The signed Roman k-domatic number of digraphs

Let k ≥ 1 be an integer. A signed Roman k-dominating function on a digraph D is a function f : V (D) −→ {−1, 1, 2} such that ∑x∈N−[v] f(x) ≥ k for every v ∈ V (D), where N−[v] consists of v and all in-neighbors of v, and every vertex u ∈ V (D) for which f(u) = −1 has an in-neighbor w for which f(w) = 2. A set {f1, f2, . . . , fd} of distinct signed Roman k-dominating functions on D with the property that ∑d i=1 fi(v) ≤ k for each v ∈ V (D), is called a signed Roman k-dominating family (of functions) on D. The maximum number of functions in a signed Roman k-dominating family on D is the signed Roman k-domatic number of G, denoted by dsR(D). In this paper we initiate the study of signed Roman k-domatic numbers in digraphs, and we present sharp bounds for dsR(D). In particular, we derive some Nordhaus-Gaddum type inequalities. In addition, we determine the signed Roman k-domatic number of some digraphs. 1 Terminology and introduction For notation and graph theory terminology, we in general follow Haynes, Hedetniemi and Slater [3]. In this paper we continue the study of Roman dominating functions in graphs and digraphs. Specifically, let G be a simple graph with vertex set V = V (G) and edge set E = E(G). The order |V | of G is denoted by n = n(G). For every vertex v ∈ V , the open neighborhood NG(v) = N(v) is the set {u ∈ V (G) | uv ∈ E(G)} and the closed neighborhood of v is the set NG[v] = N [v] = N(v) ∪ {v}. The degree of a vertex v ∈ V is d(v) = |N(v)|. The minimum and maximum degree of a graph G are denoted by δ = δ(G) and Δ = Δ(G), respectively. A graph G is regular or r-regular if d(v) = r for each vertex v of G. The complement of a graph G is denoted by G. We write Kn for the complete graph of order n, Kp,p for the complete bipartite graph of order 2p with equal size of partite sets, and Cn for the cycle of length n. L. VOLKMANN/AUSTRALAS. J. COMBIN. 64 (3) (2016), 444–457 445 If k ≥ 1 is an integer, then the signed Roman k-dominating function (SRkDF) on a graph G is defined in [4] as a function f : V (G) −→ {−1, 1, 2} such that ∑ u∈N [v] f(u) ≥ k for each v ∈ V (G), and every vertex u ∈ V (G) for which f(u) = −1 is adjacent to at least one vertex w for which f(w) = 2. The weight of an SRkDF f is the value ω(f) = ∑ v∈V f(v). The signed Roman k-domination number of a graph G, denoted by γ sR(G), equals the minimum weight of an SRkDF on G. The special case k = 1 was introduced and investigated in [1]. For γ sR(G) we also write γsR(G). A concept dual in a certain sense to the domination number is the domatic number, introduced by Cockayne and Hedetniemi [2]. They have defined the domatic number d(G) of a graph G by means of sets. A partition of V (G), all of whose classes are dominating sets in G, is called a domatic partition. The maximum number of classes of a domatic partition of G is the domatic number d(G) of G. But Rall has defined a variant of the domatic number of G, namely the fractional domatic number of G, using functions on V (G). (This was mentioned by Slater and Trees in [9].) Analogous to the fractional domatic number we may define the signed Roman k-domatic number. A set {f1, f2, . . . , fd} of distinct signed Roman k-dominating functions on G with the property that ∑d i=1 fi(v) ≤ k for each v ∈ V (G), is called in [10] a signed Roman k-dominating family (of functions) on G. The maximum number of functions in a signed Roman k-dominating family (SRkD family) on G is the signed Roman kdomatic number of G, denoted by dsR(G). If k = 1, then we write d 1 sR(G) = dsR(G). This case was introduced and investigated in [6]. The signed Roman k-domatic number is well-defined and dsR(G) ≥ 1 for all graphs G with δ(G) ≥ k− 1, since the set consisting of any SRkDF forms an SRkD family on G. Now let D be a finite and simple digraph with vertex set V (D) and arc set A(D). The integers n = n(D) = |V (D)| and m = m(D) = |A(D)| are the order and size of the digraph D, respectively. We write d+D(v) = d (v) for the out-degree of a vertex v and dD(v) = d −(v) for its in-degree. The minimum and maximum in-degree are δ−(D) = δ− and Δ−(D) = Δ− and the minimum and maximum out-degree are δ(D) = δ and Δ(D) = Δ. The sets N D (v) = N (v) = {x|(v, x) ∈ A(D)} and N− D (v) = N −(v) = {x|(x, v) ∈ A(D)} are called the out-neighborhood and in-neighborhood of the vertex v. Likewise, N D [v] = N [v] = N(v) ∪ {v} and N− D [v] = N −[v] = N−(v) ∪ {v}. If X ⊆ V (D), then D[X ] is the subdigraph induced by X. For an arc (x, y) ∈ A(D), the vertex y is an out-neighbor of x and x is an in-neighbor of y, and we also say that x dominates y or y is dominated by x. A digraph D is out-regular or r-out-regular if δ(D) = Δ(D) = r. A digraph D is in-regular or r-in-regular if δ−(D) = Δ−(D) = r. A digraph D is regular or r-regular if δ−(D) = Δ−(D) = δ(D) = Δ(D) = r. The complement D of a digraph D is the digraph with vertex set V (D) such that for any two distinct vertices u, v the arc (u, v) belongs to D if and only if (u, v) does not belong to D. If k ≥ 1 is an integer, then the signed Roman k-dominating function (SRkDF) on a digraph D is defined in [11] as a function f : V (D) −→ {−1, 1, 2} such that ∑ u∈N−[v] f(u) ≥ k for each v ∈ V (D), and such that every vertex u ∈ V (D) for L. VOLKMANN/AUSTRALAS. J. COMBIN. 64 (3) (2016), 444–457 446 which f(u) = −1 has an in-neighbor w for which f(w) = 2. The weight of an SRkDF f is the value ω(f) = ∑ v∈V (D) f(v). The signed Roman k-domination number of a digraph D, denoted by γ sR(D), equals the minimum weight of an SRkDF on D. A γ sR(D)-function is an SRkDF on D with weight γ k sR(D). If k = 1, then we write γ sR(D) = γsR(D). This case was introduced and studied in [8]. A set {f1, f2, . . . , fd} of distinct SRkDF on a digraph D with the property that ∑d i=1 fi(v) ≤ k for each v ∈ V (D), is called a signed Roman k-dominating family (of functions) on D. The maximum number of functions in a signed Roman kdominating family (SRkD family) on D is the signed Roman k-domatic number of D, denoted by dsR(D). If k = 1, then we write d 1 sR(G) = dsR(G). This case was introduced and investigated in [7]. The signed Roman k-domination number exists when δ− ≥ k 2 − 1. However, for investigations of the signed Roman k-dominating number and the signed Roman k-domatic number it is reasonable to claim that δ−(D) ≥ k − 1. Thus we assume throughout this paper that δ−(D) ≥ k − 1. The signed Roman k-domatic number is well-defined and dsR(D) ≥ 1 for all digraphs D, since the set consisting of the SRkDF with constant value 1 forms an SRkD family on D. Our purpose in this paper is to initiate the study of the signed Roman k-domatic number in digraphs. We first derive basic properties and bounds for the signed Roman k-domatic number of a digraph. In particular, we obtain the NordhausGaddum type result dsR(D) + d k sR(D) ≤ n + 1, and we discuss the equality in this inequality. In addition, we determine the signed Roman k-domatic number of some classes of digraphs. Some of our results are extensions of known properties of the signed Roman k-domatic number of graphs, given in [10]. We make use of the following results in this paper. Proposition A. ([8]) LetD be a digraph of order n. Then γsR(D) ≤ n with equality if and only if D is the disjoint union of isolated vertices and oriented triangles C3. Proposition B. ([11]) IfD is a digraph of order n with minimum in-degree δ−(D) ≥ k − 1, then γ sR(D) ≤ n. Proposition C. ([1, 4]) If Kn is the complete graph of order n ≥ k ≥ 1, then γ sR(Kn) = k, unless k = 1 and n = 3 in which case γsR(K3) = 2. Proposition D. ([6, 10]) If Kn is the complete graph of order n ≥ k ≥ 1, then dsR(Kn) = n, unless k = 1 and n = 3 in which case dsR(K3) = 1 and unless n = k = 2 in which case dsR(K2) = 1. Proposition E. ([11]) If D is a digraph of order n with δ−(D) ≥ k + 1, then γ sR(D) ≤ n− 1. L. VOLKMANN/AUSTRALAS. J. COMBIN. 64 (3) (2016), 444–457 447 Proposition F. ([11]) If D is an δ-out-regular digraph of order n with δ ≥ k − 1, then γ sR(D) ≥ ⌈ kn δ + 1 ⌉ . Proposition G. ([4]) If k ≥ 2, then γ sR(Kk,k) = 2k. Proposition H. ([10]) If k ≥ 4 is an even integer, then dsR(Kk,k) = k. The associated digraph G∗ of a graph G is the digraph obtained from G when each edge e of G is replaced by two oppositely oriented arcs with the same ends as e. Since N− G∗ [v] = NG[v] for each vertex v ∈ V (G) = V (G∗), the following useful observation is valid. Observation 1. If G∗ is the associated digraph of the graph G, then γ sR(G ∗) = γ sR(G) and d k sR(G ∗) = dsR(G). Let K∗ n be the associated digraph of the complete graph Kn. Using Observation 1 and Propositions C, D, we obtain the signed Roman k-domination number and the signed Roman k-domatic number of the complete digraph K∗ n. Corollary 2. If K∗ n is the complete digraph of order n ≥ k ≥ 1, then γ sR(K n) = k, unless k = 1 and n = 3 in which case γsR(K ∗ 3 ) = 2. Corollary 3. If K∗ n is the complete digraph of order n ≥ k ≥ 1, then dsR(K n) = n, unless k = 1 and n = 3 in which case dsR(K ∗ 3) = 1 and unless n = k = 2 in which case dsR(K ∗ 2) = 1. Let K∗ p,p be the associated digraph of the complete bipartite graph Kp,p. Observation 1, Propositions G and H lead to the next results immediately. Corollary 4. If k ≥ 2, then γ sR(K k,k) = 2k. Corollary 5. If k ≥ 4 is an even integer, then dsR(K k,k) = k. 2 Bounds on the signed Roman k-domatic number In this section we present basic properties of dsR(D) and sharp bounds on the signed Roman k-domatic number of a graph. Theorem 2.1. If D is a digraph with δ−(D) ≥ k − 1, then dsR(D) ≤ δ−(D) + 1. Moreover, if dsR(D) = δ −(D) + 1, then for each SRkD family {f1, f2, . . . , fd} on D with d = dsR(D) and each vertex v of minimum in-degree, ∑ x∈N−[v] fi(x) = k for each function fi and ∑d i=1 fi(x) = k for all x ∈ N−[v]. L. VOLKMANN/AUSTRALAS. J. COMBIN. 64 (3) (2016), 444–457 448 Proof. Let {f1, f2, . . . , fd} be an SRkD family on D such that d = dsR(D). If v is a vertex of minimum in-degree δ−(D), then we deduce that kd ≤ d ∑ i=1 ∑ x∈N−[v] fi(x) = ∑ x∈N−[v] d ∑