The (strong) rainbow connection numbers of Cayley graphs of Abelian groups

A path in an edge-colored graph G, where adjacent edges may have the same color, is called a rainbow path if no two edges of the path are colored the same. The rainbow connection number $rc(G)$ of G is the minimum integer i for which there exists an i-edge-coloring of G such that every two distinct vertices of G are connected by a rainbow path. The strong rainbow connection number $src(G)$ of G is the minimum integer i for which there exists an i-edge-coloring of G such that every two distinct vertices u and v of G are connected by a rainbow path of length $d(u,v)$. In this paper, we give upper and lower bounds of the (strong) rainbow connection Cayley graphs of Abelian groups. Moreover, we determine the (strong) rainbow connection numbers of some special cases.