly edge colored graph G with |V (G)| ≥ 2δ contains a rainbow matching of size at least δ − 2δ(G)2/3. This result extends (with a ∗2000 Mathematics Subject Classification: 05B15, 05D15, 05C15. †Research supported in part by the National Science Foundation under Grant No. DMS-0968699.