For a given graph H and n ≥ 1, let f (n, H) denote the maximum number c for which there is a way to color the edges of the complete graph K n with c colors such that every subgraph H of K n has at least two edges of the same color. Equivalently, any edge-coloring of K n with at least rb(n, H) = f (n, 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(n, H) is called the rainbow number of H. Erdős, Simonovits and Sós showed that rb(n, K 3 ) = n. In 2004, Schiermeyer used some counting technique and determined the rainbow numbers rb(n, kK 2 ) for k ≥ 2 and n ≥ 3k + 3. It is easy to see that n must be at least 2k. So, for 2k ≤ n < 3k + 3, the rainbow numbers remain not determined. In this paper we will use the Gallai-Edmonds structure theorem for matchings to determine the exact values for rainbow numbers rb(n, kK 2 ) for all k ≥ 2 and n ≥ 2k, giving a complete solution for the rainbow numbers of matchings.
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