The signed Roman k-domatic number of a graph

A signed Roman dominating function (SRDF) on a graph G is a function f : V (G) → {−1, 1, 2} such that u∈N [v] f(u) ≥ 1 for every v ∈ V (G), and every vertex u ∈ V (G) for which f(u) = −1 is adjacent to at least one vertex w for which f(w) = 2. A set {f1, f2, . . . , fd} of distinct signed Roman dominating functions on G with the property that ∑d i=1 fi(v) ≤ 1 for each v ∈ V (G), is called a signed Roman dominating family (of functions) on G. The maximum number of functions in a signed Roman dominating family on G is the signed Roman domatic number of G, denoted by dsR(G). In this paper we initiate the study of signed Roman domatic number in graphs and we present some sharp bounds for dsR(G). In addition, we determine the signed Roman domatic number of some graphs.