The double Roman domatic number of a digraph

A double Roman dominating function on a digraph D with vertex set V (D) is defined in [G. Hao, X. Chen and L. Volkmann, Double Roman domination in digraphs, Bull. Malays. Math. Sci. Soc. (2017).] as a function f : V (D) → {0, 1, 2, 3} having the property that if f(v) = 0, then the vertex v must have at least two in-neighbors assigned 2 under f or one in-neighbor w with f(w) = 3, and if f(v) = 1, then the vertex v must have at least one in-neighbor u with f(u) ≥ 2. A set {f1, f2, . . ., fd} of distinct double Roman dominating functions on D with the property that ∑i=1dfi(v)≤3 \sum\nolimits_{i = 1}^d {{f_i}\left( v \right)} \le 3 for each v ∈ V (D) is called a double Roman dominating family (of functions) on D. The maximum number of functions in a double Roman dominating family on D is the double Roman domatic number of D, denoted by ddR(D). We initiate the study of the double Roman domatic number, and we present different sharp bounds on ddR(D). In addition, we determine the double Roman domatic number of some classes of digraphs.