Bounds on the signed domatic number

Let G be a finite and simple graph with the vertex set V ( G ) , and let f : V ( G ) → { − 1 , 1 } be a two-valued function. If ∑ x ∈ N [ v ] f ( x ) ≥ 1 for each v ∈ V ( G ) , where N [ v ] is the closed neighborhood of v , then f is a signed dominating function on G . A set { f 1 , f 2 , … , f d } of signed dominating functions on G with the property that ∑ i = 1 d f i ( x ) ≤ 1 for each x ∈ V ( G ) is called a signed dominating family (of functions) on G . The maximum number of functions in a signed dominating family on G is the signed domatic number of G , denoted by d S ( G ) . s note we present upper bounds on d S ( G ) for regular graphs, and we give the Nordhaus–Gaddum type result d S ( G ) + d S ( G ¯ ) ≤ n + 1 for any graph G of order n , where G ¯ is the complement of G . In addition, we show that d S ( G ) + d S ( G ¯ ) = n + 1 if and only if n is odd and G = K n or G ¯ = K n , where K n is the complete graph of order n .