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

Let f be a function that assigns to each vertex a subset of colors chosen from a set C = {1, 2, . . . , k} of k colors. If  u∈N(v) f (u) = C for each vertex v ∈ V with f (v) = ∅, then f is called a k-rainbow dominating function (kRDF) of G where N(v) = {u ∈ V | uv ∈ E}. The weight of f , denoted by w(f ), is defined as w(f ) =  v∈V |f (v)|. Given a graph G, the minimum weight among all weights of kRDFs, denoted by γrk(G), is called the k-rainbow domination number of G. Bres̆ar and S̆umenjak (2007) [5] gave an upper bound and a lower bound for γr2(GP(n, k)). They showed that ⌈ 4n 5 ⌉ 6 γr2(GP(n, k)) 6 n. In this paper, we propose a tight upper bound for γr2(GP(n, k)) when n > 4k + 1. © 2013 Elsevier B.V. All rights reserved.