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Let $k$ be a positive integer, and let $G$ be a simple graph with vertex set $V\left(G\right)$. A of the graph $G$ is a subset $D$ of $V\left(G\right)$ such that every vertex of $V\left(G\right)-D$ is adjacent to at least $k$ vertices in $D$. A of $G$ is a partition of $V\left(G\right)$ into $k$-dominating sets. The maximum number of dominating sets in a $k$-domatic partition of $G$ is called the $d_k\left(G\right)$. In this paper, we present upper and lower bounds for the $k$-domatic number, and we establish Nordhaus-Gaddum-type results. Some of our results extend those for the classical...