The signed Roman kk-domatic number of a graph

Let k≥1k≥1 be an integer. A signed Roman  kk-dominating function   on a graph GG is a function f:V(G)⟶{−1,1,2}f:V(G)⟶{−1,1,2} such that ∑u∈N[v]f(u)≥k∑u∈N[v]f(u)≥k for every v∈V(G)v∈V(G), where N[v]N[v] is the closed neighborhood of vv, and every vertex u∈V(G)u∈V(G) for which f(u)=−1f(u)=−1 is adjacent to at least one vertex ww for which f(w)=2f(w)=2. A set {f1,f2,…,fd}{f1,f2,…,fd} of distinct signed Roman kk-dominating functions on GG with the property that ∑i=1dfi(v)≤k for each v∈V(G)v∈V(G), is called a signed Roman  kk-dominating family   (of functions) on GG. The maximum number of functions in a signed Roman kk-dominating family on GG is the signed Roman  kk-domatic number   of GG, denoted by dsRk(G). In this paper we initiate the study of signed Roman kk-domatic numbers in graphs, and we present sharp bounds for dsRk(G). In particular, we derive some Nordhaus–Gaddum type inequalities. In addition, we determine the signed Roman kk-domatic number of some graphs.