Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21284120190601The Roman domination and domatic numbers of a digraph47591384110.22049/cco.2019.26356.1101ENZhihongXieCollege of Science, East China University of Technology, Nanchang, P. R. ChinaGuoliangHaoCollege of Science, East China University of Technology, Nanchang, P. R. ChinaShouliuWeiDepartment of Mathematics, Minjiang University, Fuzhou, ChinaJournal Article20181022A Roman dominating function (RDF) on a digraph $D$ is a function $f: V(D)rightarrow {0,1,2}$ satisfying the condition that every vertex $v$ with $f(v)=0$ has an in-neighbor $u$ with $f(u)=2$. The weight of an RDF $f$ is the value $sum_{vin V(D)}f(v)$. The Roman domination number of a digraph $D$ is the minimum weight of an RDF on $D$. A set ${f_1,f_2,dots,f_d}$ of Roman dominating functions on $D$ with the property that $sum_{i=1}^df_i(v)le2$ for each $vin V(D)$, 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 paper we continue the investigation of the Roman domination number, and we initiate the study of the Roman domatic number in digraphs. We present some bounds for $d_{R}(D)$. In addition, we determine the Roman domatic number of some digraphs.http://comb-opt.azaruniv.ac.ir/article_13841_9bb2c1e1cb7db1916a327f1866d037b2.pdf