$k$-tuple total restrained domination/domatic in graphs

For any integer $kgeq 1$, a set $S$ of vertices in a graph $G=(V,E)$ is a $k$- tuple total dominating set of $G$ if any vertex of $G$ is adjacent to at least $k$ vertices in $S$, and any vertex of $V-S$ is adjacent to at least $k$ vertices in $V-S$. The minimum number of vertices of such a set in $G$ we call the $k$-tuple total restrained domination number of $G$. The maximum number of classes of a partition of $V$ such that its all classes are $k$-tuple total restrained dominating sets in $G$ we call the $k$-tuple total restrained domatic number of $G$. In this paper, we give some sharp bounds for the $k$-tuple total restrained domination number of a graph, and also calculate it for some of the known graphs. Next, we mainly present basic properties of the $k$-tuple total restrained domatic number of a graph.