The anti-Ramsey number of perfect matching

An rr-edge coloring   of a graph GG is a mapping h:E(G)→[r]h:E(G)→[r], where h(e)h(e) is the color assigned to edge e∈E(G)e∈E(G). An exact  rr-edge coloring   is an rr-edge coloring hh such that there exists an e∈E(G)e∈E(G) with h(e)=ih(e)=i for all i∈[r]i∈[r]. Let hh be an edge coloring of GG. We say GG is rainbow   if no two edges in GG are assigned the same color by hh. The anti-Ramsey number  , AR(G,n)AR(G,n), is the smallest integer rr such that for any exact rr-edge coloring of KnKn there exists a subgraph isomorphic to GG that is rainbow. In this paper we confirm a conjecture of Fujita, Kaneko, Schiermeyer, and Suzuki that states AR(Mk,2k)=max{2k−32+3,k−22+k2−2}, where MkMk is a matching of size k≥3k≥3.