Abstract
In this study, we used grids and wheel graphs , which are simple, finite, plane, and undirected graphs with as the vertex set, as the edge set, and as the face set. The article addresses the problem to find the face irregularity strength of some families of generalized plane graphs under -labeling of type . In this labeling, a graph is assigning positive integers to graph vertices, graph edges, or graph faces. A minimum integer for which a total label of all verteices and edges of a plane graph has distinct face weights is called -labeling of a graph. The integer is named as total face irregularity strength of the graph and denoted as . We also discussed a special case of total face irregularity strength of plane graphs under -labeling of type (1, 1, 0). The results will be verified by using figures and examples.
1. Introduction
This article is based on simple, plane, finite, and undirected graphs . Graph labeling is a mapping that maps graph elements into positive integers, and we name these positive integers as labels. Suppose that and is a positive integer, then a branch of labeling, named as, -labeling of type, is a mapping from the set of graph elements into the set of positive integers . A labeling of type of grid graph means that vertices and edges are labeled but face is not labeled. We will work on labeling of type for the grid graphs , in which the vertices and edges will be labeled but our ultimate focus will be on calculating distinct face weights. A detailed review of graph labeling can be seen in [1].
If the domain of -labeling of type is vertex set, edge set, face set, or vertex-edge set, then we name this as vertex -labeling of type , edge -labeling of type , face -labeling of type , or total -labeling of type , respectively. The other possible cases are vertex-face set, edge-face set, and vertex-edge-face set which we call as vertex-face -labeling of type , edge-face -labeling of type , and entire -labeling of type , respectively. The trivial case is not accepted. The weight of any vertex in a graph is the sum of labels of that particular vertex and its adjacent edges. The weight of any edge of a graph is the sum of lables of its adjacent vertices. The weight of any face in a graph is the sum of labels of that particular face and its surrounding vertices and edges. For a deep survey on weights of graph elements, reader can go through [2–4]. The weight of a face of a plane graph under -labeling of type can be defined as follows:
A -labeling of type of the plane graph is called face irregular -labeling of type of the plane graph if every two different faces have distinct weights; that is, for graph faces and , we have
Face irregularity strength of type of any plane graph is the minimum integer for which the graph admits a face irregular -labeling of type . For a vertex-edge labeled graph , the minimum integer for which the graph admits a face irregular -labeling of type is called the total face irregularity strength of type of the plane graph , and it is denoted by . A detailed work on irregularity strength of graphs can be seen in [4–12].
Gary Ebert et al. worked on the irregularity strength of grid in their research “Irregularity Strength for Certain Graphs,” [13]. Baca et al. determined total irregularity strength of graphs and calculated bounds and exact values for different families of graphs [14]. Baca et al. investigated face irregular evaluations of plane graphs and calculated face irregularity strength of type for ladder graphs [15].
By motivating from all abovementioned, we are working on grid graphs with rows and columns. Labeling of a grid graph has many stages, depending on the size of graph, on the selection of rows and columns, and sometimes on the smaller and larger values of labeling. We will calculate the total face irregularity strength of grid graphs under labeling of type , and this work is a modification of abovementioned articles. Grid graphs are constructed by the graph Cartesian products of path graphs, that is, .
We will prove the exact value for the total face irregularity strength under -labeling of type of grid graphs with the property where .
We will prove the exact value for the total face irregularity strength under -labeling of type of wheel graph .
Baca et al. determined a lower bound for the face irregularity strength of type when a -connected plane graph has more than one faces of the largest sizes [14, 16]. They presented the following theorem.
Theorem 1 (see [14, 16]). Let be a -connected plane graph with -sided faces, . Let , , and . Then, the face irregularity strength of type of the plane graph is
Proof. Suppose that face irregularity strength under a -labeling of type of the plane graph is .
The smallest face weight under the face irregular -labeling admits the value at least . Since , it follows that the largest face weight attains the value at least and at most . Hence,
This lower bound can be improved when a -connected plane graph contains only one face of the largest size, that is, and . So, we present the following theorem to calculate the lower bounds for grid graphs .
2. Main Results
In this research, we will demonstrate the tight lower bound for the total face irregular strength of type for the plan graph particularly grid and wheel graphs. It is sufficient to prove tight lower bound of grid graph that the exact value of exists and differences in weights of the horizontal faces must be 1 and the differences in weights of the vertical faces is .
Theorem 2. Let be a -connected plane graph with -sided faces, . Let , and , , and . Then, the total face irregularity strength of type of the plane graph is
Proof. We suppose that total face irregularity strength of any -connected plane graph under -labeling of type is equal to , that is,Given that the lagest face for . So, the smallest face weight under the face irregular -labeling of type will have the minimum value . The total number of faces of the graph can be obtained by adding all the number of -sided faces where . Hence, the largest face weight can have the minimum value and maximum value . So, we can construct the following results:Hence,
From the above result, we see that if a -connected plane graph contains only one largest face, then the lower bound for the face irregularity strength of type can be calculated as
In this research, we will prove the tight lower bound for the total face irregularity strength of type for the grid graph and wheel graph . To prove the tight lower bound of the grid graph, it will be sufficient to show that the exact value of exists. The exact value of , that is, calculated from grid graph under a graph -labeling of type , exists if the differences in weights of the horizontal faces are 1 and the differences in weights of the vertical faces are . Generalized grid graphs can be written as .
The vertex set and the edge set of the grid graph can be defined as follows:
Theorem 3. Let be positive integers and be generalized grid graph, then
In order to prove this, it will be sufficient to show that the exact value of exists.
The vertices for the generalized graph under a -labeling of type in different intervals of and can be defined as follows:
The horizontal edges for the generalized graph under a -labeling of type in different intervals of and can be defined as follows:
The vertical edges for the generalized graph under a -labeling of type in different intervals of and can be defined as follows:
Figure 1 represents the generalized formula for face weights. The generalization of weights over the face under a -labeling of type for the graph can be defined as follows:
Horizontal differences in weights among different intervals of and can be calculated as follows:
Vertical differences in weights among different intervals of and can be calculated as follows:
Example 1. The total face irregularity strength of grid graph , under a -labeling of type is 4.
Proof. The graph under consideration is . Figure 2 is a -labeling of type for the grid graph , and it will help us in calculating total face irregularity strength in different intervals of the grid graph.
Here, , , , and
In order to show that , it is sufficient to prove that all the horizontal differences in face weights are 1 and all the vertical differences in face weights are 3. Now, we prove these results.
Horizontal differences in face weights can be calculated as follows:Vertical differences in face weights can be calculated as follows:It shows that all the differences of horizontal faces are equal to one and all the differences of vertical faces are equal to . Hence, total face irregularity strength of grid graph is 4.
Theorem 4. . Let be a wheel graph with vertices, where . Then, under a total -labeling of type , we have
Proof. Let be a wheel graph with vertices, then by the definition of wheel graph, the total number of edges will be and the total number of faces will be , that is,As we see that a wheel graph has sided internal faces and external face, so by using Theorem 2, we haveIn Figure 3, is the vertex in the center of wheel graph which is connecting to all the vertices for . Similarly, for , the edges of the wheel graph can be constructed as . Also for , there will be exterior face, the interior face can be written as , and other all -sided interior faces can be written as . Let us define a total -labeling .
In Figure 4, we consider a finite wheel graph which is labeled under a -labeling of type . So, for , we haveWeight of exterior face will beWeight of interior faces will beNow, let us talk about the graphs except for which we define the labeling as :(i)For , we have (ii)For , we have (iii)For , we have (iv)For , we have (v)For , we have (vi)For , we have (vii)For , where in which , we have (viii)For where in which , we have (xi)For , we have For weights of the wheel graph , we proceed as follows:(i)For , we have (ii)For , we have (iii)For , we have (iv)For external weight, we will use We can easily observe that under the -labeling of type , the minimum for which the wheel graph admits total face irregular strength is . Hence,
3. Conclusion
We investigated total face irregularity strength of generalized plane grid graphs and wheel graphs under a graph -labeling of type where . This work was based on the bright idea of finding face irregularity strength of ladder graphs by Martin Baca et al. [14]. In this article, we worked on the total face irregularity strength of grid and wheel graphs. We labeled graph vertices and graph edges but focussed on estimating face weights of graphs to prove the sharpness of -labeling. We derived generalized formulas by considering graphs with different values of , , , and . Also, we verified the final results with example. In future, total and entire face irregular strength of some more products of different plane graphs can be investigated under -labeling of type .
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.