kk-Rainbow domatic numbers

A kk-rainbow dominating function of a graph is a function ff from the vertices V(G)V(G) to 2[k]2[k] such that, for all v∈V(G)v∈V(G), either f(v)≠0̸f(v)≠0̸ or ⋃u∈N[v]f(u)={1,…,k}⋃u∈N[v]f(u)={1,…,k}. The kk-rainbow domatic number drk(G)drk(G) is the maximum integer dd such that there exists a set of kk-rainbow dominating functions f1,f2,…,fdf1,f2,…,fd with ∑i=1d|fi(v)|≤k for all v∈V(G)v∈V(G). We study thekk-rainbow domatic number by finding this number for some classes of graphs and improving upon some known general bounds.