Communications on Applied Nonlinear Analysis 

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Power Dominator Equitable Coloring for some Standard Graphs 

 

G. Navamani 1, L. Jacquline 2* 
1,2* Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences 

Saveetha University, Chennai 602105, Tamil Nadu, India. 

1 Email: g_navamani@yahoo.co.in, 2* Email: jackinfanci@gmail.com 

 

Article History: 

Received: 23-10-2024 

Revised: 07-12-2024 

Accepted: 16-12-2024 

Abstract:  

A power dominator coloring of a graph 𝐺 is a proper coloring where each vertex in 𝑉(𝐺) 

power dominates at least one complete color class. The power dominator chromatic 

number of 𝐺 is represented by 𝜒𝑝𝑑(𝐺). A graph 𝐺 is said to be equitably 𝑘-colorable if it 

can be properly colored with 𝑘 colors such that the size of any two color classes 

𝐶1, 𝐶2, … , 𝐶𝑘 of 𝐺 is differ by at most one, i.e, ||𝐶𝑖| − |𝐶𝑗|| ≤ 1,1 ≤ 𝑖, 𝑗 ≤ 𝑘 and 𝜒𝑒(𝐺) 

represents an equitable chromatic number of 𝐺. The power dominator equitable coloring 

of a graph 𝐺 is a proper 𝑘 - colorable if each vertex of 𝐺 power dominates each and every 

vertex of some color class 𝐶1, 𝐶2, … 𝐶𝑘 for which ||𝐶𝑖| − |𝐶𝑗|| ≤ 1,1 ≤ 𝑖, 𝑗 ≤ 𝑘. In this 

paper, we obtain the power dominator equitable chromatic number 𝜒𝑝𝑑𝑒  for some standard 

graphs. 

Keywords: Proper coloring, color class, equitable coloring, power dominator coloring, 

standard graphs. 

AMS Subject Classification: 05C15, 05C69 

1. Introduction 

In graph theory, Haynes introduced the important and extensively studied idea of domination in graphs 

[8]. In a graph 𝐺, a dominating set is a subset 𝑆 of its vertices 𝑉(𝐺), where each vertex outside of 𝑆 

shares at least one adjacency with a vertex inside 𝑆. The smallest size among all such dominating sets 

is termed as the “domination number”, represented by 𝛾(𝐺). 𝑆 is designated as a 𝛾-set of 𝐺 if it attains 

this minimal cardinality [5]. Haynes et al [7] introduced a groundbreaking concept in domination 

known as power domination. This concept finds application within the framework of electric power 

systems, where a graph 𝐺 represents the system, with vertices symbolizing electrical nodes and edges 

representing transmission lines between these nodes. Finding the smallest possible collection of 

“Phasor Measurement Units” (PMUs) required for efficient system monitoring is the main goal. A 

connected graph 𝐺 and a subset 𝑋 of its vertices are considered, where the set monitored by 𝑋 denoted 

by 𝑀(𝑋) is defined as follows: 

1. Initialize 𝑀(𝑋) by adding the vertices in 𝑋 along with their neighbors. 

2. Iterate: While ∃ 𝑦 ∈ 𝑀(𝑋) such that all its neighbors except one, denoted by 𝑥, are                    

already in 𝑀(𝑋) then add 𝑥 to 𝑀(𝑋).  

After this process, 𝑀(𝑋) represents the set monitored by 𝑋. A power dominating set 𝑋 of 𝐺 is such 

that 𝑀(𝑋) covers all vertices in 𝐺. The smallest size of such a power dominating set is termed as the 

“power domination number” 𝛾𝑝(𝐺). Vast research is going in [3, 7, 9, 13].  



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The vertex 𝑑 power dominates the vertices within the set 𝑀(𝑑) if they satisfy the following conditions: 

1. Each vertex in 𝑁(𝑑) ∪ {𝑑} is included in 𝑀(𝑑). 

2. For any vertex 𝑐 ∈ 𝑉(𝐺), 𝑐 is added to 𝑀(𝑑) if there exists a neighbor 𝑏 in 𝑀(𝑑) such that all 

neighbors of 𝑏 except 𝑐 are already in 𝑀(𝑑) 

3. Step 2 is repeated for all vertices in the graph. 

The process of giving colors to vertices in a graph 𝐺 so that no two adjacent vertices have the same 

color in a suitable coloring of the graph is known as graph coloring, and it is useful in many graph 

theory applications. Let 𝐶𝑖 be the color class 𝑖, signifying the collection of all vertices that possess the 

color 𝑖 [12]. Each vertex in 𝑉(𝐺) dominates every vertex of some color class is said to be a dominator 

coloring of graph 𝐺 and 𝜒𝑑(𝐺) represents the dominator chromatic number of 𝐺 [10, 2, 12]. A “power 

dominator coloring” (PDC) of a graph 𝐺 entails a proper coloring where each vertex 𝑣 ∈ 𝑉(𝐺) power 

dominates all the vertices of at least one-color class. The term 𝜒𝑝𝑑(𝐺) signifies the “power dominator 

chromatic number of 𝐺”, as elucidated in reference [1, 10]. A proper coloring of a graph G is said to 

be equitably 𝑘-colorable if the number of vertices of any two-color classes 𝐶1, 𝐶2, … 𝐶𝑘 of 𝐺 is differ 

by at most one. That is ||𝐶𝑖| − |𝐶𝑗|| ≤ 1,1 ≤ 𝑖, 𝑗 ≤ 𝑘. The term 𝜒𝑒(𝐺) represents an “equitable 

chromatic number of 𝐺” [4, 11]. The power dominator equitable coloring (PDEC) of a graph 𝐺 defines 

each vertex 𝑣 ∈ 𝑉(𝐺) power dominates all the vertices of at least one color class and also satisfies the 

inequality ||𝐶𝑖| − |𝐶𝑗|| ≤ 1,1 ≤ 𝑖, 𝑗 ≤ 𝑘 The notation 𝜒𝑝𝑑𝑒(𝐺) represents the “power dominator 

equitable chromatic number of 𝐺”. The power dominator equitable chromatic number 𝜒𝑝𝑑𝑒 for a few 

common graphs is obtained in this study. 

2. Motivation 

The motivation behind this paper lies in addressing a significant problem in graph theory, finding 

efficient and balanced colorings for graphs that satisfy specific domination properties. However, in 

certain applications, such as network design or resource allocation, additional constraints need to be 

considered. The concept of power domination introduces the idea that each vertex in a graph should 

have influence over all other vertices within its neighborhood. This leads to the notion of PDC, where 

every vertex power dominates at least one-color class. This ensures a level of connectivity and 

influence within the graph. Equitable coloring, on the other hand, seeks to balance the sizes of color 

classes. This is particularly important in scenarios where fairness or resource allocation is a concern. 

By combining the principles of power dominator coloring and equitable coloring, the paper presented 

the concept of PDEC. This approach aims to find coloring where every vertex not only dominates at 

least one-color class but also ensures that the sizes of the color classes are balanced. The research 

presented in this paper commences a study on this parameter by exploring its properties and 

determining the “power dominator equitable chromatic number” for some standard graphs. 

Understanding this parameter can have implications in various real-world applications, such as 

network communication, social network analysis, and resource allocation in distributed systems. 

Ultimately, this research contributes to advancing our understanding of graph coloring with additional 

domination constraints, paving the way for more efficient and equitable solutions in practical 

scenarios. 



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3. Preliminaries 

Consider an undirected, connected, and simple graph 𝐺 = (𝑉, 𝐸) comprising two non-empty sets 𝑉 

and 𝐸, where edges of 𝐺 are the elements of 𝐸, whereas vertices are the components of 𝑉. We use 

Harary's [6] graph theoretic notations. If a path exists between any two vertices in the graph, denoted 

by 𝑢 and 𝑣, then the graph is said to be connected. The open neighborhood of vertex 𝑐 comprises all 

vertices adjacent to 𝑐, denoted as 𝑁(𝑐). The union of 𝑁(𝑐) with the vertex 𝑐 itself is known as the 

closed neighborhood of 𝑐, or 𝑁[𝑐]. A path 𝑃𝑛 is a sequence of 𝑛 vertices, denoted as 𝑎1, 𝑎2, … 𝑎𝑛 that 

are connected by 𝑛 − 1 edges, ensuring that no vertex or edge is repeated within the sequence. A cycle 

𝐶𝑛 is a closed path having 𝑛 edges and 𝑛 vertices with first and last vertex being the same. All pairs of 

vertices in a complete graph of order 𝑛, represented as 𝐾𝑛, are adjacent. In a bipartite graph 𝐾𝑚,𝑛, there 

are two separate set of vertices 𝑀1 and 𝑀2. Set  𝑀1 contains 𝑚 vertices, while 𝑀2 contains 𝑛 number 

of vertices. Each vertex in 𝑀1 is exclusively connected to every other vertex in 𝑀2, and the same holds 

true in reverse. There are no connections between vertices within the same set. A wheel graph 𝑊𝑛 with 

an order of 𝑛 + 1, is formed by joining every vertex in cycle 𝐶𝑛 to one universal vertex (is a vertex 

that shares an edge with every other vertex present in the graph). A helm graph 𝐻𝑛 is constructed by 

adding a leaf edge to each vertex of an 𝑛-wheel graph, which consists of a cycle of 𝑛 vertices connected 

to a single central vertex. 

4. Main Results 

Theorem 4.1. For path 𝑃𝑛, 𝑛 ≥ 2, 𝜒𝑝𝑑𝑒(𝑃𝑛) = 2.  

Proof. Let 𝑎𝑖, 1 ≤ 𝑖 ≤ 𝑛 be the 𝑛 vertices of path 𝑃𝑛 and 𝑎1𝑎2, 𝑎2𝑎3, … , 𝑎𝑛−1𝑎𝑛 be the edges of 𝑃𝑛. The 

odd indices receive color 1 i.e, the vertices 𝑎2𝑗+1, 0 ≤ 𝑗 ≤ ⌊(𝑛 − 1)/2⌋ receives color 1 and color 2 to 

the vertices with the even indices, i.e., to the vertices 𝑎2𝑗, 1 ≤ 𝑗 ≤ ⌊𝑛/2⌋. Evidently, each vertex 𝑎𝑖, 1 ≤

𝑖 ≤ 𝑛, power dominates each and every vertex of 𝑃𝑛. Thus, each vertex of 𝑃𝑛 power dominates both 

the color classes 𝐶1 and 𝐶2. Furthermore, the discrepancy occurs whenever 𝑛 is odd, ||𝐶𝑖| − |𝐶𝑗|| = 1 

and whenever 𝑛 is even, ||𝐶𝑖| − |𝐶𝑗|| = 0. Therefore, a minimum of 2 colors is necessary for achieving 

power dominator equitable coloring. Thus, we get the desired result 𝜒𝑝𝑑𝑒(𝑃𝑛) = 2. 

Theorem 4.2. For cycle 𝐶𝑛, 𝑛 ≥ 3, 

  𝜒𝑝𝑑𝑒(𝐶𝑛) = {
2,      𝑖𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛
3,        𝑖𝑓 𝑛 𝑖𝑠 𝑜𝑑𝑑

 

Proof. We analyze two separate cases for 𝐶𝑛, based on  𝑛 is even or odd. 

Case 1: 𝑛 is even 

Let 𝐶2𝑘, 𝑘 ≥ 2 be an even cycle with the vertices 𝑎1, 𝑎2, … , 𝑎2𝑘, 𝑘 ≥ 2 and let 

𝑎1𝑎2, 𝑎2𝑎3, … 𝑎2𝑘−1𝑎2𝑘, 𝑎2𝑘𝑎1 be the edges of 𝐶2𝑘. We assign color 1 to the vertices with the odd 

indices, i.e., to the vertices 𝑎2𝑗+1, 0 ≤ 𝑗 ≤ ⌊(2𝑘 − 1)/2⌋ and color 2 to the vertices with the even 

indices, i.e., to the vertices 𝑎2𝑗, 1 ≤ 𝑗 ≤ ⌊(2𝑘)/2⌋. Each vertex 𝑎𝑖, 1 ≤ 𝑖 ≤ 𝑛 power dominates all the 



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vertices of 𝐶2𝑘 and hence each vertex of 𝐶2𝑘 power dominates both the color classes 𝐶1 and 𝐶2. Here 

||𝐶1| − |𝐶2|| = 0. Therefore ||𝐶𝑖| − |𝐶𝑗|| ≤ 1,1 ≤ 𝑖, 𝑗 ≤ 2. Hence 𝜒𝑝𝑑𝑒(𝐶2𝑘) = 2. 

Case 2: 𝑛 is odd 

Consider the odd cycle 𝐶2𝑘+1, 𝑘 ≥ 1 with the vertices 𝑎2𝑖+1, 0 ≤ 𝑖 ≤ 𝑘. Assigning color 1 and 2 

alternatively results violation in assigning color 1 and 2 to 𝑎𝑛, since neighbors of vertex 𝑎𝑛 receives 

color 1 and 2. For odd values of 𝑛 at least 3 colors are required. Assign the colors 1, 2 and 3 repeatedly 

for the vertices 𝑎𝑖+1, 0 ≤ 𝑖 ≤ 2𝑘 in the following manner, 

Subcase (i): 𝑛 ≡ 0,2 (𝑚𝑜𝑑 3) 

The vertex set {𝑎3𝑘−2: 1 ≤ 𝑘 ≤
𝑛

3
}  receives color 1. The vertex set {𝑎3𝑘−1: 1 ≤ 𝑘 ≤

𝑛

3
} receives color 

2. The vertices {𝑎3𝑘: 1 ≤ 𝑘 ≤
𝑛

3
} receives color 3. 

Subcase (ii): 𝑛 ≡ 1 (𝑚𝑜𝑑 3) 

The vertex set {𝑎3𝑘−2: 1 ≤ 𝑘 ≤
𝑛−1

3
} receives color 1. The vertex set {𝑎3𝑘−1: 1 ≤ 𝑘 ≤

𝑛−1

3
} ∪ {𝑎𝑛} 

receives color 2. The vertex set {𝑎3𝑘: 1 ≤ 𝑘 ≤
𝑛−1

3
} receives color 3. Specifically, vertex 𝑎𝑛 receives 

color 2, since neighbors of 𝑎𝑛 receives color 1 and 3. 

From the above subcases, we clearly see that the graph 𝐶2𝑘+1 receives proper coloring and each vertex 

of 𝐶2𝑘+1 power dominates all the vertices of 𝐶2𝑘+1 and hence each vertex power dominates all the color 

classes 𝐶1, 𝐶2 and 𝐶3 and also it satisfies the equitable condition that ||𝐶𝑖| − |𝐶𝑗|| ≤ 1,1 ≤ 𝑖, 𝑗 ≤ 3. 

Hence 𝜒𝑝𝑑𝑒(𝐶2𝑘+1) = 3. 

Theorem 4.3. For complete graph 𝐾𝑛, 𝑛 ≥ 1, 𝜒𝑝𝑑𝑒(𝐾𝑛) = 𝑛. 

Proof. Examine the entire graph 𝐾𝑛, 𝑛 ≥ 1, consisting of the vertices 𝑎𝑖, 1 ≤ 𝑖 ≤ 𝑛. Since all the 

vertices in 𝑉(𝐺) are next to each other, assign color 𝑖, 1 ≤ 𝑖 ≤ 𝑛 to the corresponding vertex 𝑎𝑖, 1 ≤

𝑖 ≤ 𝑛 respectively. Each vertex in the complete graph 𝐾𝑛 power dominates every other vertex in 𝐾𝑛, 

as well as all color classes. Furthermore, the inequality ||𝐶𝑖| − |𝐶𝑗|| = 0 is satisfied for all pairs of 

color classes 𝐶𝑖 and 𝐶𝑗, where 1 ≤ 𝑖, 𝑗 ≤ 𝑛. Hence  𝜒𝑝𝑑𝑒(𝐾𝑛) = 𝑛. 

Theorem 4.4. For complete bipartite graph 𝐾𝑚,𝑛, 𝑚, 𝑛 ≥ 1 

𝜒𝑝𝑑𝑒(𝐾𝑚,𝑛) = {
2,                  𝑖𝑓 (𝑚 − 𝑛) < 2

⌈
𝑚

𝑛 + 1
⌉ + 1,        𝑖𝑓 (𝑚 − 𝑛) ≥ 2

 

Proof. Let (𝑀1, 𝑀2) be the partitions of 𝐾𝑚,𝑛. Let 𝑚 be the cardinality of 𝑀1 and 𝑛 be the cardinality 

of 𝑀2. 

Case 1: (𝑚 − 𝑛) < 2 

Each vertex 𝑎𝑖, 1 ≤ 𝑖 ≤ 𝑚 of 𝑀1 power dominates 𝑁[𝑎𝑖], 1 ≤ 𝑖 ≤ 𝑚. Each vertex 𝑎𝑗 , 1 ≤ 𝑗 ≤ 𝑛 of 𝑀2 

power dominates 𝑁[𝑎𝑗], 1 ≤ 𝑗 ≤ 𝑛. By assigning color 1 to vertices in 𝑀1 and color 2 to vertices in 



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𝑀2, we ensure that each vertex in 𝑀1 power dominates the color class 𝐶2, while each vertex in 𝑀2 

power dominates the color class 𝐶1. Therefore ||𝐶1| − |𝐶2|| ≤ 1. Hence 𝜒𝑝𝑑𝑒(𝐾𝑚,𝑛) = 2. 

Case 2: (𝑚 − 𝑛) ≥ 2 

Without loss of generality, let 𝑚 > 𝑛. Each vertex 𝑎𝑖, 1 ≤ 𝑖 ≤ 𝑚, of partition 𝑀1 power dominates 

𝑁[𝑎𝑖], 1 ≤ 𝑖 ≤ 𝑚 and each vertex 𝑎𝑗, 1 ≤ 𝑗 ≤ 𝑛, 𝑛 > 2 of partition 𝑀2 power dominates 𝑁[𝑎𝑗], 1 ≤

𝑗 ≤ 𝑛. Divide the vertices of 𝑀1 into ⌈
𝑚

𝑛+1
⌉ number of sets and utilize ⌈

𝑚

𝑛+1
⌉  colors to color the vertices 

in 𝑀1 and assign color 1 to all vertices of 𝑀2. This ensures that every vertex power dominates at least 

one-color class and maintains the equitable condition. Hence 𝜒𝑝𝑑𝑒(𝐾𝑚,𝑛) = ⌈
𝑚

𝑛+1
⌉ + 1. 

Theorem 4.5. For wheel graph 𝑊𝑛, 𝑛 ≥ 4,  𝜒𝑝𝑑𝑒(𝑊𝑛) = ⌈
𝑛

2
⌉ + 1 

Proof. Consider 𝑎1 as the central vertex and 𝑎𝑖, where 2 ≤ 𝑖 ≤ 𝑛 as the vertices located on the cycle 

of 𝑊𝑛. Assign color 1 to 𝑎1. Each vertex of 𝑛 wheel graph power dominates the color class 𝐶1. Assign 

⌈
𝑛

2
⌉ colors to the remaining vertices equitably, so that  ||𝐶𝑖| − |𝐶𝑗|| ≤ 1 ∀ 𝑖 and 𝑗. Therefore 

𝜒𝑝𝑑𝑒(𝑊𝑛) = ⌈
𝑛

2
⌉ + 1. 

Theorem 4.6. For helm graph 𝐻𝑛, 𝑛 ≥ 3,  𝜒𝑝𝑑𝑒(𝐻𝑛) = 𝑛 + ⌈
𝑛+1

2
⌉. 

Proof. Let 𝑎𝑖, 1 ≤ 𝑖 ≤ 𝑛 be the vertices on the cycle of 𝐻𝑛, 𝑢𝑖, 1 ≤ 𝑖 ≤ 𝑛 representing the pendent 

vertices and 𝑎𝑛+1 denotes the central vertex within the graph 𝐻𝑛. The vertex 𝑎𝑛+1 power dominates 

over all vertices of 𝐻𝑛. The pendent vertices 𝑢𝑖 for 1 ≤ 𝑖 ≤ 𝑛 where 𝑛 ≥ 4 power dominates 𝑁[𝑢𝑖] for 

1 ≤ 𝑖 ≤ 𝑛 where 𝑛 ≥ 4. Moreover, the vertices on the cycle 𝑎𝑖, 1 ≤ 𝑖 ≤ 𝑛 power dominates 𝑁[𝑎𝑖], 1 ≤

𝑖 ≤ 𝑛 where 𝑛 ≥ 4. So, assign color 𝑖 to {𝑎𝑖}, 1 ≤ 𝑖 ≤ 𝑛 respectively. Assign ⌈
𝑛+1

2
⌉ colors to the 

leftover 𝑛 + 1 vertices equitably. The central vertex 𝑎𝑛+1 power dominates all the color classes. Vertex 

𝑢𝑖 , 1 ≤ 𝑖 ≤ 𝑛 and 𝑎𝑖, 1 ≤ 𝑖 ≤ 𝑛 power dominates the color class 𝐶𝑖 , 1 ≤ 𝑖 ≤ 𝑛 respectively and also it 

holds the inequality ||𝐶𝑖| − |𝐶𝑗|| ≤ 1, ∀ 𝑖 and 𝑗. Hence 𝜒𝑝𝑑𝑒(𝐻𝑛) = 𝑛 + ⌈
𝑛+1

2
⌉ 

5. Conclusion 

The inequality 𝜒(𝐺) ≤ 𝜒𝑝𝑑(𝐺) ≤ 𝜒𝑒(𝐺) ≤ 𝜒𝑝𝑑𝑒(𝐺) provides a concise framework for understanding 

the precise values of power dominator equitable chromatic numbers across various standard graphs, 

such as 𝑃𝑛, 𝐶𝑛, 𝐾𝑛, 𝐾𝑚,𝑛, 𝑊𝑛 and 𝐻𝑛. The graph 𝐺, exhibiting 𝜒(𝐺) = 𝜒𝑝𝑑(𝐺) = 𝜒𝑒(𝐺) = 𝜒𝑝𝑑𝑒(𝐺) 

encompasses 𝑃𝑛, 𝐶𝑛 and 𝐾𝑛 whereas for 𝐾𝑚,𝑛, all these parameters are equal when (𝑚 − 𝑛) < 2. For 

wheel graph 𝑊𝑛,  𝜒𝑒(𝑊𝑛) = 𝜒𝑝𝑑𝑒(𝑊𝑛). For helm graph 𝐻𝑛, 𝜒(𝐻𝑛) = 𝜒𝑒(𝐻𝑛) and 𝜒𝑝𝑑𝑒(𝐻𝑛) does not 

coincide with other three parameters. Determining the power dominator equitable chromatic numbers 

for diverse graphs remains a task reserved for future investigation. 

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