On the Roman {2}-domatic number of graphs.

Let G = (V , E) be a graph. A Roman { 2 } -dominating function f : V → { 0 , 1 , 2 } has the property that for every vertex v ∈ V with f (v) = 0 , either v is adjacent to a vertex assigned 2 under f , or v is adjacent to at least two vertices assigned 1 under f. A set { f 1 , f 2 , ... , f d } of distinct Roman { 2 } -dominating functions on G with the property that ∑ i = 1 d f i (v) ≤ 2 for each v ∈ V (G) is called a Roman { 2 } -domination family (or functions) on G. The maximum number of functions in a Roman { 2 } -dominating family on G is the Roman { 2 } -domatic number of G , denoted by d { R 2 } (G). In this paper, we answer two conjectures of Volkman [L. Volkmann, The Roman { 2 } -domatic number of graphs, Discrete Appl. Math. 258 (2019) 235–241] about Roman { 2 } -domatic number of graphs and we study this parameter for join of graphs and complete bipartite graphs. [ABSTRACT FROM AUTHOR]