The anti-Ramsey number of perfect matching

An r-edge coloring of a graph G is a mapping h : E(G) → [r], where h(e) is the color assigned to edge e ∈ E(G). An exact r-edge coloring is an r-edge coloring h such that there exists an e ∈ E(G) with h(e) = i for all i ∈ [r]. Let h be an edge coloring of G. We say G is rainbow if no two edges in G are assigned the same color by h. The anti-Ramsey number, AR(G,n), is the smallest integer r such that for any exact r-edge coloring of Kn there exists a subgraph isomorphic to G that is rainbow. In this paper we confirm a conjecture of Fujita, Kaneko, Schiermeyer, and Suzuki that states AR(Mk, 2k) = max{ ( 2k−3 2 ) +3, ( k−2 2 ) +k2−2}, where Mk is a matching of size k ≥ 3.