Anti-Ramsey numbers for matchings in 3-regular bipartite graphs

The anti-Ramsey number AR(Kn, H) was introduced by Erdős, Simonovits and Sós in 1973, which is defined to be the maximum number of colors in an edge coloring of the complete graph Kn without any rainbow H. Later, the anti-Ramsey numbers for several special graph classes in complete are determined. Moreover, researchers generalized the host graph Kn to other graphs, in particular, to complete bipartite graphs and regular bipartite graphs. Li and Xu (2009) [18] proved that: Let G be a k-regular bipartite graph with n   vertices in each partite set, then AR(G,mK2)=k(m−2)+1AR(G,mK2)=k(m−2)+1 for all m ≥ 2, k   ≥ 3 and n>3(m−1)n>3(m−1). In this paper, we consider the anti-Ramsey number for matchings in 3-regular bipartite graphs. By using the known result that the vertex cover equals the size of maximum matching in bipartite graphs, we prove that AR(G,mK2)=3(m−2)+1AR(G,mK2)=3(m−2)+1 for n>32(m−1) when G is a 3-regular bipartite graph with n vertices in each partite set.