Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21284120190601The Italian domatic number of a digraph61701384510.22049/cco.2019.26360.1102ENLutzVolkmannRWTH Aachen University0000-0003-3496-277XJournal Article20181026An 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 Italian dominating family (of functions) on $D$. The maximum number of functions in an Italian dominating family on $D$ is the 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.http://comb-opt.azaruniv.ac.ir/article_13845_b207051edc37d9a82ecd605da8ed79b4.pdf