A note on the Roman domatic number of a digraph

A Roman dominating function on a digraph D with vertex set V (D) is a labeling f : V (D) ! f0; 1; 2g such that every vertex with label 0 has an in-neighbor with label 2. A set ff1; f2; : : : ; fdg of Roman dominating functions on D with the property that Pd i=1 fi(v) ≤ 2 for each v 2 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 dR(D). In this note, we study the Roman domatic number in digraphs, and we present some sharp bounds for dR(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.