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

Let GG be a finite and simple graph with the vertex set V(G)V(G), and let f:V(G)→{−1,1}f:V(G)→{−1,1} be a two-valued function. If ∑x∈N[v]f(x)≥1∑x∈N[v]f(x)≥1 for each v∈V(G)v∈V(G), where N[v]N[v] is the closed neighborhood of vv, then ff is a signed dominating function on GG. A set {f1,f2,…,fd}{f1,f2,…,fd} of signed dominating functions on GG with the property that ∑i=1dfi(x)≤1 for each x∈V(G)x∈V(G) is called a signed dominating family (of functions) on GG. The maximum number of functions in a signed dominating family on GG is the signed domatic number of GG, denoted by dS(G)dS(G). If vv is a vertex of a graph GG, then dG(v)dG(v) is the degree of the vertex vv.In this note we show that dS(G)=1dS(G)=1 if either GG contains a vertex of degree 3 or GG contains a cycle Cp=u1u2…upu1Cp=u1u2…upu1 of length p≥4p≥4 such that p≢0(mod3) and dG(ui)≤3dG(ui)≤3 for 1≤i≤p−11≤i≤p−1. In particular, dS(G)=1dS(G)=1 for each grid and each cylinder different from the cycle CpCp with the property that p≡0(mod3).