The Italian domatic number of a digraph

An {em Italian dominating function} on a digraph $D$ with vertex set $V(D)$ is defined as a function $fcolon V(D)to {0, 1, 2}$ such that every vertex $vin V(D)$ with $f(v)=0$ has at least two in-neighbors assigned 1 under $f$ or one in-neighbor $w$ with $f(w)=2$. A set ${f_1,f_2,ldots,f_d}$ of distinct Italian dominating functions on $D$ with the property that $sum_{i=1}^d f_i(v)le 2$ for each $vin V(D)$, is called an {em Italian dominating family} (of functions) on $D$. The maximum number of functions in an Italian dominating family on $D$ is the {em Italian domatic number} of $D$, denoted by $d_{I}(D)$. In this paper we initiate the study of the Italian domatic number in digraphs, and we present some sharp bounds for $d_{I}(D)$. In addition, we determine the Italian domatic number of some digraphs.