A note on the Roman domatic number of a digraph

Abstract

Roman dominating function} on a digraph $D$ with vertex set $V(D)$ is a labeling$fcolon V(D)to {0, 1, 2}$such that every vertex with label $0$ has an in-neighbor with label $2$. A set ${f_1,f_2,ldots,f_d}$ ofRoman dominating functions on $D$ with the property that $sum_{i=1}^d f_i(v)le 2$ for each $vin V(D)$,is called a {em Roman dominating family} (of functions) on $D$. The maximum number of functions in aRoman dominating family on $D$ is the {em 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 sharpbounds 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 ofundirected graphs.