The Roman domatic number of a graph

A Roman dominating function   on a graph GG is a labeling f:V(G)⟶{0,1,2}f:V(G)⟶{0,1,2} such that every vertex with label 0 has a neighbor with label 2. A set {f1,f2,…,fd}{f1,f2,…,fd} of Roman dominating functions on GG with the property that ∑i=1dfi(v)≤2 for each v∈V(G)v∈V(G) is called a Roman dominating family   (of functions) on GG. The maximum number of functions in a Roman dominating family on GG is the Roman domatic number   of GG, denoted by dR(G)dR(G). In this work we initiate the study of the Roman domatic number in graphs and we present some sharp bounds for dR(G)dR(G). In addition, we determine the Roman domatic number of some graphs.