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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.

In this paper, we study the signed total domination number in graphs and present new sharp lower and upper bounds for this parameter. For example by making use of the classic theorem of Turan, we present a sharp lower bound on this parameter for graphs with no complete graph of order r+1 as a subgraph. Also, we prove that n-2(s-s') is an upper bound on the signed total domination number of any tree of order n with s support vertices and s' support vertives of degree two. Moreover, we characterize all trees attainig this bound., Comment: This paper contains 11 pages and one figure