O ct 2 00 6 Bipartite rainbow numbers of matchings ∗

Given two graphs G and H, let f(G,H) denote the maximum number c for which there is a way to color the edges of G with c colors such that every subgraph H of G has at least two edges of the same color. Equivalently, any edge-coloring of G with at least rb(G,H) = f(G,H) + 1 colors contains a rainbow copy of H, where a rainbow subgraph of an edge-colored graph is such that no two edges of it have the same color. The number rb(G,H) is called the rainbow number of H with respect to G, and simply called the bipartite rainbow number of H if G is the complete bipartite graph Km,n. Erdős, Simonovits and Sós showed that rb(Kn,K3) = n. In 2004, Schiermeyer determined the rainbow numbers rb(Kn,Kk) for all n ≥ k ≥ 4, and the rainbow numbers rb(Kn, kK2) for all k ≥ 2 and n ≥ 3k+3. In this paper we will determine the rainbow numbers rb(Km,n, kK2) for all k ≥ 1.