Total kk-Rainbow domination numbers in graphs

Let $k\geq 1$ 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 $v\in V(G)$ with‎ ‎$f(v)=\emptyset$‎, ‎the condition $\bigcup_{u\in‎ ‎N_{G}(v)}f(u)=\{1,2,\ldots,k\}$ 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 $\omega (f)=\sum _{v\in V(G)}|f(v)|$‎. ‎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 $\{v \in V(G) \mid f (v) \not =\emptyset\}$ 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 the total‎ ‎$k$-rainbow domination number of some classes of graphs‎.