Abstract

In this paper, we study the total product and total edge product cordial labeling for dragonfly graph . We also define generalized dragonfly graph and find product cordial and total product cordial labeling for this family of graphs.

1. Introduction

In this paper, all graphs are simple and finite connected with order and size . We will give some definitions and other information, which are useful for this research. Terms that are not defined here, we refer to West [1]. Let function be a vertex labeling of graph and , an edge labeling of graph . Let (respectively ) denote the number of vertices (edges) labeled with .

The cordial labeling was introduced in 1987 by Cahit [2], which he defines that a graph is said to be cordial graph if there exists a vertex labeling such that induces an edge labeling defined by and satisfied and . In [2], Cahit proved many result for cordial labeling. Prime cordial labeling, A-cordial labeling, product cordial labeling, H-cordial labeling, etc. are some variations of labeling schemes introduced after cordial labeling. For product cordial labeling, it was introduced in 2004 by Sundaram, et al. [3], which on cordial labeling is replaced by . In this paper we investigate the total product and total edge product cordial labelings of dragonfly graph .

The product cordial labeling is defined in Definition 1.1.

Definition 1.1. A graph is said to be the product cordial if there exists a vertex labeling such that induces an edge labeling defined by and satisfied and .

In [3], Sundaram et al. proved that unicycle graphs with odd order, trees, helms, triangular snakes, dragons, and unicon with two paths are product cordial. Furthermore, Vaidya and Barasara [4] discussed product cordial labeling of graph fans , with one chord, and with two chord. Gao et al. [5] discussed product cordial labeling of graph .

Motivated by definition of product cordial labeling, in [6], Sundaram et al. introduce a total product cordial labeling and investigate the total product cordial of some standard graphs. The total product cordial labeling is defined in Definition 1.2.

Definition 1.2. A graph is said to be the total product cordial if there exists a vertex labeling such that induces an edge labeling defined by and satisfied .

The total product cordial labeling of cycle is shown in Figure 1.

In [6, 7], Sundaram et al. proved that tree graph , fans graph , graph , except , wheels graph , helms graph , and graph with edges appended at each vertex are total product cordial graph. They also proved that every product cordial graph is a total product cordial if has either even size and even order or odd order.

In [8], Vaidya and Barasara introduce the concept of edge product cordial labeling, which is defined in Definition 1.3.

Definition 1.3 (see [8]). A graph is said to be edge product cordial if there exits an edge labeling such that it induces a vertex labeling defined by for and satisfies and .

In [810], Vaidya and Barasara have investigated several results related to edge product cordial labeling.

In, Vaidya and Barasara introduce the concept of total edge product cordial labeling, which is defined in Definition 1.4.

Definition 1.4 (see [10]). A graph is said to be the total edge product cordial if there exits an edge labeling , such that it induces a vertex labeling defined by for and satisfies .

The total edge product cordial labeling of graph is shown in Figure 2.

In [4], Vaidya and Barasara have investigated total edge product cordial labeling in the context of various graph operations.

Proposition 1.5 (see [10]). If every edge product cordial graph has either even size or even order, then graph is the total edge product cordial.

In this paper, we determine the total product and total edge product cordial labelings of dragonfly graph, denoted by , which is defined in Definition 1.6. Also, we generalized dragonfly graph, defined in Definition 3.1, and present two family of graphs in that, which are product and total product cordial graph.

Definition 1.6. For an integer , the dragonfly graph is the graph with vertex set: and edge set

In Figure 3, we give a representation of our definition.

2. Main Results

Theorem 2.1. The dragonfly is product cordial graph.

Proof. Let is the dragonfly graph. Define the function , we consider following two cases.
Case 1. Let be even. By of the above labeling, we have and . On the other hand, the edges of with labels one are the following: and the edges of with labels zero are the following: By of the above labeling, we have and . Hence, and . Thus, the graph is product cordial labeling.
Case 2. Let be odd. By of the above labeling, we have and . On the other hand, the edges of with labels one are the following: and the edges of with labels zero are the following: By of the above labeling, we have and . Hence, and . Thus, the graph is product cordial labeling. Therefore, considering two cases above, we prove that graph is product cordial graph.

Theorem 2.2. The dragonfly is a total product cordial.

Proof. By: Theorem 2.1, . Thus, the graph is a total product cordial.
The total product cordial labeling of is shown in Figure 4.

Theorem 2.3. The dragonfly is an edge product cordial.

Proof. Let is dragonfly graph. Define the function , we consider following two cases.
Case 1. Let be even. By of the above labeling, we have and . On the other hand, the vertices of with labels zero are the following: and the vertices of with labels one are the following: By of the above labeling, we have and . Hence, and . Thus, the graph is an edge product cordial labeling.
Case 2. Let be odd. By of the above labeling, we have and . On the other hand, the vertices of with labels zero are the following: and the vertices of with labels one are the following: By of the above labeling, we have and . Hence, and . Thus, the graph is an edge product cordial labeling. Therefore, considering two cases above, we prove that graph is edge product cordial.

Corollary 2.4. The dragonfly is a total edge product cordial graph.

Proof. Let is dragonfly graph. Here, graph has even size and in Theorem 2.3, is edge product cordial. Then, by proposition 3.2, the result holds.

The total edge product cordial labeling of Dg5 is shown in Figure 5.

3. The Generalized Dragonfly Graphs

In this section, we present a generalization of dragonfly graph and show that some of those graphs are total product cordial graphs.

Definition 3.1. For every and , the generalized dragonfly graph, denoted by , is the graph with vertex set and edge set

It is clear that (see Figure 6, for the case ).

Theorem 3.2. For , graph is a product cordial graph.

Proof. We define vertex labeling , of vertices as follow. By the above labeling, we have and . On the other hand, the edges of with labels one are the following: and the edges of with labels zero are the following: By the above labeling, we have and . Hence, and . Thus, labeling is a product cordial labeling for , and the proof is completed.

Corollary 3.3. For , graph is a total product cordial.

Proof. By: Theorem 3.2, . Therefore, the graph is a total product cordial.

Theorem 3.4. For , graph is a product cordial graph.

Proof. We define vertex labeling , of vertices as follow. By the above labeling, we have and . On the other hand, the edges of with labels one are the following: and the edges of with labels zero are the following: By the above labeling, we have and . Hence, and . Thus, labeling is a product cordial labeling for , and the proof is completed.

Corollary 3.5. For , graph is a total product cordial.

It is interesting to find all values , and such that generalized dragonfly is cordial product graph. We end the paper with the following question.

Question. Find all values , and , such that is (edge) cordial product graph.

Data Availability

Data sharing is not applicable to this article as no data were collected or analyzed in this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

We would like to acknowledge the support and help of the late Budi Harianto in the preparation and contribution of this paper. This study was supported by the center of research and publication of State Islamic University Syarif Hidayatullah Jakarta.