Lin Hu and Xueliang Li

Let G=(V,E) be an (edge-)colored graph, i.e., G is assigned a mapping C:E

{1,2,...,r}, the set of colors. A matching of G is called heterochromatic if its any two edges have different colors. Unlike uncolored matchings for which the maximum matching problem is solvable in polynomial time, the maximum heterochromatic matching problem is NP-complete. This means that to find both sufficient and necessary good conditions for the existence of perfect heterochromatic matchings should be not easy. In this paper, we obtain sufficient conditions of Hall-type and Tutte-type for the existence of perfect heterochromatic matchings in colored bipartite graphs and general colored graphs by showing that a colored bipartite graph B=(X,Y) contains a heterochromatic matching that saturates all vertices in X if |N_c(S)|

|S| for all S X, and a general colored graph G contains a perfect heterochromatic matching if |N_c(S)|

|S| for all S V such that 0

|S||V|/2 and |N(S)\ S| |S|. We also obtain a sufficient and necessary condition of Berge-type to verify if a heterochromatic matching M of G is maximum, i.e., if and only if there does not exist any heterochromatic M-augmenting path system in G.