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et $D$ be a finite and simple digraph with vertex set $V(D)$, and let $f:V(D)rightarrow{-1,1}$ be a two-valued function. If $kge 1$ is an integer and $sum_{xin N^-[v]}f(x)ge k$ for each $vin V(D)$, where $N^-[v]$ consists of $v$ and all vertices of $D$ from which arcs go into $v$, then $f$ is a signed $k$-dominating function on $D$. A set ${f_1,f_2,ldots,f_d}$ of distinct signed $k$-dominating functions on $D$ with the property that $sum_{i=1}^df_i(x)le k$ for each $xin V(D)$, is called a signed $(k,k)$-dominating family (of functions) on $D$. The maximum number of functions in a signed $(k,k)$-dominating family on $D$ is the signed $(k,k)$-domatic number on $D$, denoted by $d_{S}^{k}(D)$. In this paper, we initiate the study of the signed $(k,k)$-domatic number of digraphs, and we present different bounds on $d_{S}^{k}(D)$. Some of our results are extensions of well-known properties of the signed domatic number $d_S(D)=d_{S}^{1}(D)$ of digraphs $D$ as well as the signed $(k,k)$-domatic number $d_S^k(G)$ of graphs $G$.

J. Combin. Math. Combin Comput 77 (2011), 89-101