Total k-Rainbow domination numbers in graphs

Let k≥1 be an integer, and let G be a graph. A k-rainbow dominating function (or a {k-RDF) of G is a function f from the vertex set V(G) to the family of all subsets of {1,2,…,k} such that for every v∈V(G) with f(v)=∅, the condition ⋃u∈NG(v)f(u)={1,2,…,k} is fulfilled, where NG(v) is the open neighborhood of v. The weight of a k-RDF f of G is the value ω(f)=∑v∈V(G)|f(v)|. A k-rainbow dominating function f in a graph with no isolated vertex is called a total k-rainbow dominating function if the subgraph of G induced by the set {v∈V(G)∣f(v)≠∅} has no isolated vertices. The total k-rainbow domination number of G, denoted by γtrk(G), is the minimum weight of a total k-rainbow dominating function on G. The total 1-rainbow domination is the same as the total domination. In this paper we initiate the study of total k-rainbow domination number and we investigate its basic properties. In particular, we present some sharp bounds on the total k-rainbow domination number and we determine the total k-rainbow domination number of some classes of graphs.