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.