double roman domination and domatic numbers of graphs

A double roman dominating function on a graph g with vertex set v(g) is defined in cite{bhh} as a function f:v(g)→{0,1,2,3} having the property that if f(v)=0, then the vertex v must have at least two neighbors assigned 2 under f or one neighbor w with f(w)=3, and if f(v)=1, then the vertex v must have at least one neighbor u with f(u)≥2. the weight of a double roman dominating function f is the sum ∑v∈v(g)f(v), and the minimum weight of a double roman dominating function on g is the double roman domination number γdr(g) of g. a set {f1,f2,…,fd} of distinct double roman dominating functions on g with the property that ∑di=1fi(v)≤3 for each v∈v(g) is called a double roman dominating family (of functions) on g. the maximum number of functions in a double roman dominating family on g is the double roman domatic number of g. in this note we continue the study the double roman domination and domatic numbers. in particular, we present a sharp lower bound on γdr(g), and we determine the double roman domination and domatic numbers of some classes of graphs.