The Roman domatic number of a graph

A 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 { f 1 , f 2 , … , f d } of Roman dominating functions on G with the property that ∑ i = 1 d f i ( 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 d R ( G ) . In this work we initiate the study of the Roman domatic number in graphs and we present some sharp bounds for d R ( G ) . In addition, we determine the Roman domatic number of some graphs.