Signed domatic number of a graph

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 on G, denoted by dS(G).The properties of the signed domatic number dS(G) are studied in this paper. In particular, we determine the signed domatic number of complete graphs, cycles, fans, and wheels