Nordhaus–Gaddum bounds on the -rainbow domatic number of a graph

For a positive integer k , a k -rainbow dominating function of a graph G is a function f from the vertex set V ( G ) to the set of all subsets of the set { 1 , 2 , … , k } such that for any vertex v ∈ V ( G ) with f ( v ) = 0̸ the condition ⋃ u ∈ N ( v ) f ( u ) = { 1 , 2 , … , k } is fulfilled, where N ( v ) is the neighborhood of v . The 1 -rainbow domination is the same as the ordinary domination. A set { f 1 , f 2 , … , f d } of k -rainbow dominating functions on G with the property that ∑ i = 1 d | f i ( v ) | ≤ k for each v ∈ V ( G ) is called a k -rainbow dominating family (of functions) on G . The maximum number of functions in a k -rainbow dominating family on G is the k -rainbow domatic number of G , denoted by d r k ( G ) . Note that d r 1 ( G ) is the classical domatic number d ( G ) . If G is a graph of order n and G ¯ is the complement of G , then we prove in this note for k ≥ 2 the Nordhaus–Gaddum inequality d r k ( G ) + d r k ( G ¯ ) ≤ n + 2 k − 2 . This improves the Nordhaus–Gaddum bound given by Sheikholeslami and Volkmann recently.