The (Strong) Rainbow Connection Number of Stellar Graphs

Let G = (V, E) be a simple, connected, and finite graph. A function c from E to {1, 2, ... , k} is said rainbow k-coloring of G, if for any pair of vertices u and v in V, there exists au - vpath whose edges have different colors. The rainbow connection number of G, denoted by rc(G), is the smallest positive integer k such that Ghas a rainbow k-coloring. Furthermore, such the function c is said strong rainbow k-coloring, if for any pair of vertices u and v in V, there exists a rainbow u-v path with its length is equal to distance betweenu and v. The smallest positive integer k such that G has a strong rainbow k-coloring is defined as the strong rainbow connection number, denoted by src(G). In this paper, we introduce a new class of graphs, namely stellar graphs. A stellar graph on 2mn+1 vertices, denoted by St(m,n), is the corona product of a trivial graph and mcopies ladder graph on 2n vertices (K-1 circle dot m.L-n). We determine the (strong) rainbow connection number of stellar graphs.