The signed domatic number of some regular graphs

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 ∑ 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 Gwith the property that ∑d i=1 fi(x) ≤ 1 for each x ∈ V (G), is called a signed dominating family (of functions) onG. Themaximumnumber of functions in a signed dominating family on G is the signed domatic number on G. In this paper, we investigate the signed domatic number of some circulant graphs and of the torus Cp × Cq. © 2008 Elsevier B.V. All rights reserved. 1. Terminology and introduction Weconsider finite, undirected and simple graphsGwith vertex set V (G). If v is a vertex of the graphG, thenN(v) = NG(v) is the open neighborhood of v, i.e., the set of all vertices adjacent with v. The closed neighborhood N[v] = NG[v] of a vertex v consists of the vertex set N(v) ∪ {v}. The number dG(v) = d(v) = |N(v)| is the degree of the vertex v ∈ V (G), and δ(G) is the minimum degree of G. The cycle of order n is denoted by Cn. If A ⊆ V (G) and f is a mapping from V (G) into some set of numbers, then f (A) = ∑ x∈A f (x). The signed dominating function is defined in [2] as a two-valued function f : V (G)→ {−1, 1} such that ∑ x∈N[v] f (x) ≥ 1 for each v ∈ V (G). The sum f (V (G)) is called the weight w(f ) of f . The minimum of weights w(f ), taken over all signed dominating functions f on G, is called the signed domination number of G, denoted by γS(G). Signed domination has been studied in [2–4,7,8,11]. Further information on this parameter can be found in themonographs [5,6] by Haynes, Hedetniemi and Slater. A set {f1, f2, . . . , fd} of signed dominating functions on Gwith the property that ∑d i=1 fi(x) ≤ 1 for each vertex x ∈ 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 of G, denoted by dS(G). The signed domatic number was introduced by Volkmann and Zelinka [10]. Volkmann andZelinka [10] andVolkmann [9] have determined the signeddomatic number of complete graphs and complete bipartite graphs, respectively. In addition, Volkmann and Zelinka [10] presented the following two basic results, which are useful for our investigations. Theorem 1.1 (Volkmann, Zelinka [10] 2005). If G is a graph, then 1 ≤ dS(G) ≤ δ(G)+ 1. ∗ Corresponding author. Tel.: +49 241 80 94999; fax: +49 241 8092 136. E-mail addresses:[email protected] (D. Meierling), [email protected] (L. Volkmann). 0166-218X/$ – see front matter© 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.dam.2008.11.012 1906 D. Meierling et al. / Discrete Applied Mathematics 157 (2009) 1905–1912 Theorem 1.2 (Volkmann, Zelinka [10] 2005). The signed domatic number is an odd integer. Next we derive a structural result on 2r-regular graphs with maximal possible signed domatic number. Theorem 1.3. Let G be a 2r-regular graph, and let u be an arbitrary vertex of G. If d = dS(G) = 2r + 1 and {f1, f2, . . . , fd} is a signed domatic family of G, then ∑d i=1 fi(u) = 1 and ∑ x∈N[u] fi(x) = 1 for each u ∈ V (G) and each i ∈ {1, 2, . . . , 2r + 1}. Proof. Let u be an arbitrary vertex of G. Because of ∑d i=1 fi(u) ≤ 1, this sum contains at least r summands which have the value−1. Using the fact that ∑ x∈N[u] fi(x) ≥ 1 for each i ∈ {1, 2, . . . , 2r + 1}, we observe that each of these sums contains at least r + 1 summands which have the value 1. Consequently, the sum ∑