Anti-Ramsey number was introduced by Erd(o)s et al.in 1973.The anti-Ramsey number AR(Kn, H) for the graph H in Kn is defined to be the maximum number of colors in an edge coloring of Kn without rainbow H.It has been shown that the anti Ramsey number AR(Kn, H) is closely related to Turán number ex(n,-) of the family H ={H-e : e ∈ E(H)} in Kn , i.e., AR(Kn,H)-ex(n,-) =o(n2).It follows from the asymptotic of Turán numbers that AR(Kn,H)/(n2) → 1-(1/d) as n → +∞,where d + 1 =min{x(H-e) : e ∈ E(H)}.So the anti-Ramsey number AR(Kn, H) is determined asymptotically for graphs H with min{x(H-e) : e ∈ E(H)} ≥ 3.The case min{x(H-e) : e ∈ E(H)} =2 still remains harder.Also, the bipartite version of the anti-Ramsey number was introduced and studied during the last decades.Besides this, the anti-Ramsey numbers for several special graph classes in complete and complete bipartite graphs have been determined exactly.Firstly, Li and Xu considered the anti Ramsey number for matchings in a general bipartite graph.