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

Abstract

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)=0̸, 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. Bres˘ar and S˘umenjak (2007) [5] gave an upper bound and a lower bound for γr2(GP(n,k)). They showed that ⌈4n5⌉⩽γr2(GP(n,k))⩽n. In this paper, we propose a tight upper bound for γr2(GP(n,k)) when n⩾4k+1.