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

Let ff be a function that assigns to each vertex a subset of colors chosen from a set C={1,2,…,k}C={1,2,…,k} of kk colors. If ⋃u∈N(v)f(u)=C⋃u∈N(v)f(u)=C for each vertex v∈Vv∈V with f(v)=0̸f(v)=0̸, then ff is called a kk-rainbow dominating function   (kkRDF) of GG where N(v)={u∈V∣uv∈E}N(v)={u∈V∣uv∈E}. The weight   of ff, denoted by w(f)w(f), is defined as w(f)=∑v∈V|f(v)|w(f)=∑v∈V|f(v)|. Given a graph GG, the minimum weight among all weights of kkRDFs, denoted by γrk(G), is called the kk-rainbow domination number   of GG. 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+1n⩾4k+1.