The signed domatic number of some regular graphs

Let GG be a finite and simple graph with 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 on GG. In this paper, we investigate the signed domatic number of some circulant graphs and of the torus Cp×CqCp×Cq.