Roman k-tuple domination number of a graph

For any integer $k\ge 1$ and any graph $G=(V,E)$ with minimum degree at least $k-1$, we define a function $f:V\to \{0,1,2\}$ as a Roman $k$-tuple dominating function on $G$ if for any vertex $v$ with $f(v)=0$ there exist at least $k$ and for any vertex $v$ with $f(v)\mathrm{\ne }0$ at least $k-1$ vertices in its neighborhood with $f(w)=2$. The minimum weight of a Roman $k$-tuple dominating function $f$ on $G$ is called the Roman $k$-tuple domination number of the graph where the weight of $f$ is $f(V)=\sum_{v\in V}f(v)$. In this paper, we initiate to study the Roman $k$-tuple domination number of a graph, by giving some sharp bounds for the Roman $k$-tuple domination number of a garph, the Mycieleskian of a graph, and the corona graphs. Also finding the Roman $k$-tuple domination number of some known graphs is our other goal. Some of our results extend these one given by Cockayne and et al.} in 2004 for the Roman domination number.