Some remarks on the signed domatic number of graphs with small minimum degree

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 ) . If v is a vertex of a graph G , then d G ( v ) is the degree of the vertex v . s note we show that d S ( G ) = 1 if either G contains a vertex of degree 3 or G contains a cycle C p = u 1 u 2 … u p u 1 of length p ≥ 4 such that p ≢ 0 ( mod 3 ) and d G ( u i ) ≤ 3 for 1 ≤ i ≤ p − 1 . In particular, d S ( G ) = 1 for each grid and each cylinder different from the cycle C p with the property that p ≡ 0 ( mod 3 ) .