Color Degree Condition for Large Heterochromatic Matchings in Edge-Colored Bipartite Graphs

Let G = (V,E) be an edge-colored graph, i.e., G is assigned a surjective function C : E → {1, 2, · · · , r}, the set of colors. A matching of G is called heterochromatic if its any two edges have different colors. Let (B,C) be an edge-colored bipartite graph and d(v) be color degree of a vertex v. We show that if d(v) ≥ k for every vertex v of B, then B has a heterochromatic matching of cardinality at least k − 1. By taking B = Kn,n and k = n, our result solves a conjecture by Stein and Brualdi on the transversals of latin squares, since a latin square of order n corresponds to a proper edge-coloring of the complete bipartite Kn,n with n colors, a transversal of a latin square corresponds to a perfect matching of Kn,n and a latin transversal of a latin square corresponds to a perfect heterochromatic matching of Kn,n.