The rainbow connection number of a watermill graph

Abstract

Let G be a simple, finite and undirected connected graph. An edge-colored graph G is called rainbow connected, if any two vertices in the graph are connected by a path which the edges have distinct colors. Such a path is called rainbow path. An edge-coloring on a graph G is a map . The smallest number k of colors needed in order to make G rainbow connected is called the rainbow connection number of G, denoted by rc(G). The concepts of rainbow connection were introduced by Chartrand et. al. in 2008. Since then many classes of graphs are studied to find its rainbow connection number. The corona product is called a sun graph with 2n vertices. Let Pm be a path with m vertices. The watermill graph is a Cartesian product of graphs and Pm, which is denoted by W M(m, n). In this paper, we determine the rainbow connection number of a watermill graph W M(m, n).