weak signed roman $k$-domatic number of a digraph

Let $d$ be a digraph with vertex set $v(d)$, and let $kge 1$ be an integer. a weak signed roman $k$-dominating function on a digraph $d$ is a function  $f:v (d)longrightarrow {-1, 1, 2}$ such that $sum_{uin n^-[v]}f(u)ge k$ for every $vin v(d)$, where $n^-[v]$ consists of $v$ and all vertices of $d$ from which arcs go into $v$. a set ${f_1,f_2,ldots,f_d}$ of distinct weak signed roman $k$-dominating functions on $d$ with the property that $sum_{i=1}^df_i(v)le k$ for each $vin v(d)$, is called a  weak signed roman $k$-dominating family (of functions) on $d$. the maximum number of functions in a  weak signed roman $k$-dominating family on $d$ is the  weak signed roman $k$-domatic number of $d$, denoted by $d_{wsr}^k(d)$. in this paper we initiate the study of the weak signed roman $k$-domatic number in digraphs, and we present sharp bounds for $d_{wsr}^k(d)$. in addition, we determine the weak signed roman $k$-domatic number of some digraphs.