# On Domatic Number of Some Rotationally Symmetric Graphs

Authors: Hassan Raza; Sunny Kumar Sharma; Muhammad Azeem
Journal: Journal of Mathematics
 (2023)

---

## Abstract

Domination is a well-known graph theoretic concept due to its significant real-world applications in several domains, such as design and communication network analysis, coding theory, and optimization. For a connected graphΓ=V,E, a subset U of VΓ is called a dominating set if every member present in V−U is adjacent to at least one member in U. The domatic partition is the partition of the vertices VΓ into the disjoint dominating set. The domatic number of the graph Γ is the maximum cardinality of the disjoint dominating sets. In this paper, we improved the results for the middle and central graphs of a cycle, respectively. Furthermore, we discuss the domatic number for some other cycle-related graphs and graphs of convex polytopes.


---

## Body

## 1. Introduction

The paper considers connected simple, nontrivial, and undirected graphs. For the notations and terminologies in graph theory, we follow the book by Diestel [1]. Let Γ=V,E be a graph with the vertex set V and the edge set E. The neighborhood of a vertex k∈V, denoted by NΓk, is defined as the set NΓk=l∈V|kl∈E. The degree (valency) of a vertex k is denoted by dΓk=NΓk. If the text is clear and there is no ambiguity, we omit the subscript Γ. The minimum and maximum degrees of a vertex in Γ are represented by δΓ and ΔΓ, respectively, where δΓ=mindk|k∈V and ΔΓ=maxdk|k∈V.The scarcity of energy, the most critical resource, is one of the key aspects of wireless ad hoc and sensor networks. Sensor nodes are typically battery-powered and can only be active for a short time before the battery runs out. Improved network lifetime (i.e., the time interval during which the network is capable of executing its intended activity) is an essential aspect of the system design in sensor networks, and the techniques for energy-saving are highly desirable.One of the most well-known strategies is to plan vertex (node) activity so that redundant vertices (nodes) can enter some sleep mode as frequently and for as long as possible. Vertices cannot send or receive messages while in this sleeping mode, although they consume hardly any energy. The amount of energy used in active mode when the CPU is working with total capacity is often orders of magnitude more than in sleep mode.Clustering the network, likewise, data collection has shown to be one of the most effective strategies when working with the complexity of sensor and wireless ad hoc networks. Clustering realizes spatial multiplexing in nonoverlapping clusters and enhances the use of scarce resources such as energy and bandwidth. Different types of clustering have been proposed in recent years, depending on the unique network organization problem under investigation. One popular technique is to select cluster heads so that each vertex is either a cluster head or has at least one cluster head in its general neighborhood. This type of clustering relates to the well-known dominating set problem when the given network is represented as a graphΓ=V,E. A set D⊆V is the dominating set of the graph Γ if u∈V/D occupies some vertex in the set D. The minimum cardinality of the dominating set D of Γ is called the domination (dominating) number. It is denoted as γΓ.Graph theory has attracted the attention of researchers from almost all domains, and the notion of domination possesses an essential role in graph theory in terms of the applicability of domination and its related parameters in various fields, see Haynes et al. [2, 3] for comprehensive treatment and some detailed surveys on (earlier) domination results in graphs. Node failure is a nonnegligible probability event in sensor and ad hoc wireless networks, one of their essential characteristics. A basic dominating set may not be the ideal type of clustering for applications where fault tolerance is crucial. The concept of a k-dominating set accounts for this increased fault tolerance by requiring that every vertex in its proximity has at least k cluster heads. A k-dominating set is a subset of Dk⊆V in which each vertex x∈V has at least k dominators in its immediate proximity (including itself) in Dk.The concept adopted in this paper is that, by having many disjoint dominating sets, we may activate them sequentially. For instance, the only vertices in the active set take over information-collecting tasks. In contrast, the rest of the vertices go into low-energy sleep mode for several plane networks. The maximum domatic partition problem is a classical graph-theoretic problem in which the main objective is to determine the maximum number of disjoint dominating sets. The domatic partition of the graphΓ is the partition of V∈Γ into a disjoint dominating set of Γ. The maximum cardinality of the disjoint dominating set of Γ is called the domatic number of the graph Γ. The domatic number can be denoted as domΓ. To elaborate this point, a domatic partition of the graph Γ is the partition Dℓ=D1,D2,…,Dh⊆V1≤ℓ≤h into a pairwise disjoint dominating set of the graph Γ. The domatic number can be considered in the following way as well. The domatic number of the graph Γ is the maximum number of classes of a partition of V such that each class is a dominating set of the graph Γ. In particular, the domatic problem is analogous to allocating resources in the computer network and also locating facilities in a communication network.Cockayne and Hedetniemi [4] were the first to establish the concept of domatic partition. This concept later proved to be quite beneficial in numerous situations, for instance, locating facilities in a network by Fujita et al. [14], cluster head rotation in sensor networks by Misra and Mandal [5], prolonging the life and conserving the energy of networks by Islam et al. [6] and Pemmaraju and Pirwani [7], respectively.The following result is due to Cockayne and Hedetniemi [4].Theorem 1.
For any graphΓ of order n, we have(1)domΓ≤δΓ+1.The graphs, which satisfy equation (1), are known as domatically full graphs.The paper is organized as follows. In Section2, the improved results are depicted for the middle and central graphs of cycles, respectively. In Section 3, we consider some more families of cycle-related graphs and obtain their domatic numbers. In Section 4, graphs of convex polytopes are considered for the domatic number. The improved result for the middle graph of a cycle is then utilized to study domatic numbers in some classes of convex polytopes.


## 2. Improved Results

The result for the middle graph of cycleMCn (where Cn represents the cycle graph on n vertices) was proved in [8], and is.Theorem 2.
[8] Let G=MCn and k≥1. Then,(i)
domG=3 for n=3k(ii)
domG=2 for n=3k+1 and n=3k+2
The improved result is shown for the middle graph of cycleMCn in the following section.


### 2.1. Middle Graph of Cycle

LetCn be the cycle graph on n vertices. The vertex set VCn=c1,c2,…,cn. When the definition of the middle graph of cycle is employed, then the set with the newly added vertices is as follows:(2)VMCn=c1,c2,…,cn∪d1,d2,…,dn.The corresponding edge set for the middle graph of cycle is given as follows:(3)EMCn=cici+1,cidi|i=1,…,n∪ci+1di|i=1,…,n−1.The order and size of the graphMCn are 2n and 3n, respectively. For the sake of simplicity, the the vertex set c1,c2,…,cn are called the inner vertices, and d1,d2,…,dn are called the outer vertices. The graph of MCn is shown in Figure 1. Moreover, where necessary cn+1=c1.Figure 1
The graph ofMCn.Theorem 3.
The domatic number for the middle graph of cycleMCn, when n≥1, is(4)domMCn=3,n≡0,2mod3;2,n≡1mod3.Proof.
In order to show this, we consider the following three cases:Case 1.
Whenn=3k and k≥2, the partition of set V of Γ into disjoint dominating sets is shown as follows.
LetD1=c3i+1,d3i+2|i=0,…,k−1. D2=c3i+2,d3i+1|i=0,…,k−1. D3=c3i+1,d3i+1|i=0,…,k−1.Case 2.
Whenn=3k+1 and k≥2, the partition of set V of Γ into disjoint dominating set is shown as follows.
LetD1=c3i+2,d3i+1∪c3k+1|i=0,…,k−1. D2=c3i+1|i=0,…,k−1∪d3i+1|i=0,…,k.Case 3.
Whenn=3k+2 and k≥2, the partition of set V in Γ into disjoint dominating set is shown as follows.
LetD1=c3i+1|i=0,…,k∪d3i+2|i=0,…,k−1. D2=c3i+2|i=0,…,k∪d3i+1|i=0,…,k−1. D3=c3i+1|i=0,…,k−1∪d3i+1∪d3k+2|i=0,…,k. It is not hard to observe that when n=3k,3k+2, D1∩D2∩D3=∅. Moreover, when n=3k+1, we have D1∩D2=∅. Furthermore, to show that the classes of the domatic partition set are also the dominating sets, Table 1 is exhibited.
It is quite clear from Table1, and the classes of the domatic partition set are also dominating.Table 1
The table for middle graph of cycleMCn.
nDℓv∈V/DℓDℓ∩Nv3kD1c3i+2c3i+1,d3i+2c3i+1c3i+4d3i+1c3i+1d3i+1c3i+4D2c3i+4i=0,…,k−2c3i+5,d3i+1c3i+1c3i+2,d3i+1d3i+1c3i+2d3i+2c3i+2c1c2,dnD3c3i+4i=0,…,k−2c3i+1,d3i+4c3i+2c3i+1,d3i+1d3i+2c3i+1d3i+1c3i+1c1cn,d13k+1D1c3i+4i=0,…,k−2c3i+2,d3i+1c3i+1i=0,…,k−1c3i+2,d3i+1d3i+1c3i+2d3i+2c3i+2c1c2,c3k+1d3k+1c3k+13k+2D1c3i+2i=0,…,k−1c3i+1,d3i+2c3i+1i=0,…,k−1c3i+4,d3i+2d3i+1i=0,…,kc3i+1d3i+1i=0,…,k−1c3i+1c3k+2c3k+1,c1d3k+2c1D2c3i+4i=0,…,k−1c3i+2,d3i+1c3i+1i=0,…,k−1c3i+2,d3i+1d3i+1i=0,…,kc3i+2d3i+2i=0,…,kc3i+2c1c2,c3k+2D3c3i+4i=0,…,k−1c3i+1,d3i+4c3i+2i=0,…,k−1c3i+1,d3i+1d3i+2i=0,…,k−1c3i+1d3i+1i=0,…,k−1c3i+1c1d1,d3k+2c3k+2d3k+1,d3k+2Observation 1.
LetΓ be the graph of middle graph of cycle MCn, then Γ is domatically full for n=3k and 3k+2.Example 1.
For better understanding, the casen=3k+2 for k=2 is discussed graphically. The graph of MC8 is exhibited, which shows that the domMC8=3. Let us consider a partition dominating set Dℓ=D1,D2,D31≤ℓ≤3. Now, let us consider the classes as D1=green color vertices, D2=red color vertices, and D3=black color vertices vertices, respectively. Now, it is obvious from Figure 2 that D1∩D2∩D3=∅. Furthermore, the classes of disjoint dominating set are dominating.
The improved result is shown for the central graph of cycleCCn in the following section.Figure 2
The graph ofMC8.


### 2.2. Central Cycle Graph

This graph is constructed by considering the graph of a cycle onn vertices. Then, VCn=c1,c2,…,cn be the vertex set. The corresponding edge set is ECn=cici+1|1≤i≤n−1. From the definition of central graph [8], the vertex set of central graph of cycle VCCn=ci,di|1≤i≤n. In the central graph of cycle, it is imperative to note that the vertex ci is not adjacent to the vertices ci+1 and ci−1 for 1≤i≤n.The result for the central graph of cycleCCn (where Cn represents the cycle graph on n vertices) was proved in [8], and is.Theorem 4 (see [8]).
For anyn≥1,(i)
domCC3n=3 and(ii)
domCC3n+1=domCC3n+2=2Proof.
We first examine the statement of Theorem4, before improved result is presented.(i)
domCC3n=3, holds for even n, but not for odd n(ii)
domCC3n+1=domCC3n+2=2, holds for odd n, but not for even n
As an illustration, we show graphically in Figure3, when n is even domCC3n+2≠2, but domCC3n+2=3. Let us consider a domating partition set Dℓ=D1,D2,D31≤ℓ≤3. Now, let us consider the classes as D1=red color vertices, D2=black color vertices, and D3=green color vertices vertices, respectively. Thus, D1∩D2∩D3=∅, and this is the maximum disjoint dominating sets, and these classes of domatic partition show domination.
An improved result is shown for the central cycle graphCCn.Figure 3
The graph ofCC8.Theorem 5.
The domatic number for the central cycle graphCCn when n≥2 is(5)domCCn=3,n≡0,2,4mod6;2,n≡1,3,5mod6.Proof.
The proof is divided into six cases as illustrated.Case 4.
Whenn=6k and k≥1, the partition of the set V of Γ into disjoint dominating set is shown as follows.
LetD1=c6i+1,c6i+3,c6i+5|i=0,…,k−1. D2=c6i+2,c6i+4,c6i+1|i=0,…,k−1. D3=d1,d2,…,dn.Case 5.
Whenn=6k+1 and k≥2, the partition of the set V of Γ into disjoint dominating set is shown as follows.
LetD1=d1,d2,…,dn.(6)D2=c6i+1|i=0,…,k∪c6i+3,c6i+5|i=0,…,k−1.Case 6.
Whenn=6k+2 and k≥2, the partition of the set V of Γ into disjoint dominating set is shown as follows.
LetD1=d1,d2,…,dn.(7)D2=c6i+1|i=0,…,k∪c6i+3,c6i+5|i=0,…,k−1,D3=c6i+2|i=0,…,k∪c6i+4,c6i+1|i=0,…,k−1.Case 7.
Whenn=6k+3 and k≥2, the partition of the set V of Γ into disjoint dominating set is shown as follows.
LetD1=d1,d2,…,dn.(8)D2=c6i+1,c6i+3|i=0,…,k∪c6i+5|i=0,…,k−1.Case 8.
Whenn=6k+4 and k≥2, the partition of the set V of Γ into disjoint dominating set is shown as follows.
LetD1=d1,d2,…,dn,(9)D2=c6i+1,c6i+3|i=0,…,k∪c6i+5|i=0,…,k−1,D3=c6i+2,c6i+4|i=0,…,k∪c6i+1|i=0,…,k−1.Case 9.
Whenn=6k+5 and k≥2, the partition of the set V of Γ into disjoint dominating set is shown as follows.
LetD1=c6i+1,c6i+3,c6i+5|i=0,…,k. D2=d1,d2,…,dn.
It is not difficult to observe that whenn=6k,6k+2,6k+4, D1∩D2∩D3=∅, and when n=6k+1,6k+3,6k+5, D1∩D2=∅, respectively.Observation 2.
LetΓ be the graph of central graph of cycle CCn, then Γ is domatically full for n=6k,6k+2,6k+4.


## 2.1. Middle Graph of Cycle

LetCn be the cycle graph on n vertices. The vertex set VCn=c1,c2,…,cn. When the definition of the middle graph of cycle is employed, then the set with the newly added vertices is as follows:(2)VMCn=c1,c2,…,cn∪d1,d2,…,dn.The corresponding edge set for the middle graph of cycle is given as follows:(3)EMCn=cici+1,cidi|i=1,…,n∪ci+1di|i=1,…,n−1.The order and size of the graphMCn are 2n and 3n, respectively. For the sake of simplicity, the the vertex set c1,c2,…,cn are called the inner vertices, and d1,d2,…,dn are called the outer vertices. The graph of MCn is shown in Figure 1. Moreover, where necessary cn+1=c1.Figure 1
The graph ofMCn.Theorem 3.
The domatic number for the middle graph of cycleMCn, when n≥1, is(4)domMCn=3,n≡0,2mod3;2,n≡1mod3.Proof.
In order to show this, we consider the following three cases:Case 1.
Whenn=3k and k≥2, the partition of set V of Γ into disjoint dominating sets is shown as follows.
LetD1=c3i+1,d3i+2|i=0,…,k−1. D2=c3i+2,d3i+1|i=0,…,k−1. D3=c3i+1,d3i+1|i=0,…,k−1.Case 2.
Whenn=3k+1 and k≥2, the partition of set V of Γ into disjoint dominating set is shown as follows.
LetD1=c3i+2,d3i+1∪c3k+1|i=0,…,k−1. D2=c3i+1|i=0,…,k−1∪d3i+1|i=0,…,k.Case 3.
Whenn=3k+2 and k≥2, the partition of set V in Γ into disjoint dominating set is shown as follows.
LetD1=c3i+1|i=0,…,k∪d3i+2|i=0,…,k−1. D2=c3i+2|i=0,…,k∪d3i+1|i=0,…,k−1. D3=c3i+1|i=0,…,k−1∪d3i+1∪d3k+2|i=0,…,k. It is not hard to observe that when n=3k,3k+2, D1∩D2∩D3=∅. Moreover, when n=3k+1, we have D1∩D2=∅. Furthermore, to show that the classes of the domatic partition set are also the dominating sets, Table 1 is exhibited.
It is quite clear from Table1, and the classes of the domatic partition set are also dominating.Table 1
The table for middle graph of cycleMCn.
nDℓv∈V/DℓDℓ∩Nv3kD1c3i+2c3i+1,d3i+2c3i+1c3i+4d3i+1c3i+1d3i+1c3i+4D2c3i+4i=0,…,k−2c3i+5,d3i+1c3i+1c3i+2,d3i+1d3i+1c3i+2d3i+2c3i+2c1c2,dnD3c3i+4i=0,…,k−2c3i+1,d3i+4c3i+2c3i+1,d3i+1d3i+2c3i+1d3i+1c3i+1c1cn,d13k+1D1c3i+4i=0,…,k−2c3i+2,d3i+1c3i+1i=0,…,k−1c3i+2,d3i+1d3i+1c3i+2d3i+2c3i+2c1c2,c3k+1d3k+1c3k+13k+2D1c3i+2i=0,…,k−1c3i+1,d3i+2c3i+1i=0,…,k−1c3i+4,d3i+2d3i+1i=0,…,kc3i+1d3i+1i=0,…,k−1c3i+1c3k+2c3k+1,c1d3k+2c1D2c3i+4i=0,…,k−1c3i+2,d3i+1c3i+1i=0,…,k−1c3i+2,d3i+1d3i+1i=0,…,kc3i+2d3i+2i=0,…,kc3i+2c1c2,c3k+2D3c3i+4i=0,…,k−1c3i+1,d3i+4c3i+2i=0,…,k−1c3i+1,d3i+1d3i+2i=0,…,k−1c3i+1d3i+1i=0,…,k−1c3i+1c1d1,d3k+2c3k+2d3k+1,d3k+2Observation 1.
LetΓ be the graph of middle graph of cycle MCn, then Γ is domatically full for n=3k and 3k+2.Example 1.
For better understanding, the casen=3k+2 for k=2 is discussed graphically. The graph of MC8 is exhibited, which shows that the domMC8=3. Let us consider a partition dominating set Dℓ=D1,D2,D31≤ℓ≤3. Now, let us consider the classes as D1=green color vertices, D2=red color vertices, and D3=black color vertices vertices, respectively. Now, it is obvious from Figure 2 that D1∩D2∩D3=∅. Furthermore, the classes of disjoint dominating set are dominating.
The improved result is shown for the central graph of cycleCCn in the following section.Figure 2
The graph ofMC8.


## 2.2. Central Cycle Graph

This graph is constructed by considering the graph of a cycle onn vertices. Then, VCn=c1,c2,…,cn be the vertex set. The corresponding edge set is ECn=cici+1|1≤i≤n−1. From the definition of central graph [8], the vertex set of central graph of cycle VCCn=ci,di|1≤i≤n. In the central graph of cycle, it is imperative to note that the vertex ci is not adjacent to the vertices ci+1 and ci−1 for 1≤i≤n.The result for the central graph of cycleCCn (where Cn represents the cycle graph on n vertices) was proved in [8], and is.Theorem 4 (see [8]).
For anyn≥1,(i)
domCC3n=3 and(ii)
domCC3n+1=domCC3n+2=2Proof.
We first examine the statement of Theorem4, before improved result is presented.(i)
domCC3n=3, holds for even n, but not for odd n(ii)
domCC3n+1=domCC3n+2=2, holds for odd n, but not for even n
As an illustration, we show graphically in Figure3, when n is even domCC3n+2≠2, but domCC3n+2=3. Let us consider a domating partition set Dℓ=D1,D2,D31≤ℓ≤3. Now, let us consider the classes as D1=red color vertices, D2=black color vertices, and D3=green color vertices vertices, respectively. Thus, D1∩D2∩D3=∅, and this is the maximum disjoint dominating sets, and these classes of domatic partition show domination.
An improved result is shown for the central cycle graphCCn.Figure 3
The graph ofCC8.Theorem 5.
The domatic number for the central cycle graphCCn when n≥2 is(5)domCCn=3,n≡0,2,4mod6;2,n≡1,3,5mod6.Proof.
The proof is divided into six cases as illustrated.Case 4.
Whenn=6k and k≥1, the partition of the set V of Γ into disjoint dominating set is shown as follows.
LetD1=c6i+1,c6i+3,c6i+5|i=0,…,k−1. D2=c6i+2,c6i+4,c6i+1|i=0,…,k−1. D3=d1,d2,…,dn.Case 5.
Whenn=6k+1 and k≥2, the partition of the set V of Γ into disjoint dominating set is shown as follows.
LetD1=d1,d2,…,dn.(6)D2=c6i+1|i=0,…,k∪c6i+3,c6i+5|i=0,…,k−1.Case 6.
Whenn=6k+2 and k≥2, the partition of the set V of Γ into disjoint dominating set is shown as follows.
LetD1=d1,d2,…,dn.(7)D2=c6i+1|i=0,…,k∪c6i+3,c6i+5|i=0,…,k−1,D3=c6i+2|i=0,…,k∪c6i+4,c6i+1|i=0,…,k−1.Case 7.
Whenn=6k+3 and k≥2, the partition of the set V of Γ into disjoint dominating set is shown as follows.
LetD1=d1,d2,…,dn.(8)D2=c6i+1,c6i+3|i=0,…,k∪c6i+5|i=0,…,k−1.Case 8.
Whenn=6k+4 and k≥2, the partition of the set V of Γ into disjoint dominating set is shown as follows.
LetD1=d1,d2,…,dn,(9)D2=c6i+1,c6i+3|i=0,…,k∪c6i+5|i=0,…,k−1,D3=c6i+2,c6i+4|i=0,…,k∪c6i+1|i=0,…,k−1.Case 9.
Whenn=6k+5 and k≥2, the partition of the set V of Γ into disjoint dominating set is shown as follows.
LetD1=c6i+1,c6i+3,c6i+5|i=0,…,k. D2=d1,d2,…,dn.
It is not difficult to observe that whenn=6k,6k+2,6k+4, D1∩D2∩D3=∅, and when n=6k+1,6k+3,6k+5, D1∩D2=∅, respectively.Observation 2.
LetΓ be the graph of central graph of cycle CCn, then Γ is domatically full for n=6k,6k+2,6k+4.


## 3. Cycle-Related Graphs

### 3.1. Wheel Graphs

The domatic number for cycle graphs is recently studied by Mary and Amutha [8] and is obtained as follows.Theorem 6.
Forn≥1,(10)domCn=3,n≡0mod3,2,n≡1,2mod3.Next, it is quite natural to study the notion of domatic number for wheel graphs. The wheel graph, denoted byWn, consists of a cycle Cn, which is called the rim, and a vertex is called the hub (or universal vertex), which is joined with all the vertices of Cn. The wheel graph Wn has order n+1, and the universal vertex is denoted as c0, and the rim as c1,c2,…,cn,c1.Theorem 7.
The domatic number for the wheel graphWn, when n≥3, is(11)domWn=4,n≡0mod3,3,n≡1,2mod3.Proof.
In order to show this, we consider the following three cases.Case 10.
Whenn=3k and k≥1, the partition of the set V of Γ into disjoint dominating set is shown as follows.
LetD1=c3i+1|i=0,…,k−1. D2=c3i+2|i=0,…,k−1. D3=c3i+1|i=0,…,k−1. D4=c0.Case 11.
Whenn=3k+1 and k≥1, the partition of the set V of Γ into disjoint dominating set is shown as follows.
LetD1=c3i+1|i=0,…,k. D2=c3i+2∪c3k|i=0,…,k−1. D3=c0.Case 12.
Whenn=3k+2 and k≥1, the partition of the set V of Γ into disjoint dominating set is shown as follows.
LetD1=c3i+1|i=0,…,k. D2=c3i+2|i=0,…,k. D3=c0.
It is not difficult to observe that whenn=3k, D1∩D2∩D3∩D4=∅ and when n=3k+1,3k+2, D1∩D2∩D3=∅, respectively.Observation 3.
LetΓ be the wheel graph Wn, then Γ is domatically full for n=3k+1,3k+2.


### 3.2. Squared Cycle Graph

The squared cycle graphs are the extension of the cycle graphs. The squared cycle graph, denoted byC2n, is of order n. The vertex set for simplicity can be written as VC2n=w1,w2,…,wn. The graph of C210 is shown in Figure 4. For each 1≤i≤n, we join wi to wi+1,wi+2 and to wi−1,wi−2. Furthermore, if these vertices are arranged cyclically w1,w2,…,wn, then each vertex wi is adjacent to the two vertices which follows that wi and two vertices that immediately precede wi. Thus, C2n is a four regular graph.Figure 4
The graph ofC102.The graph ofC52 is the complete graph so domC52=n, domC62=3, domC72=3, domC82=4, domC92=4.Theorem 8.
The domatic number for the squared cycle graphCn2, for n≥10, is(12)domCn2=5,n≡0mod5,3,n≡1,2,3mod5,4,n≡4mod5.Proof.
The proof of the theorem is split into the following five cases.Case 13.
Letn=5k and k≥2. The partition of the set V of Γ into disjoint dominating set is shown as follows.
LetD1=w5i+1|i=0,…,k−1. D2=w5i+2|i=0,…,k−1. D3=w5i+3|i=0,…,k−1. D4=w5i+4|i=0,…,k−1. D5=w5i+1|i=0,…,k−1.Case .
14. Letn=5k+1 and k≥2. The partition of the set V of Γ into disjoint dominating set is shown as follows.
LetD1=w5i+1∪w5k−1|i=0,…,k−1. D2=w5i+2∪w5k|i=0,…,k−1. D3=w5i+3∪w5k+1|i=0,…,k−1.Case 15.
Letn=5k+12 and k≥2. The partition of the set V of Γ into disjoint dominating set is shown as follows.
LetD1=w5i+1|i=0,…,k−1. D2=w5i+2|i=0,…,k−1. D3=w5i+3∪w5k|i=0,…,k−1.Case 16.
Letn=5k+3 and k≥2. The partition of the set V of Γ into disjoint dominating set is shown as follows.
LetD1=w5i+1|i=0,…,k−1. D2=w5i+2|i=0,…,k−1. D3=w5i+3|i=0,…,k−1.Case 17.
Letn=5k+4 and k≥2. The partition of the set V of Γ into disjoint dominating set is shown as follows.
LetD1=w5i+1∣i=0,…,k−1. D2=w5i+2|i=0,…,k−1. D3=w5i+3|i=0,…,k−1. D4=w5i+4∪w5k+4|i=0,…,k−1.
In all the cases considered for the squared cycle graph, there exist disjoint dominating of different cardinalities. Furthermore, it is not hard to analyze that all these mentioned partitions are dominating set as well.Observation 4.
The minimum degree for the squared cycle graph isfour. So for the case of n=5k, domCn2=5, it is domatically full graph.


## 3.1. Wheel Graphs

The domatic number for cycle graphs is recently studied by Mary and Amutha [8] and is obtained as follows.Theorem 6.
Forn≥1,(10)domCn=3,n≡0mod3,2,n≡1,2mod3.Next, it is quite natural to study the notion of domatic number for wheel graphs. The wheel graph, denoted byWn, consists of a cycle Cn, which is called the rim, and a vertex is called the hub (or universal vertex), which is joined with all the vertices of Cn. The wheel graph Wn has order n+1, and the universal vertex is denoted as c0, and the rim as c1,c2,…,cn,c1.Theorem 7.
The domatic number for the wheel graphWn, when n≥3, is(11)domWn=4,n≡0mod3,3,n≡1,2mod3.Proof.
In order to show this, we consider the following three cases.Case 10.
Whenn=3k and k≥1, the partition of the set V of Γ into disjoint dominating set is shown as follows.
LetD1=c3i+1|i=0,…,k−1. D2=c3i+2|i=0,…,k−1. D3=c3i+1|i=0,…,k−1. D4=c0.Case 11.
Whenn=3k+1 and k≥1, the partition of the set V of Γ into disjoint dominating set is shown as follows.
LetD1=c3i+1|i=0,…,k. D2=c3i+2∪c3k|i=0,…,k−1. D3=c0.Case 12.
Whenn=3k+2 and k≥1, the partition of the set V of Γ into disjoint dominating set is shown as follows.
LetD1=c3i+1|i=0,…,k. D2=c3i+2|i=0,…,k. D3=c0.
It is not difficult to observe that whenn=3k, D1∩D2∩D3∩D4=∅ and when n=3k+1,3k+2, D1∩D2∩D3=∅, respectively.Observation 3.
LetΓ be the wheel graph Wn, then Γ is domatically full for n=3k+1,3k+2.


## 3.2. Squared Cycle Graph

The squared cycle graphs are the extension of the cycle graphs. The squared cycle graph, denoted byC2n, is of order n. The vertex set for simplicity can be written as VC2n=w1,w2,…,wn. The graph of C210 is shown in Figure 4. For each 1≤i≤n, we join wi to wi+1,wi+2 and to wi−1,wi−2. Furthermore, if these vertices are arranged cyclically w1,w2,…,wn, then each vertex wi is adjacent to the two vertices which follows that wi and two vertices that immediately precede wi. Thus, C2n is a four regular graph.Figure 4
The graph ofC102.The graph ofC52 is the complete graph so domC52=n, domC62=3, domC72=3, domC82=4, domC92=4.Theorem 8.
The domatic number for the squared cycle graphCn2, for n≥10, is(12)domCn2=5,n≡0mod5,3,n≡1,2,3mod5,4,n≡4mod5.Proof.
The proof of the theorem is split into the following five cases.Case 13.
Letn=5k and k≥2. The partition of the set V of Γ into disjoint dominating set is shown as follows.
LetD1=w5i+1|i=0,…,k−1. D2=w5i+2|i=0,…,k−1. D3=w5i+3|i=0,…,k−1. D4=w5i+4|i=0,…,k−1. D5=w5i+1|i=0,…,k−1.Case .
14. Letn=5k+1 and k≥2. The partition of the set V of Γ into disjoint dominating set is shown as follows.
LetD1=w5i+1∪w5k−1|i=0,…,k−1. D2=w5i+2∪w5k|i=0,…,k−1. D3=w5i+3∪w5k+1|i=0,…,k−1.Case 15.
Letn=5k+12 and k≥2. The partition of the set V of Γ into disjoint dominating set is shown as follows.
LetD1=w5i+1|i=0,…,k−1. D2=w5i+2|i=0,…,k−1. D3=w5i+3∪w5k|i=0,…,k−1.Case 16.
Letn=5k+3 and k≥2. The partition of the set V of Γ into disjoint dominating set is shown as follows.
LetD1=w5i+1|i=0,…,k−1. D2=w5i+2|i=0,…,k−1. D3=w5i+3|i=0,…,k−1.Case 17.
Letn=5k+4 and k≥2. The partition of the set V of Γ into disjoint dominating set is shown as follows.
LetD1=w5i+1∣i=0,…,k−1. D2=w5i+2|i=0,…,k−1. D3=w5i+3|i=0,…,k−1. D4=w5i+4∪w5k+4|i=0,…,k−1.
In all the cases considered for the squared cycle graph, there exist disjoint dominating of different cardinalities. Furthermore, it is not hard to analyze that all these mentioned partitions are dominating set as well.Observation 4.
The minimum degree for the squared cycle graph isfour. So for the case of n=5k, domCn2=5, it is domatically full graph.


## 4. Convex Polytopes

A convex polytope is a polytope that is also a convex set of points in then-dimensional Euclidean space ℝn. The graph of a convex polytope is constructed by the vertices and edges with the same incidence relation. The graphs of convex polytopes were put forward by Bača [9, 10]. The author studied graceful and antigraceful labeling problems for these geometrically essential graphs. Recently, the graphs of convex polytopes are considered for the domination-related parameters. The binary locating domination number (resp. locating domination number) is studied by Simić et al. [11]. The open locating domination number in graphs of convex polytopes was studied by Savić et al. and Raza in [12, 13].


### 4.1. The Graph of Convex PolytopesDn

The graph of convex polytope defined in [10] consists of 2n5-sided faces as shown in Figure 5. From the mathematical point of view, the vertex and edge set for Dn are as follows.Figure 5
The graph ofDn.Vertex set ofDn is(13)VDn=ci,di,ei,fi|i=0,…,n−1,and the edge set of Dn is(14)EDn=cici+1,cidi,diei,di+1ei,eifi,fifi+1|i=0,…,n−1.In order to validate the vertex and edge set of graphDn, we fix n=6 and draw the graph D6 as shown in Figure 6.Figure 6
The graph ofDn.Theorem 9.
The domatic number for the graph of convex polytopeDn, when n≥6, is(15)domDn=3,n≡0mod3,2,n≡1,2mod3.Proof.
The proof is divided into three case0073 as illustrated.Case 18.
Whenn=3k and k≥2. The partition of the set V of Γ into disjoint dominating set is shown as follows.
LetD1=c3i+1,d3i+2,e3i+1,f3i+1|i=0,…,k−1, D2=c3i+2,d3i+1,e3i+2,f3i+2|i=0,…,k−1, and D3=c3i+1,d3i+1,e3i+1,f3i+1|i=0,…,k−1.Case 19.
Whenn=3k+1 and k≥2, the partition of the set V of Γ into disjoint dominating set is shown as follows.
LetD1=c3i+1,d3i+2,e3i+1,f3i+1∪c3k+1,f3k+1|i=0,…,k−1 and D2=c3i+1,d3i+1,e3i+2,f3i+2∪e3k+1,f3k|i=0,…,k−1.Case 20.
Whenn=3k+2 and k≥2, the partition of the set V of Γ into disjoint dominating set is shown as follows.
LetD1=c3i+1,d3i+2,e3i+1,f3i+2∪c3k+1,e3k+2,f3k+1|i=0,…,k−1 and D2=c3i+2,d3i+1,e3i+2,f3i+1∪c3k+2,d3k+1,f3k+1|i=0,…,k−1.
It is not hard to see that the classes of the domatic partitionD1∩D2∩D3=∅ for n=3k, and D1∩D2=∅ for n=3k+1,3k+2, respectively. To show that these are dominating sets as well, Table 2 is exhibited.
The family of convex polytopeℛn can be attained by expending the above-studied family of convex polytope Dn. The graph of a convex polytope ℛn consists of 2 n 5-sided faces, n 6-sided faces, and n-sided faces, as shown in Figure 7. The domatic number for the graph of convex polytope ℛn is the same as the graph of convex polytope Dn.Table 2
The table for graph of convex polytopeDn.
nDℓv∈V/DℓDℓ∩Nvv∈V/DℓDi∩Nvv∈V/DℓDi∩Nv3kD1c3i+2c3i+1,d3i+2c3i+1c3i+4d3i+1e3i+1d3i+4i=0,…,k−1c3i+4,e3i+1e3i+1d3i+2,f3i+1e3i+2d3i+2f3i+2f3i+1f3i+1f3i+4d1c1,enD2c3i+1c3i+2,d3i+1c3i+1c3i+2d3i+2c3i+2,e3i+2d3i+1e3i+2e3i+1d3i+1e3i+1d3i+4f3i+1f3i+2f3i+1f3i+2D3c3i+2c3i+1c3i+4i=0,…,k−1c3id3i+1e3i+1d3i+2e3i+1e3i+2d3i+1,f3i+2e3i+1d3i+1,f3i+1f3i+2f3i+1f3i+4i=0,…,k−1e3i+4,f3i+1c1cnf1e1,fn3k+1D1c3i+2c3i+1,d3i+2c3i+1c3i+4d3i+1e3i+1d3i+4i=0,…,k−1c3i+4,e3i+1e3i+1d3i+2,f3i+1e3i+2d3i+2f3i+2f3i+1f3i+1e3i+1,f3i+4d1c1d3k+1c3k+1,e3ke3k+1f3k+1D2c3i+2c3i+1c3i+4i=0,…,k−1c3i+1,d3i+4d3i+2e3i+2d3i+1c3i+1,e3i+2e3i+1d3i+1e3i+1i=0,…,k−1d3i+4f3i+1f3i+2f3i+1i=0,…,k−1f3i+2c1d1c3k+1c3kd3k+1e3k+1e3kf3kf3k+1f3k3k+2D1c3i+2c3i+1,d3i+2c3i+1c3i+4d3i+1e3i+1d3i+4i=0,…,k−1c3i+4,e3i+1e3i+1d3i+2e3i+2d3i+2,f3i+2f3i+1f3i+2f3i+1e3i+1,f3i+2c3k+2c1,c3k+1d3k+2e3k+2e3k+1f3k+1f3k+2f3k+1D2c3i+1c3i+2,d3i+1c3i+1c3i+2d3i+2c3i+2,e3i+2d3i+1e3i+2e3i+1d3i+1e3i+1d3i+4,f3i+1f3i+2e3i+2,f3i+1f3i+4i=0,…,k−1f3i+1c3k+1c3k+2,d3k+1d3k+2c3k+2e3k+1d3k+1e3k+2d1,f3k+2f3k+1f3k,f3k+2Figure 7
The graph ofℛn.Theorem 10.
The domatic number of graph of convex polytopeℛn, for n≥6, is(16)domDn=domℛn=3,n≡0mod3,2,n≡1,2mod3.Proof.
The proof is similar as shown for the graph of convex polytopeDn.
To validate the result in Theorem3 for the middle graph of a cycle, the graph of convex polytope Tn is considered, which is generated from the middle graph of a cycle. There are infinite classes of convex polytopes that show the same structural similarity.


### 4.2. The Graph of Convex PolytopeTn

The graph of convex polytopeTn, as shown in Figure 8, consists of 4 n 3-sided faces, n 4-sided faces, and a pair of n-sided faces. The vertex and edge set for Tn can be written as follows:(17)VTn=ci,di,ei,fi,ETn=cici+1,didi+1,eiei+1,fifi+1,∪cidi,diei,ci+1di,eifi,eifi+1.Figure 8
The graph ofTn.Theorem 11.
The domatic number of graph of convex polytopeTn, when n≥6, is(18)domTn=3,n≡0,2mod3,2,n≡1mod3.Proof.
In order to show this, we consider the following three cases.Case 21.
Whenn=3k and k≥2, the partition of the set V of Γ into disjoint dominating set is shown as follows.
LetD1=c3i+1,d3i+2,e3i+1,f3i+2|i=0,…,k−1, D2=c3i+2,d3i+1,e3i+2,f3i+1|i=0,…,k−1, and D3=c3i+1,d3i+1,e3i+1,f3i+1|i=0,…,k−1.Case 22.
Whenn=3k+1 and k≥2, the partition of the set V of Γ into disjoint dominating set is shown as follows.
LetD1=c3i+2,d3i+1,e3i+2,f3i+1∪c3k+1,e3k+1|i=0,…,k−1 and D2=c3i+1,d3i+1,e3i+1,f3i+1∪d3k+1,f3k+1|i=0,…,k−1.Case 23.
Whenn=3k+2 and k≥2, the partition of the set V of Γ into disjoint dominating set is shown as follows.
Let(19)D1=c3i+1|i=0,…,k∪d3i+2|i=0,…,k−1∪e3i+1|i=0,…,k∪f3i+2|i=0,…,k−1,D2=c3i+2|i=0,…,k∪d3i+1|i=0,…,k−1∪e3i+2|i=0,…,k∪f3i+1|i=0,…,k−1,D3=c3i+1|i=0,…,k−1∪d3i+1|i=0,…,k∪e3i+1|i=0,…,k−1∪f3i+1midi=0,…,k∪d3k+2,f3k+2.
It is not hard to observe that whenn=3k,3k+2, D1∩D2∩D3=∅. Moreover, when n=3k+1, we have D1∩D2=∅. Furthermore, to show that classes of the domatic partition set are also dominating set, Table 3 is presented which clearly indicates that the chosen classes are dominating.Table 3
The table for graph of convex polytopeTn.
nDℓv∈V/DℓDℓ∩Nvv∈V/DℓDℓ∩Nv3kD1c3i+2c3i+1,d3i+2c3i+1c3i+4,d3i+2d3i+1c3i+1,d3i+2,e3i+1d3i+1c3i+4,d3i+2e3i+2d3i+2,e3i+1,f3i+2e3i+1e3i+4f3i+1e3i+1,f3i+2f3i+1f3i+2D2c3i+1c3i+2,d3i+1c3i+4i=0,…,k−2c3i+5,d3i+1d3i+2c3i+2,d3i+1,e3i+2d3i+4i=0,…,k−2c3i+5,d3i+1e3i+1e3i+2e3i+1d3i+1,e3i+2,f3i+1f3i+2e3i+2,f3i+1f3i+4i=0,…,k−2e3i+2,f3i+1c1c2,dnd1c2,dnf1fnD3c3i+2c3i+1,d3i+1c3i+4i=0,…,k−2c3i+1,d3i+4d3i+2c3i+1,d3i+1d3i+1c3i+1,d3i+4,e3i+1e3i+2e3i+1e3i+4i=0,…,k−2d3i+4,e3i+1,f3i+4f3i+2f3i+1f3i+1e3i+1,f3i+4c1cn,d1e1d1,en,f13k+1D1c3i+1i=0,…,k−2c3i+2,d3i+1c3i+4i=0,…,k−2c3i+5,d3i+1d3i+2c3i+2,d3i+1,e3i+2d3i+4i=0,…,k−2c3i+5,d3i+1e3i+1d3i+1,e3i+2,f3i+1e3i+4i=0,…,k−2e3i+5f3i+2e3i+2,f3i+1f3i+4i=0,…,k−2f3i+1c1c2,c3k+1c3kc3k−1,c3k+2,d3kd1c2d3k+1c3k+1,d3ke1e2,e3k+1f1e3k+1f3k+1e3k+1,f3kD2c3i+2c3i+1,d3i+1c3i+4i=0,…,k−2c3i+1,d3i+4d3i+2c3i+1,d3i+1d3i+1c3i+1,d3i+4,e3i+1e3i+2e3i+1e3i+4i=0,…,k−2d3i+4,e3i+1,f3i+4f3i+2f3i+1f3i+1e3i+1,f3i+4c1d1,d3k+1c3k+1c3k,d1,d3k+1e1d1,f1e3k+1d3k+1,e3k,f3k+1,f13k+2D1c3i+2i=0,…,k−1c3i+1,d3i+2c3i+1i=0,…,k−1c3i+4,d3i+2d3i+1i=0,…,k−1c3i+1,d3i+2,e3i+1d3i+1i=0,…,k−1c3i+4,d3i+2e3i+2i=0,…,k−1d3i+2,e3i+1,f3i+2e3i+1i=0,…,k−1e3i+4f3i+1i=0,…,k−1e3i+1,f3i+2f3i+1i=0,…,k−1f3i+2c3k+2c3k+1,c1d3k+1c3k+1,e3k+1e3k+2e3k+1,e1f3k+1e3k+1f3k+2e3k+1D2c3i+1i=0,…,k−1c3i+2,d3i+1c3i+4i=0,…,k−1c3i+5,d3i+1d3i+1i=0,…,kc3i+2d3i+2i=0,…,k−1c3i+2,d3i+1,e3i+2e3i+1i=0,…,ke3i+2e3i+1i=0,…,k−1d3i+1,e3i+2,f3i+1f3i+2i=0,…,k−1e3i+2,f3i+1f3i+4i=0,…,k−1f3i+1c1c2,e3k+2d3k+2c3k+2,e3k+2e3k+1e3k+2f1e3k+2f3k+2e3k+2D3c3i+2i=0,…,k−1c3i+1,d3i+1c3i+4i=0,…,k−1c3i+1,d3i+4d3i+2i=0,…,k−1c3i+1,d3i+1d3i+1i=0,…,k−1c3i+1,d3i+4,e3i+1e3i+2i=0,…,k−1e3i+1e3i+4i=0,…,k−1d3i+4,e3i+1f3i+2i=0,…,k−1f3i+1f3i+1i=0,…,k−1e3i+1,f3i+4c1d1,d3k+2c3k+2d3k+1,d3k+2e1d1e3k+2d3k+2


## 4.1. The Graph of Convex PolytopesDn

The graph of convex polytope defined in [10] consists of 2n5-sided faces as shown in Figure 5. From the mathematical point of view, the vertex and edge set for Dn are as follows.Figure 5
The graph ofDn.Vertex set ofDn is(13)VDn=ci,di,ei,fi|i=0,…,n−1,and the edge set of Dn is(14)EDn=cici+1,cidi,diei,di+1ei,eifi,fifi+1|i=0,…,n−1.In order to validate the vertex and edge set of graphDn, we fix n=6 and draw the graph D6 as shown in Figure 6.Figure 6
The graph ofDn.Theorem 9.
The domatic number for the graph of convex polytopeDn, when n≥6, is(15)domDn=3,n≡0mod3,2,n≡1,2mod3.Proof.
The proof is divided into three case0073 as illustrated.Case 18.
Whenn=3k and k≥2. The partition of the set V of Γ into disjoint dominating set is shown as follows.
LetD1=c3i+1,d3i+2,e3i+1,f3i+1|i=0,…,k−1, D2=c3i+2,d3i+1,e3i+2,f3i+2|i=0,…,k−1, and D3=c3i+1,d3i+1,e3i+1,f3i+1|i=0,…,k−1.Case 19.
Whenn=3k+1 and k≥2, the partition of the set V of Γ into disjoint dominating set is shown as follows.
LetD1=c3i+1,d3i+2,e3i+1,f3i+1∪c3k+1,f3k+1|i=0,…,k−1 and D2=c3i+1,d3i+1,e3i+2,f3i+2∪e3k+1,f3k|i=0,…,k−1.Case 20.
Whenn=3k+2 and k≥2, the partition of the set V of Γ into disjoint dominating set is shown as follows.
LetD1=c3i+1,d3i+2,e3i+1,f3i+2∪c3k+1,e3k+2,f3k+1|i=0,…,k−1 and D2=c3i+2,d3i+1,e3i+2,f3i+1∪c3k+2,d3k+1,f3k+1|i=0,…,k−1.
It is not hard to see that the classes of the domatic partitionD1∩D2∩D3=∅ for n=3k, and D1∩D2=∅ for n=3k+1,3k+2, respectively. To show that these are dominating sets as well, Table 2 is exhibited.
The family of convex polytopeℛn can be attained by expending the above-studied family of convex polytope Dn. The graph of a convex polytope ℛn consists of 2 n 5-sided faces, n 6-sided faces, and n-sided faces, as shown in Figure 7. The domatic number for the graph of convex polytope ℛn is the same as the graph of convex polytope Dn.Table 2
The table for graph of convex polytopeDn.
nDℓv∈V/DℓDℓ∩Nvv∈V/DℓDi∩Nvv∈V/DℓDi∩Nv3kD1c3i+2c3i+1,d3i+2c3i+1c3i+4d3i+1e3i+1d3i+4i=0,…,k−1c3i+4,e3i+1e3i+1d3i+2,f3i+1e3i+2d3i+2f3i+2f3i+1f3i+1f3i+4d1c1,enD2c3i+1c3i+2,d3i+1c3i+1c3i+2d3i+2c3i+2,e3i+2d3i+1e3i+2e3i+1d3i+1e3i+1d3i+4f3i+1f3i+2f3i+1f3i+2D3c3i+2c3i+1c3i+4i=0,…,k−1c3id3i+1e3i+1d3i+2e3i+1e3i+2d3i+1,f3i+2e3i+1d3i+1,f3i+1f3i+2f3i+1f3i+4i=0,…,k−1e3i+4,f3i+1c1cnf1e1,fn3k+1D1c3i+2c3i+1,d3i+2c3i+1c3i+4d3i+1e3i+1d3i+4i=0,…,k−1c3i+4,e3i+1e3i+1d3i+2,f3i+1e3i+2d3i+2f3i+2f3i+1f3i+1e3i+1,f3i+4d1c1d3k+1c3k+1,e3ke3k+1f3k+1D2c3i+2c3i+1c3i+4i=0,…,k−1c3i+1,d3i+4d3i+2e3i+2d3i+1c3i+1,e3i+2e3i+1d3i+1e3i+1i=0,…,k−1d3i+4f3i+1f3i+2f3i+1i=0,…,k−1f3i+2c1d1c3k+1c3kd3k+1e3k+1e3kf3kf3k+1f3k3k+2D1c3i+2c3i+1,d3i+2c3i+1c3i+4d3i+1e3i+1d3i+4i=0,…,k−1c3i+4,e3i+1e3i+1d3i+2e3i+2d3i+2,f3i+2f3i+1f3i+2f3i+1e3i+1,f3i+2c3k+2c1,c3k+1d3k+2e3k+2e3k+1f3k+1f3k+2f3k+1D2c3i+1c3i+2,d3i+1c3i+1c3i+2d3i+2c3i+2,e3i+2d3i+1e3i+2e3i+1d3i+1e3i+1d3i+4,f3i+1f3i+2e3i+2,f3i+1f3i+4i=0,…,k−1f3i+1c3k+1c3k+2,d3k+1d3k+2c3k+2e3k+1d3k+1e3k+2d1,f3k+2f3k+1f3k,f3k+2Figure 7
The graph ofℛn.Theorem 10.
The domatic number of graph of convex polytopeℛn, for n≥6, is(16)domDn=domℛn=3,n≡0mod3,2,n≡1,2mod3.Proof.
The proof is similar as shown for the graph of convex polytopeDn.
To validate the result in Theorem3 for the middle graph of a cycle, the graph of convex polytope Tn is considered, which is generated from the middle graph of a cycle. There are infinite classes of convex polytopes that show the same structural similarity.


## 4.2. The Graph of Convex PolytopeTn

The graph of convex polytopeTn, as shown in Figure 8, consists of 4 n 3-sided faces, n 4-sided faces, and a pair of n-sided faces. The vertex and edge set for Tn can be written as follows:(17)VTn=ci,di,ei,fi,ETn=cici+1,didi+1,eiei+1,fifi+1,∪cidi,diei,ci+1di,eifi,eifi+1.Figure 8
The graph ofTn.Theorem 11.
The domatic number of graph of convex polytopeTn, when n≥6, is(18)domTn=3,n≡0,2mod3,2,n≡1mod3.Proof.
In order to show this, we consider the following three cases.Case 21.
Whenn=3k and k≥2, the partition of the set V of Γ into disjoint dominating set is shown as follows.
LetD1=c3i+1,d3i+2,e3i+1,f3i+2|i=0,…,k−1, D2=c3i+2,d3i+1,e3i+2,f3i+1|i=0,…,k−1, and D3=c3i+1,d3i+1,e3i+1,f3i+1|i=0,…,k−1.Case 22.
Whenn=3k+1 and k≥2, the partition of the set V of Γ into disjoint dominating set is shown as follows.
LetD1=c3i+2,d3i+1,e3i+2,f3i+1∪c3k+1,e3k+1|i=0,…,k−1 and D2=c3i+1,d3i+1,e3i+1,f3i+1∪d3k+1,f3k+1|i=0,…,k−1.Case 23.
Whenn=3k+2 and k≥2, the partition of the set V of Γ into disjoint dominating set is shown as follows.
Let(19)D1=c3i+1|i=0,…,k∪d3i+2|i=0,…,k−1∪e3i+1|i=0,…,k∪f3i+2|i=0,…,k−1,D2=c3i+2|i=0,…,k∪d3i+1|i=0,…,k−1∪e3i+2|i=0,…,k∪f3i+1|i=0,…,k−1,D3=c3i+1|i=0,…,k−1∪d3i+1|i=0,…,k∪e3i+1|i=0,…,k−1∪f3i+1midi=0,…,k∪d3k+2,f3k+2.
It is not hard to observe that whenn=3k,3k+2, D1∩D2∩D3=∅. Moreover, when n=3k+1, we have D1∩D2=∅. Furthermore, to show that classes of the domatic partition set are also dominating set, Table 3 is presented which clearly indicates that the chosen classes are dominating.Table 3
The table for graph of convex polytopeTn.
nDℓv∈V/DℓDℓ∩Nvv∈V/DℓDℓ∩Nv3kD1c3i+2c3i+1,d3i+2c3i+1c3i+4,d3i+2d3i+1c3i+1,d3i+2,e3i+1d3i+1c3i+4,d3i+2e3i+2d3i+2,e3i+1,f3i+2e3i+1e3i+4f3i+1e3i+1,f3i+2f3i+1f3i+2D2c3i+1c3i+2,d3i+1c3i+4i=0,…,k−2c3i+5,d3i+1d3i+2c3i+2,d3i+1,e3i+2d3i+4i=0,…,k−2c3i+5,d3i+1e3i+1e3i+2e3i+1d3i+1,e3i+2,f3i+1f3i+2e3i+2,f3i+1f3i+4i=0,…,k−2e3i+2,f3i+1c1c2,dnd1c2,dnf1fnD3c3i+2c3i+1,d3i+1c3i+4i=0,…,k−2c3i+1,d3i+4d3i+2c3i+1,d3i+1d3i+1c3i+1,d3i+4,e3i+1e3i+2e3i+1e3i+4i=0,…,k−2d3i+4,e3i+1,f3i+4f3i+2f3i+1f3i+1e3i+1,f3i+4c1cn,d1e1d1,en,f13k+1D1c3i+1i=0,…,k−2c3i+2,d3i+1c3i+4i=0,…,k−2c3i+5,d3i+1d3i+2c3i+2,d3i+1,e3i+2d3i+4i=0,…,k−2c3i+5,d3i+1e3i+1d3i+1,e3i+2,f3i+1e3i+4i=0,…,k−2e3i+5f3i+2e3i+2,f3i+1f3i+4i=0,…,k−2f3i+1c1c2,c3k+1c3kc3k−1,c3k+2,d3kd1c2d3k+1c3k+1,d3ke1e2,e3k+1f1e3k+1f3k+1e3k+1,f3kD2c3i+2c3i+1,d3i+1c3i+4i=0,…,k−2c3i+1,d3i+4d3i+2c3i+1,d3i+1d3i+1c3i+1,d3i+4,e3i+1e3i+2e3i+1e3i+4i=0,…,k−2d3i+4,e3i+1,f3i+4f3i+2f3i+1f3i+1e3i+1,f3i+4c1d1,d3k+1c3k+1c3k,d1,d3k+1e1d1,f1e3k+1d3k+1,e3k,f3k+1,f13k+2D1c3i+2i=0,…,k−1c3i+1,d3i+2c3i+1i=0,…,k−1c3i+4,d3i+2d3i+1i=0,…,k−1c3i+1,d3i+2,e3i+1d3i+1i=0,…,k−1c3i+4,d3i+2e3i+2i=0,…,k−1d3i+2,e3i+1,f3i+2e3i+1i=0,…,k−1e3i+4f3i+1i=0,…,k−1e3i+1,f3i+2f3i+1i=0,…,k−1f3i+2c3k+2c3k+1,c1d3k+1c3k+1,e3k+1e3k+2e3k+1,e1f3k+1e3k+1f3k+2e3k+1D2c3i+1i=0,…,k−1c3i+2,d3i+1c3i+4i=0,…,k−1c3i+5,d3i+1d3i+1i=0,…,kc3i+2d3i+2i=0,…,k−1c3i+2,d3i+1,e3i+2e3i+1i=0,…,ke3i+2e3i+1i=0,…,k−1d3i+1,e3i+2,f3i+1f3i+2i=0,…,k−1e3i+2,f3i+1f3i+4i=0,…,k−1f3i+1c1c2,e3k+2d3k+2c3k+2,e3k+2e3k+1e3k+2f1e3k+2f3k+2e3k+2D3c3i+2i=0,…,k−1c3i+1,d3i+1c3i+4i=0,…,k−1c3i+1,d3i+4d3i+2i=0,…,k−1c3i+1,d3i+1d3i+1i=0,…,k−1c3i+1,d3i+4,e3i+1e3i+2i=0,…,k−1e3i+1e3i+4i=0,…,k−1d3i+4,e3i+1f3i+2i=0,…,k−1f3i+1f3i+1i=0,…,k−1e3i+1,f3i+4c1d1,d3k+2c3k+2d3k+1,d3k+2e1d1e3k+2d3k+2


## 5. Conclusion

This article first improves the results for the domatic number of a middle graph and central graphs of a cycle graph. Next, the domatic number of squared cycle graphs and wheel graphs is determined. The graphs of convex polytopes are also considered, and for these, the concept of domatic number is studied. We also found that the two families of convex polytopes, namely,Dn and ℛn are with the same domatic number, i.e., domDn=domℛn. Future research can be directed towards studying domatic numbers for more complex planar and chemical graphs.



---

*Source: 3816772-2023-02-22.xml*