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, A R ( G , n ) , is the smallest integer r such that for any exact r -edge coloring of K n 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 A R ( M k , 2 k ) = max { 2 k − 3 2 + 3 , k − 2 2 + k 2 − 2 } , where M k is a matching of size k ≥ 3 .