Rainbow 2-Connection Numbers of Cayley Graphs

• Establish a lemma for bounding the rainbow 2-connection numbers of some special graphs.
• Provide an upper bound for the rainbow 2-connection numbers of abelian Cayley graphs.
• Characterize the rainbow 2-connection numbers of cubic dihedrants.

A path in an edge colored graph is said to be a rainbow path if no two edges on this path share the same color. For an l-connected graph Γ and an integer k   with 1≤k≤l1≤k≤l, the rainbow k-connection number of Γ is the minimum number of colors required to color the edges of Γ such that any two distinct vertices of Γ are connected by k internally disjoint rainbow paths. In this paper, a method is provided for bounding the rainbow 2-connection numbers of graphs with certain structural properties. Using this method, we consider the rainbow 2-connection numbers of Cayley graphs, especially, those defined on abelian groups and dihedral groups.