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

Abstract

A path in an edge-colored graph G , whose 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 r c ( 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 s r c ( 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 numbers of Cayley graphs on Abelian groups. Moreover, we determine the (strong) rainbow connection numbers of some special cases.