The Roman domatic number of a graph

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