On Rainbow k-Connection Number of Special Graphs and It’s Sharp Lower Bound

Abstract

Let G = (V, E) be a simple, nontrivial, finite, connected and undirected graph. Let c be a coloring c : E(G) → {1, 2, …, s}, s ∈ N. A path of edge colored graph is said to be a rainbow path if no two edges on the path have the same color. An edge colored graph G is said to be a rainbow connected graph if there exists a rainbow u − v path for every two vertices u and v of G. The rainbow connection number of a graph G, denoted by rc(G), is the smallest number of k colors required to edge color the graph such that the graph is rainbow connected. Furthermore, for an l-connected graph G and an integer k with 1 ≤ k ≤ l, the rainbow k-connection number rck(G) of G is defined to be the minimum number of colors required to color the edges of G such that every two distinct vertices of G are connected by at least k internally disjoint rainbow paths. In this paper, we determine the exact values of rainbow connection number of some special graphs and obtain a sharp lower bound.