%0 Journal Article %T A note on the Roman domatic number of a digraph %J Communications in Combinatorics and Optimization %I Azarbaijan Shahid Madani University %Z 2538-2128 %A Volkmann, Lutz %A Meierling, D. %D 2020 %\ 06/01/2020 %V 5 %N 1 %P 19-26 %! A note on the Roman domatic number of a digraph %K Digraphs %K Roman dominating function %K Roman domination number %K Roman domatic number %R 10.22049/cco.2019.26419.1107 %X A  Roman dominating function on a digraph $D$ with vertex set $V(D)$ is a labeling $f\colon V(D)\to \{0, 1, 2\}$ such that every vertex with label $0$ has an in-neighbor with label $2$. A set $\{f_1,f_2,\ldots,f_d\}$ of Roman dominating functions on $D$ with the property that $\sum_{i=1}^d f_i(v)\le 2$ for each $v\in V(D)$, is called a Roman dominating family (of functions) on $D$. The maximum number of functions in a Roman dominating family on $D$ is the  Roman domatic number of $D$, denoted by $d_{R}(D)$. In this note, we study the Roman domatic number in digraphs, and we present some sharp bounds for $d_{R}(D)$. In addition, we determine the Roman domatic number of some digraphs. Some of our results are extensions of well-known properties of the Roman domatic number of undirected graphs. %U http://comb-opt.azaruniv.ac.ir/article_13884_bf374c8fd79d776bfc11bd95660ff3b1.pdf