A tight upper bound for 2-rainbow domination in generalized Petersen graphs

Let! be a function that assigns to each vertex a subset of colors chosen from a set l = [1, 2, ... , k) of k colors. If boolean OR(u is an element of N(v))f(u) = l for each vertex v E V with f (v) = phi, then f is called a k-rainbow dominating function (kRDF) of G where N(v) = {u is an element of V vertical bar uv is an element of E}. The weight off, denoted by in(f), is defined as w(f) = Sigma(u is an element of v). vertical bar f(v)vertical bar Given a graph G, the minimum weight among all weights of kRDFs, denoted by y(rk)(G), is called the k-rainbow domination number of G. Bresar and Sumenjak (2007) 151gave an upper bound and a lower bound for gamma(r2)(GP(n, k)). They showed that [4n/5] <= gamma(r2)(GP(n, k)) <= n. In this paper, we propose a tight upper bound for gamma(r2)(GP(n. k)) when n >= 4k + 1. (C) 2013 Elsevier B.V. All rights reserved.