Total $k$-Rainbow domination numbers in graphs

Let $kgeq 1$ be an integer, and let $G$ be a graph. A {it $k$-rainbow dominating function} (or a {it $k$-RDF}) of $G$ is a function $f$ from the vertex set $V(G)$ to the family of all subsets of ${1,2,ldots ,k}$ such that for every $vin V(G)$ with $f(v)=emptyset $, the condition $bigcup_{uin N_{G}(v)}f(u)={1,2,ldots,k}$ is fulfilled, where $N_{G}(v)$ is the open neighborhood of $v$. The {it weight} of a $k$-RDF $f$ of $G$ is the value $omega (f)=sum _{vin V(G)}|f(v)|$. A $k$-rainbow dominating function $f$ in a graph with no isolated vertex is called a {em total $k$-rainbow dominating function} if the subgraph of $G$ induced by the set ${v in V(G) mid f (v) not = {color{blue}emptyset}}$ has no isolated vertices. The {em total $k$-rainbow domination number} of $G$, denoted by $gamma_{trk}(G)$, 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 {color{blue}the} total $k$-rainbow domination number of some classes of graphs.