Nordhaus–Gaddum bounds on the kk-rainbow domatic number of a graph

For a positive integer kk, a kk-rainbow dominating function   of a graph GG is a function ff from the vertex set V(G)V(G) to the set of all subsets of the set {1,2,…,k}{1,2,…,k} such that for any vertex v∈V(G)v∈V(G) with f(v)=0̸f(v)=0̸ the condition ⋃u∈N(v)f(u)={1,2,…,k}⋃u∈N(v)f(u)={1,2,…,k} is fulfilled, where N(v)N(v) is the neighborhood of vv. The 11-rainbow domination is the same as the ordinary domination. A set {f1,f2,…,fd}{f1,f2,…,fd} of kk-rainbow dominating functions on GG with the property that ∑i=1d|fi(v)|≤k for each v∈V(G)v∈V(G) is called a kk-rainbow dominating family   (of functions) on GG. The maximum number of functions in a kk-rainbow dominating family on GG is the kk-rainbow domatic number   of GG, denoted by drk(G)drk(G). Note that dr1(G)dr1(G) is the classical domatic number d(G)d(G). If GG is a graph of order nn and G¯ is the complement of GG, then we prove in this note for k≥2k≥2 the Nordhaus–Gaddum inequality drk(G)+drk(G¯)≤n+2k−2. This improves the Nordhaus–Gaddum bound given by Sheikholeslami and Volkmann recently.