Heterochromatic Matchings in Edge-Colored Graphs

Let G be an (edge-)colored graph. A heterochromatic matching of G is a matching in which no two edges have the same color. For a vertex v, let d(v) be the color degree of v. We show that if d(v) ≥ k for every vertex v of G, then G has a heterochromatic matching of size ⌈ 5k−3 12 ⌉ . For a colored bipartite graph with bipartition (X,Y ), we prove that if it satisfies a Hall-like condition, then it has a heterochromatic matching of cardinality ⌈ |X| 2 ⌉ , and we show that this bound is best possible.