The Roman (k,k)-domatic number of a graph

Let $k$ be a positive integer. A {\em Roman $k$-dominating function} on a graph $G$ is a labeling $f:V (G)\longrightarrow \{0, 1, 2\}$ such that every vertex with label 0 has at least $k$ neighbors with label 2. A set $\{f_1,f_2,\ldots,f_d\}$ of distinct Roman $k$-dominating functions on $G$ with the property that $\sum_{i=1}^df_i(v)\le 2k$ for each $v\in V(G)$, is called a {\em Roman $(k,k)$-dominating family} (of functions) on $G$. The maximum number of functions in a Roman $(k,k)$-dominating family on $G$ is the {\em Roman $(k,k)$-domatic number} of $G$, denoted by $d_{R}^k(G)$. Note that the Roman $(1,1)$-domatic number $d_{R}^1(G)$ is the usual Roman domatic number $d_{R}(G)$. In this paper we initiate the study of the Roman $(k,k)$-domatic number in graphs and we present sharp bounds for $d_{R}^k(G)$. In addition, we determine the Roman $(k,k)$-domatic number of some graphs. Some of our results extend those given by Sheikholeslami and Volkmann in 2010 for the Roman domatic number.