Domatic Number Of Graphs Research Articles

Weak signed Roman k-domatic number of a graph

Let k≥ 1 be an integer. A weak signed Roman k-dominating function on a graph G isa function f:V (G)→ {-1, 1, 2} such that ΣueN[v] f(u)≥ k for everyve V(G), where N[v] is the closed neighborhood of v.A set {f1,f2, ... ,fd} of distinct weak signed Roman k-dominatingfunctions on G with the property that Σ1≤i≤d fi(v)≤ k for each ve V(G), is called a weak signed Roman k-dominating family (of functions) on G. The maximum number of functionsin a weak signed Roman k-dominating family on G is the weak signed Roman k-domatic number} of Gdenoted by dwsR k(G). In this paper we initiate the study of the weak signed Roman $k$-domatic numberin graphs, and we present sharp bounds for dwsR k(G). In addition, we determine the weak signed Roman k-domatic number of some graphs.

Regular graphs with large Italian domatic number

For a graph $G$, an Italian dominating function is a function $f: V(G) rightarrow {0,1,2}$ such that for each vertex $v in V(G)$ either $f(v) neq 0$, or $sum_{u in N(v)} f(u) geq 2$.If a family $mathcal{F} = {f_1, f_2, dots, f_t}$ of distinct Italian dominating functions satisfy $sum^t_{i = 1} f_i(v) leq 2$ for each vertex $v$, then this is called an Italian dominating family.In [L. Volkmann, The {R}oman {${2}$}-domatic number of graphs, Discrete Appl. Math. {bf 258} (2019), 235--241], Volkmann defined the textit{Italian domatic number} of $G$, $d_{I}(G)$, as the maximum cardinality of any Italian dominating family. In this same paper, questions were raised about the Italian domatic number of regular graphs. In this paper, we show that two of the conjectures are false, and examine some exceptions to a Nordhaus-Gaddum type inequality.

On the Roman {2}-domatic number of graphs

Let [Formula: see text] be a graph. A Roman[Formula: see text]-dominating function [Formula: see text] has the property that for every vertex [Formula: see text] with [Formula: see text], either [Formula: see text] is adjacent to a vertex assigned [Formula: see text] under [Formula: see text], or [Formula: see text] is adjacent to at least two vertices assigned [Formula: see text] under [Formula: see text]. A set [Formula: see text] of distinct Roman [Formula: see text]-dominating functions on [Formula: see text] with the property that [Formula: see text] for each [Formula: see text] is called a Roman[Formula: see text]-domination family (or functions) on [Formula: see text]. The maximum number of functions in a Roman [Formula: see text]-dominating family on [Formula: see text] is the Roman[Formula: see text]-domatic number of [Formula: see text], denoted by [Formula: see text]. In this paper, we answer two conjectures of Volkman [L. Volkmann, The Roman [Formula: see text]-domatic number of graphs, Discrete Appl. Math. 258 (2019) 235–241] about Roman [Formula: see text]-domatic number of graphs and we study this parameter for join of graphs and complete bipartite graphs.

Total Roman domatic number of a graph

A total Roman dominating function on a graph [Formula: see text] is a labeling [Formula: see text] such that every vertex with label 0 has a neighbor with label 2 and the subgraph of [Formula: see text] induced by the set of all vertices of positive weight has no isolated vertex. A set [Formula: see text] of total Roman dominating functions on [Formula: see text] with the property that [Formula: see text] for each [Formula: see text], is called a total Roman dominating family (of functions) on [Formula: see text]. The maximum number of functions in a total Roman dominating family on [Formula: see text] is the total Roman domatic number of [Formula: see text], denoted by [Formula: see text]. In this paper, we investigate the properties of total Roman domatic number in graphs. In particular, we present some sharp bounds for [Formula: see text] and we determine the total Roman domatic number of some special graphs.

English

English

On the signed total Roman domination and domatic numbers of graphs

A signed total Roman dominating function on a graph G is a function f:V(G)⟶{−1,1,2} such that ∑u∈N(v)f(u)≥1 for every vertex v∈V(G), where N(v) is the neighborhood of v, 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 signed total Roman domination number of a graph G, denoted by γstR(G), equals the minimum weight of a signed total Roman dominating function. A set {f1,f2,…,fd} of distinct signed total Roman dominating functions on G with the property that ∑i=1dfi(v)≤1 for each v∈V(G), is called a signed total Roman dominating family (of functions) on G. The maximum number of functions in a signed total Roman dominating family on G is the signed total Roman domatic number of G, denoted by dstR(G). In this paper we continue the investigation of the signed total Roman domination number, and we initiate the study of the signed total Roman domatic number in graphs. We present sharp bounds for γstR(G) and dstR(G). In addition, we determine the total signed Roman domatic number of some graphs.

Boundary Domination of Line and Middle Graph of Wheel Graph Families

Let G = (V;E) be a connected graph. A subset S of V (G) is called a boundary dominating set if every vertex of V S is boundary dominated by some vertex of S. The minimum taken over all boundary dominating sets of a graphG is called the boundary domination number ofG and is denoted by b(G). We define the boundary domatic number in graphs. Exact values of of Wheel Graph Families are obtained and some other interesting results are established.

Upper bounds on the signed total (k,k)-domatic number of graphs

Upper bounds on the signed total (k,k)-domatic number of graphs

The distance Roman domatic number of a graph

Let k be a positive integer, and let G be a simple graph with vertex set V(G). A k-distance Roman dominating function on G is a labeling f: V(G) → {0, 1, 2} such that every vertex with label 0 has a vertex with label 2 within distance k from each other. A set {f1, f2, …, fdd} of k-distance Roman dominating functions on G with the property that eqfi(v) ≤ 2 for each v ∊ V(G), is called a k-distance Roman dominating family (of functions) on G. The maximum number of functions in a k-distance Roman dominating family on G is the k-distance Roman domatic number of G, denoted by . In this paper we initiate the study of k-distance Roman domatic number in graphs and we present some sharp bounds for . In addition, we determine the k-distance Roman domatic number of some graphs.

The {k}-domatic number of a graph

For a positive integer k, a {k}-dominating function of a graph G is a function f from the vertex set V(G) to the set {0, 1, 2, . . . , k} such that for any vertex $${v\in V(G)}$$ , the condition $${\sum_{u\in N[v]}f(u)\ge k}$$ is fulfilled, where N[v] is the closed neighborhood of v. A {1}-dominating function is the same as ordinary domination. A set {f 1, f 2, . . . , f d } of {k}-dominating functions on G with the property that $${\sum_{i=1}^df_i(v)\le k}$$ for each $${v\in V(G)}$$ , is called a {k}-dominating family (of functions) on G. The maximum number of functions in a {k}-dominating family on G is the {k}-domatic number of G, denoted by d {k}(G). Note that d {1}(G) is the classical domatic number d(G). In this paper we initiate the study of the {k}-domatic number in graphs and we present some bounds for d {k}(G). Many of the known bounds of d(G) are immediate consequences of our results.

The (j, k)-domatic number of a graph

Let k ≥ j ≥ 1 be two integers, and let G be a simple graph such that j(δ(G)+1) ≥ k, where δ(G) is the minimum degree of G. A (j, k)-dominating function of a graph G is a function f from the vertex set V(G) to the set {0, 1, 2, …, j} such that for any vertex v ∊ V(G), the condition σu∊N[v] f(u) ≥ k is fulfilled, where N [v] is the closed neighborhood of v. A set {f1, f2, …, fd} of (j, k)-dominating functions on G with the property that σdi=1 fi(v ≤ j for each v ∊ V(G), is called a (j, k)-dominating family (of functions) on G. The maximum number of functions in a (j, k)-dominating family on G is the (j, k)-domatic number of G, denoted by d(j, k)(G). Note that d(1, 1)(G) is the classical domatic number d(G). In this paper we initiate the study of the (j, k)-domatic number in graphs and we present some bounds for d(j, k) (G). Many of the known bounds of d(G) are immediate consequences of our results.

The total {k}-domatic number of a graph

For a positive integer k, a total {k}-dominating function of a graph G is a function f from the vertex set V(G) to the set {0,1,2,?,k} such that for any vertex v?V(G), the condition ? u?N(v) f(u)?k is fulfilled, where N(v) is the open neighborhood of v. A set {f 1,f 2,?,f d } of total {k}-dominating functions on G with the property that $\sum_{i=1}^{d}f_{i}(v)\le k$ for each v?V(G), is called a total {k}-dominating family (of functions) on G. The maximum number of functions in a total {k}-dominating family on G is the total {k}-domatic number of G, denoted by $d_{t}^{\{k\}}(G)$ . Note that $d_{t}^{\{1\}}(G)$ is the classic total domatic number d t (G). In this paper we initiate the study of the total {k}-domatic number in graphs and we present some bounds for $d_{t}^{\{k\}}(G)$ . Many of the known bounds of d t (G) are immediate consequences of our results.

The Roman domatic number of a graph

A Roman dominating function on a graph G is a labeling f : V ( G ) ⟶ { 0 , 1 , 2 } such that every vertex with label 0 has a neighbor with label 2. A set { f 1 , f 2 , … , f d } of Roman dominating functions on G with the property that ∑ i = 1 d f i ( v ) ≤ 2 for each v ∈ V ( G ) is called a Roman dominating family (of functions) on G . The maximum number of functions in a Roman dominating family on G is the Roman domatic number of G , denoted by d R ( G ) . In this work we initiate the study of the Roman domatic number in graphs and we present some sharp bounds for d R ( G ) . In addition, we determine the Roman domatic number of some graphs.

A fast localized algorithm for scheduling sensors

Given n sensors and m targets, a monitoring schedule is a partition of the sensor set such that each subset of the partition can monitor all the targets. Monitoring schedules are used for maximizing the time all the targets are monitored when there is no possibility of replacing the batteries of the sensors. Each subset of the partition is used for one unit of time, and thus the goal is to maximize the number of subsets in the partition. We make the assumption that any two sensors for which there is a target both can monitor can communicate in one hop. The Monitoring Schedule problem is closely related to the domatic number in graphs and from previous work one can obtain a logarithmic approximation with one round of communication. We consider the special case when the targets are on a curve (such as a road) and each sensor can monitor an interval of the curve. For this case an optimum schedule can be computed in centralized manner. However, in the worst case a localized algorithm requires a linear number of communication rounds to compute a solution which is better than a 2-approximation. We present a ( 2 + ε ) -approximate randomized localized algorithm with polylogarithmic number of communication rounds.