Rainbow connection numbers of Cayley digraphs on abelian groups

A directed path in an edge colored digraph is said to be a rainbow path if no two edges on this path share the same color. An edge colored digraph Γ is rainbow connected if any two distinct vertices can be reachable from each other through rainbow paths. The rc-number of a digraph Γ is the smallest number of colors that are needed in order to make Γ rainbow connected. In this paper, we investigate the rc-numbers of Cayley digraphs on abelian groups and present an upper bound for such digraphs. In addition, we consider the rc-numbers of bi-Cayley graphs on abelian groups.