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

AbstractLet 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 {f1,f2,…,fd} of signed dominating functions on G with the property that ∑i=1dfi(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 dS(G). If v is a vertex of a graph G, then dG(v) is the degree of the vertex v.In this note we show that dS(G)=1 if either G contains a vertex of degree 3 or G contains a cycle Cp=u1u2…upu1 of length p≥4 such that p≢0(mod3) and dG(ui)≤3 for 1≤i≤p−1. In particular, dS(G)=1 for each grid and each cylinder different from the cycle Cp with the property that p≡0(mod3)