Signed domatic numbers of the complete bipartite graphs

Abstract

Let G be a finite and simple graph with vertex set, V(G), and let f: V(G) → {-1, 1} be a two-valued function. If Σ_{xε N[v]} f (x) ≥ 1 for each vε v(G), where N[v] is the closed neighborhood of v, then f is a signed dominating function on G. A set {f₁, f₂, ..., f_d} of signed dominating functions on G with the property that Σ_i=1 ^d fi(x) ≤ 1 for each x ε V(G), is called a signed dominating family on G. The maximum number of functions in a signed dominating family on G is the signed domatic number on G, denoted by d _s(G). In this paper we determine the signed domatic number d _s(K_p,q) for all complete bipartite graphs K_p,q.