The heterochromatic matchings in edge-colored bipartite graphs

Let (G,C) be an edge-colored bipartite graph with bipartition (X,Y ). A heterochromatic matching of G is such a matching in which no two edges have the same color. Let N c(S) denote a maximum color neighborhood of S ⊆ V (G). We show that if |N c(S)| ≥ |S| for all S ⊆ X, then G has a heterochromatic matching with cardinality at least d |X| 3 e. We also obtain that if |X| = |Y | = n and |N c(S)| ≥ |S| for all S ⊆ X or S ⊆ Y , then G has a heterochromatic matching with cardinality at least d3n−1 8 e.