Total Face Irregularity Strength of Grid and Wheel Graph under K-Labeling of Type (1, 1, 0)

In this study, we used grids and wheel graphs G � (V, E, F), which are simple, finite, plane, and undirected graphs with V as the vertex set, E as the edge set, and F as the face set. )e article addresses the problem to find the face irregularity strength of some families of generalized plane graphs under k-labeling of type (α, β, c). In this labeling, a graph is assigning positive integers to graph vertices, graph edges, or graph faces. A minimum integer k for which a total label of all verteices and edges of a plane graph has distinct face weights is called k-labeling of a graph. )e integer k is named as total face irregularity strength of the graph and denoted as tfs(G). We also discussed a special case of total face irregularity strength of plane graphs under k-labeling of type (1, 1, 0). )e results will be verified by using figures and examples.


Introduction
is article is based on simple, plane, finite, and undirected graphs G � (V, E, F). Graph labeling is a mapping that maps graph elements (V, E, F) into positive integers, and we name these positive integers as labels. Suppose that α, β, c ∈ 0, 1 { } and k is a positive integer, then a branch of labeling, named as, k-labeling of type (α, β, c), is a mapping ϕ from the set of graph elements (V, E, F) into the set of positive integers 1, 2, 3, . . . , k { }. A labeling of type (1, 1, 0) of grid graph G m n means that vertices and edges are labeled but face is not labeled. We will work on labeling of type (1, 1, 0) for the grid graphs G m n , 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 k-labeling of type (α, β, c) is vertex set, edge set, face set, or vertex-edge set, then we name this as vertex k-labeling of type (1, 0, 0), edge k-labeling of type (0, 1, 0), face k-labeling of type (0, 0, 1), or total k-labeling of type (1, 1, 0), respectively. e other possible cases are vertex-face set, edge-face set, and vertex-edge-face set which we call as vertex-face k-labeling of type (1, 0, 1), edge-face k-labeling of type (0, 1, 1), and entire k-labeling of type (1, 1, 1), respectively. e trivial case (α, β, c) � (0, 0, 0) is not accepted. e weight of any vertex in a graph is the sum of labels of that particular vertex and its adjacent edges. e weight of any edge of a graph is the sum of lables of its adjacent vertices. e 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][3][4]. e weight of a face f of a plane graph G under k-labeling ϕ of type (α, β, c) can be defined as follows: (1) A k-labeling ϕ of type (α, β, c) of the plane graph G is called face irregular k-labeling of type (α, β, c) of the plane graph G if every two different faces have distinct weights; that is, for graph faces f, g ∈ G and f ≠ g, we have Wt ϕ (α,β,c) (f) ≠ Wt ϕ (α,β,c) (g). (2) Face irregularity strength of type (α, β, c) of any plane graph G is the minimum integer k for which the graph G admits a face irregular k-labeling of type (α, β, c). For a vertex-edge labeled graph G, the minimum integer k for which the graph G admits a face irregular k-labeling of type (α, β, c) is called the total face irregularity strength of type (α, β, c) of the plane graph G, and it is denoted by tfs (α,β,c) (G). A detailed work on irregularity strength of graphs can be seen in [4][5][6][7][8][9][10][11][12].
Gary Ebert et al. worked on the irregularity strength of 2 × n 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 (α, β, c) for ladder graphs [15].
By motivating from all abovementioned, we are working on grid graphs G m n with n rows and m 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 ϕ of type (α, β, c), and this work is a modification of abovementioned articles. Grid graphs are constructed by the graph Cartesian products of path graphs, that is, G m n � P n+1 □P m+1 . We will prove the exact value for the total face irregularity strength under k-labeling ϕ of type (α, β, c) of grid graphs with the property ⌊(m We will prove the exact value for the total face irregularity strength under k-labeling ϕ of type (α, β, c) of wheel graph W n .
Baca et al. determined a lower bound for the face irregularity strength of type (α, β, c) when a 2-connected plane graph G has more than one faces of the largest sizes [14,16]. ey presented the following theorem.
Theorem 1 (see [14,16]). Let G � (V, E, F) be a 2-connected plane graph with n i i-sided faces, i ≥ 3. Let α, β, c ∈ 0, 1 { }, a � min i: n i ≠ 0 , and b � max i: n i ≠ 0 . en, the face irregularity strength of type (α, β, c) of the plane graph G is Proof. Suppose that face irregularity strength under a klabeling ϕ of type α, β, c of the plane graph G is k. e smallest face weight under the face irregular k-labeling ϕ admits the value at least (α + β)a + c. Since |F(G)| � b i�3 n i , it follows that the largest face weight attains the value at least (α + β)a + c + |F(G)| − 1 and at most ((α + β)b + c)k. Hence, □ is lower bound can be improved when a 2-connected plane graph G contains only one face of the largest size, that is, n b � 1 and c � max i: n i ≠ 0, i < b . So, we present the following theorem to calculate the lower bounds for grid graphs G m n .

Main Results
In this research, we will demonstrate the tight lower bound for the total face irregular strength of type (1; 1; 0) 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 tfs(G m n) exists and differences in weights of the horizontal faces must be 1 and the differences in weights of the vertical faces is m.
Proof. We suppose that total face irregularity strength of any 2-connected plane graph G under k-labeling ϕ of type α, β, c is equal to k, that is, Given that the lagest face n b � 1 for i < b. So, the smallest face weight under the face irregular k-labeling ϕ of type (α, β, c) will have the minimum value (α + β)c + c. e total number of faces of the graph can be obtained by adding all the number of i-sided faces where i ≥ 3. Hence, the largest face weight can have the minimum value (α + β)a + c + |F(G)| − 2 and maximum value ((α + β)c + c)k. So, we can construct the following results: Hence,

□
From the above result, we see that if a 2-connected plane graph G contains only one largest face, then the lower bound for the face irregularity strength of type (1, 1, 0) can be calculated as tfs (1,1,0) 2

Journal of Mathematics
In this research, we will prove the tight lower bound for the total face irregularity strength of type (1, 1, 0) for the grid graph G m n and wheel graph W n . To prove the tight lower bound of the grid graph, it will be sufficient to show that the exact value of tfs(G m n ) exists. e exact value of tfs(G m n ), that is, calculated from grid graph G m n under a graph k-labeling of type (1, 1, 0), exists if the differences in weights of the horizontal faces are 1 and the differences in weights of the vertical faces are m. Generalized grid graphs can be written as G m n � P n+1 □ P m+1 . e vertex set and the edge set of the grid graph can be defined as follows: Theorem 3. Let n, m ≥ 2 be positive integers and G m n � P n+1 □ P m+1 be generalized grid graph, then In order to prove this, it will be sufficient to show that the exact value of tfs(G m n ) exists. e vertices for the generalized graph G m n under a klabeling ϕ of type (1, 1, 0) in different intervals of i and j can be defined as follows: e horizontal edges for the generalized graph G m n under a k-labeling ϕ of type (1, 1, 0) in different intervals of i and j can be defined as follows: e vertical edges for the generalized graph G m n under a k-labeling ϕ of type (1, 1, 0) in different intervals of i and j can be defined as follows: Figure 1 represents the generalized formula for face weights. e generalization of weights over the face f under a k-labeling ϕ of type (1, 1, 0) for the graph G m n can be defined as follows: Horizontal differences in weights among different intervals of i and j can be calculated as follows: Journal of Mathematics � 1, for every value of j, Figure 1: Construction of weights over the face f under k-labeling of type (1, 1, 0).

Journal of Mathematics
� 1, for every even value of j,

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For i � 7 and j � 1, 2, . . . , m + 1, Vertical differences in face weights can be calculated as follows: For i � 1, 2, 3, 4, 5, 6 and j � 1, 2, 3, 4, For i � 7; j � 1, 2, 3, 4, 14 Journal of Mathematics It shows that all the differences of horizontal faces are equal to one and all the differences of vertical faces are equal to m. Hence, total face irregularity strength of grid graph G 3 6 is 4. □ Theorem 4. . Let W n be a wheel graph with n + 1 vertices, where n ≥ 3. en, under a total k-labeling of type (1, 1, 0), we have Proof. Let W n be a wheel graph with n + 1 vertices, then by the definition of wheel graph, the total number of edges will be 2n and the total number of faces will be n + 1, that is, As we see that a wheel graph has 3− sided internal faces and external face, so by using eorem 2, we have tfs W n ≥ n + 4 5 .
In Figure 3, v is the vertex in the center of wheel graph W n which is connecting to all the vertices v i for 1 ≤ i ≤ n. Similarly, for 1 ≤ i ≤ n − 1, the edges of the wheel graph can be constructed as E(W n ) � vv i , v i v i+1 , vv n , v 1 v n . Also for 1 ≤ i ≤ n − 1, there will be exterior face, the nth interior face can be written as f(W n ) � vv 1 v n v , and other all 3-sided interior faces can be written as f(W n ) � vv i v i+1 v . Let us define a total k-labeling ϕ: V ∪ E ⟶ 1, 2, 3, . . . , ⌈(n + { 4)/5⌉}.
In Figure 4, we consider a finite wheel graph W 3 which is labeled under a 2-labeling of type (1, 1, 0). So, for 1 ≤ i ≤ 3, we have Weight of exterior face will be Wt f exterior � 10.

Conclusion
We investigated total face irregularity strength of generalized plane grid graphs G m n and wheel graphs W n under a graph k-labeling of type (α, β, c) where α, β ∈ 0, 1 { }. is 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 k-labeling. We derived generalized formulas by considering graphs with different values of n, m, ⌊(m + 1)/3⌋, and m − 2⌊(m + 1)/3⌋. 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 k-labeling of type (α, β, c).

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.

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Journal of Mathematics