Upper bound on 3-rainbow domination in graphs with minimum degree 2

Let k≥1 be an integer, and let G be a graph. A function f:V(G)→2{1,…,k} is a k-rainbow dominating function of G if every vertex x∈V(G) with f(x)=∅ satisfies ⋃y∈NG(x)f(y)={1,…,k}. The k-rainbow domination number of G, denoted by γrk(G), is the minimum weight w(f)=∑x∈V(G)|f(x)| of a k-rainbow dominating function f of G. In this paper, we prove that for every connected graph G of order n≥8 with δ(G)≥2, γr3(G)≤5n6.