Abstract
The edge -irregularity strength, , of a graph is the smallest integer , such that has an -irregular edge -labeling. In this study, we compute the exact value of edge -irregularity strength of hexagonal and octagonal grid graphs.
1. Introduction
Let be a connected graph. A mapping that assigns numbers to graph elements (vertices and edges) is called labeling. A graph labeling is called vertex labeling or edge labeling if its domain set is the vertex set or the edge set of the graph, respectively. The concept of graph labeling plays an important role to construct models for a wide range of engineering applications such as coding theory, X-rays crystallography, astronomy, radar, circuit design, wired communication, and wireless communications. For further detail related to graph labeling, refer to [1].
For an edge -labeling , , the corresponding weight of is . The edge -labeling is called irregular if , . The irregularity strength of , , is the minimum , such that there exists an edge irregular -labeling of . The concept of the irregularity strength of a graph was introduced by Chartrand et al. [2]. There has been a flurry of research work on the irregularity strength in the last few years [3–12].
A vertex -labeling of is called an edge irregular -labeling if for every two distinct edges and , , where . The edge irregularity strength of , , is the minimum , such that the graph has an edge irregular -labeling. The concept of the edge irregularity strength was given by Ahmad et al. [13]. Ashraf et al. [14] have introduced two new graph parameters, i.e., vertex (edge) -irregularity strength of a graph. These parameters are considered as extensions of the irregularity strength and the edge irregularity strength of .
A family of subgraphs of a graph , such that each edge of belonging to at least one member of is called an edge-covering of . If every member of is isomorphic to a graph , then admits an -covering. An edge -labeling is called an -irregular edge -labeling of the graph that admits -covering if for every two distinct subgraphs and , which are isomorphic to , . The edge -irregularity strength of a graph , , is the smallest integer , such that has an -irregular edge -labeling.
In proving lower bound concerning a graph admitting -covering, the following result is useful.
Theorem 1 (see [14]). Let be a graph admitting -covering given by subgraphs isomorphic to . Then, .
In this study, we compute the exact values of the edge -irregularity strength of the hexagonal grid graph and octagonal grid graph.
2. Motivation and Applications of -Covering of Graph
The topic of -covering of the graph is newly addressed by Ashraf et al. in [14] and found the -covering for different graphs. Still now, the problem of -covering was addressed on grids by any author. As the topic is related to the networking and communication system, it is needed to work on some networks. So, we choose these two graphs as these graphs are related to network. The -covering has a wide range of applications in X-rays, circuit design, and especially in communication and networks. The graphs studied in the present study are hexagonal and octagonal grids. These graphs consist of 6 and 8 vertices in each face of the graph, and each face made a cycle consisting of 6 and 8 vertices and edges as well. These graphs and their faces could be served as models for surveillance or security systems, electrical switchboards, circuit design, and communication networks. These networks can be extended in both horizontal and vertical directions, so that the extension in the graphs and networks can be handled and made easy. Moreover, these networks are used especially in communication networks, and the efficiency of the networks may be improved as the weights of the each face have distinct numbers.
3. Hexagonal Grid Graph
For finite , the hexagonal grid graph (honeycomb), , is a graph with rows and columns of hexagons [15]. The vertex and the edge sets of this graph are defined as . The face set of consists of 6-sided faces and one external face which is infinite. Also, and , while . For , the hexagonal grid graph is shown in Figure 1.
In the following theorem, we determine the edge -irregularity strength of , where are finite.
Theorem 2. For hexagonal grid graph , admitting an -covering given by subgraphs isomorphic to , .
Proof. The graph obviously admits a -covering with exactly subcovers of . Set ; then, is the lower bound of by Theorem 1. Now, to prove the converse inequality, we have to describe a irregular edge -labeling as follows: Case 1: when is odd, Case 2: when is even,Figure 2 shows the weights of the edge -covering of the hexagonal grid graph, and the formula for the weights of the edge -covering is given as follows: . Observe that all the weights of subcovers of are distinct. Hence, , which completes the proof of this theorem.
4. Octagonal Grid Graph
For finite , the octagonal grid graph, , is a graph with rows and columns of octagons [16]. The vertex and the edge sets of this graph are defined as . The face set of consists of faces as 8-sided faces and one external face which is infinite. Also, and , while .
For , the octagonal grid graph is shown in Figure 3.
The following result establishes the edge -irregularity strength of for and .
Theorem 3. Let be an octagonal grid graph admitting -covering given by subgraphs isomorphic to ; then, .
Proof. It is clear that the graph for and and admits -covering with exactly subcovers of . Set ; then, is the lower bound of by Theorem 1. Now, to prove the converse inequality, we have to describe a irregular edge -labeling given as follows: Case 1: when and , , Case 2: when , and , , Case 3: when and , ,Figure 4 shows the weights of the edge -covering of the octagonal grid graph, and the formula for the weights of the edge -covering is given as follows:Observe that all the weights of subcovers of are distinct. Hence, , which completes the proof of this theorem.
5. Conclusion
In this study, we have determined the edge -irregularity strength for the hexagonal grid graphs for and the octagonal grid graphs for . We have tried to find the edge -irregularity strength of octagonal grid graph for ( is even) and ( is odd), but so far without success. Hence, we conclude the study with the following open problems.
Open Problem 1. Let be an octagonal grid graph admitting -covering. Then, for and for .
Open Problem 2. Let be an octagonal grid graph admitting -covering. Then, for , for any choice of and .
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.