Rainbow matchings and partial transversals of Latin squares

Abstract: 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 \it rainbow \rm if its edges have different colors. The minimum degree of a graph is denoted by $\delta(G)$. We show that properly edge colored graphs $G$ with $|V(G)|\ge 4\delta(G)-3$ have rainbow matchings of size $\delta(G)$, this gives the best known estimate to a recent question of Wang. Since one obviously needs at least $2\delta(G)$ vertices to guarantee a rainbow matching of size $\delta(G)$, we investigate what happens when $|V(G)|\ge 2\delta(G)$. We show that any properly edge colored graph $G$ with $|V(G)|\ge 2\delta$ contains a rainbow matching of size at least $\delta - 2\delta(G)^{2/3}$. 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 {\em cycle-free partial transversal} of size $n-o(n)$.