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

Authors: Aleem Mughal; Noshad Jamil
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this study, we used grids and wheel graphsG=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. The article addresses the problem to find the face irregularity strength of some families of generalized plane graphs under k-labeling of type α,β,γ. 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. The integer k is named as total face irregularity strength of the graph and denoted as tfsG. We also discussed a special case of total face irregularity strength of plane graphs under k-labeling of type (1, 1, 0). The results will be verified by using figures and examples.


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## Body

## 1. Introduction

This article is based on simple, plane, finite, and undirected graphsG=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,γ∈0,1 and k is a positive integer, then a branch of labeling, named as, k-labeling of typeα,β,γ, 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 Gnm 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 Gnm, 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 ofk-labeling of type α,β,γ 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. The 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. The trivial case α,β,γ=0,0,0 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 f of a plane graph G under k-labeling ϕ of type α,β,γ can be defined as follows:(1)Wtϕα,β,γf=α∑v∼fϕv+β∑e∼fϕe+γϕf.Ak-labeling ϕ of type α,β,γ of the plane graph G is called face irregular k-labeling of type α,β,γ 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(2)Wtϕα,β,γf≠Wtϕα,β,γg.Face irregularity strength of typeα,β,γ 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,γ. 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,γ is called the total face irregularity strength of type α,β,γ of the plane graph G, and it is denoted by tfsα,β,γG. A detailed work on irregularity strength of graphs can be seen in [4–12].Gary Ebert et al. worked on the irregularity strength of2×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,γ for ladder graphs [15].By motivating from all abovementioned, we are working on grid graphsGnm 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,γ, and this work is a modification of abovementioned articles. Grid graphs are constructed by the graph Cartesian products of path graphs, that is, Gnm=Pn+1□Pm+1.We will prove the exact value for the total face irregularity strength underk-labeling ϕ of type α,β,γ of grid graphs with the property ⌊m+1/3⌋=m−2⌊m+1/3⌋ where 1<m<n.We will prove the exact value for the total face irregularity strength underk-labeling ϕ of type α,β,γ of wheel graph Wn.Baca et al. determined a lower bound for the face irregularity strength of typeα,β,γ when a 2-connected plane graph G has more than one faces of the largest sizes [14, 16]. They presented the following theorem.Theorem 1 (see [14, 16]).
LetG=V,E,F be a 2-connected plane graph with nii-sided faces, i≥3. Let α,β,γ∈0,1, a=mini:ni≠0, and b=maxi:ni≠0. Then, the face irregularity strength of type α,β,γ of the plane graph G is(3)fsα,β,γG≥α+βa+γ+FG−1α+βb+γ.Proof.
Suppose that face irregularity strength under ak-labeling ϕ of type α,β,γ of the plane graph G is k.
The smallest face weight under the face irregulark-labeling ϕ admits the value at least α+βa+γ. Since FG=∑i=3bni, it follows that the largest face weight attains the value at least α+βa+γ+FG−1 and at most α+βb+γk. Hence,(4)α+βa+γ+FG−1≤α+βb+γk,k≥α+βa+γ+FG−1α+βb+γ.This lower bound can be improved when a2-connected plane graph G contains only one face of the largest size, that is, nb=1 and c=maxi:ni≠0,i<b. So, we present the following theorem to calculate the lower bounds for grid graphs Gnm.


## 2. Main Results

In this research, we will demonstrate the tight lower bound for the total face irregular strength of type1;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 tfsGmn exists and differences in weights of the horizontal faces must be 1 and the differences in weights of the vertical faces is m.Theorem 2.
LetG=V,E,F be a 2-connected plane graph with nii-sided faces, i≥3. Let α,β,γ∈0,1, a=mini:ni≠0 and b=maxi:ni≠0, nb=1, and c=maxi:ni≠0,i≤b. Then, the total face irregularity strength of type α,β,γ of the plane graph G is(5)fsα,β,γG≥α+βa+γ+FG−2α+βc+γ.Proof.
We suppose that total face irregularity strength of any2-connected plane graph G under k-labeling ϕ of type α,β,γ is equal to k, that is,(6)tfsα,β,γG=k.
Given that the lagest facenb=1 for i<b. So, the smallest face weight under the face irregular k-labeling ϕ of type α,β,γ will have the minimum value α+βc+γ. The 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+βa+γ+FG−2 and maximum value α+βc+γk. So, we can construct the following results:(7)α+βa+γ+FG−2≤α+βb+γk⇒k≥α+βa+γ+FG−2α+βc+γ.
Hence,(8)tfsα,β,γG≥α+βa+γ+FG−2α+βc+γ.From the above result, we see that if a2-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(9)tfs1,1,0G≥2a+FG−22c.In this research, we will prove the tight lower bound for the total face irregularity strength of type1,1,0 for the grid graph Gnm and wheel graph Wn. To prove the tight lower bound of the grid graph, it will be sufficient to show that the exact value of tfsGnm exists. The exact value of tfsGnm, that is, calculated from grid graph Gnm 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 Gnm=Pn+1□Pm+1.The vertex set and the edge set of the grid graph can be defined as follows:(10)VPn+1□Pm+1=vij:i=1,2,…,n+1,j=1,2,…,m+1,EPn+1□Pm+1=vijvi+1j:i=1,2,…,n,j=1,2,…,m+1∪vijvij+1:i=1,2,…,n+1,j=1,2,…,m.Theorem 3.
Letn,m≥2 be positive integers and Gnm=Pn+1□Pm+1 be generalized grid graph, then(11)tfs1,1,0Pn+1□Pm+1=mn+78.In order to prove this, it will be sufficient to show that the exact value oftfsGnm exists.The vertices for the generalized graphGnm under a k-labeling ϕ of type 1,1,0 in different intervals of i and j can be defined as follows:(12)ϕvij=1+m+13i−12,fori=1,2,3,…,2km+1/3andj=1,2,…,m+1,k,fori=2km+1/3+1,…,n+1andj=1,2,…,m+1.The horizontal edges for the generalized graphGnm under a k-labeling ϕ of type 1,1,0 in different intervals of i and j can be defined as follows:(13)ϕvijvij+1=1+m+13i−12,fori=1,2,3,…,2km+1/3andj=1,2,…,m,k,fori=2km+1/3+1,…,n+1andj=1,2,…,m.The vertical edges for the generalized graphGnm under a k-labeling ϕ of type 1,1,0 in different intervals of i and j can be defined as follows:(14)ϕvijvi+1j=j2,fori=1,2,…,2km+1/3−1andj=1,2,…,m+1,ϕvijvi+1j=j2+12m−3k+3+3m+13km+1/3−112i−2km+1/3+1+12m−3k+3+3m+13km+1/3−112i−2km+1/3+1fori=2km+1/3,2km+1/3+1andj=1,3,…,m;m≡1mod2orj=1,3,…,m+1;m≡0mod2,j2+12m−3k+3+3m+13km+1/3−112i−2km+1/3+1+12m−3k+3+3m+13km+1/3−112i−2km+1/3+1fori=2km+1/3,2km+1/3+1andj=2,4,…,m+1;m≡1mod2orj=2,4,…,m;m≡0mod2,ϕvijvi+1j=j2+m−3k+3+3m+13km+1/3−1+m212i−2km+1/3−1+m212i−2km+1/3−1,fori=2km+1/3+2,…,nandj=1,3,…,m;m≡1mod2orj=1,3,…,m+1;m≡0mod2,j2+m−3k+3+3m+13km+1/3−1+m212i−2km+1/3−1+m212i−2km+1/3−1,fori=2km+1/3+2,…,nandj=2,4,…,m+1;m≡1mod2orj=2,4,…,m;m≡0mod2.Figure1 represents the generalized formula for face weights. The generalization of weights over the face f under a k-labeling ϕ of type 1,1,0 for the graph Gnm can be defined as follows:(15)Wt1,1,0fij=∑v∼fijfv+∑e∼fijfe=ϕvij+ϕvij+1+ϕvi+1j+ϕvi+1j+1+ϕvijvij+1+ϕvijvi+1j+ϕvi+1jvi+1j+1+ϕvij+1vi+1j+1.Figure 1
Construction of weights over the facef under k-labeling of type (1, 1, 0).Horizontal differences in weights among different intervals ofi and j can be calculated as follows:(16)Fori=1,2,3,…,2km+1/3−2andj=1,2,…,m+1,Wt1,1,0fij+1−Wt1,1,0fij=ϕvij+1+ϕvij+2+ϕvi+1j+1+ϕvi+1j+2+ϕvij+1vij+2+ϕvij+1vi+1j+1+ϕvi+1j+1vi+1j+2+ϕvij+2vi+1j+2−ϕvij−ϕvij+1−ϕvi+1j−ϕvi+1j+1−ϕvijvij+1−ϕvijvi+1j−ϕvi+1jvi+1j+1−ϕvij+1vi+1j+1=1+m+13i−12+1+m+13i2+1+m+13i−12+1+m+13i2+j+22−1−m+13i−12−1−m+13i2−1−m+13i−12−j2−1−m+13i2=j+22−j2=1,for every value ofj,Fori=2km+1/3−1andj=1,2,…,m+1,Wt1,1,0fij+1−Wt1,1,0fij=ϕvij+1+ϕvij+2+ϕvi+1j+1+ϕvi+1j+2+ϕvij+1vij+2+ϕvij+1vi+1j+1+ϕvi+1j+1vi+1j+2+ϕvij+2vi+1j+2−ϕvij−ϕvij+1−ϕvi+1j−ϕvi+1j+1−ϕvijvij+1−ϕvijvi+1j−ϕvi+1jvi+1j+1−ϕvij+1vi+1j+1=1+m+13i−12+1+m+13i2+1+m+13i−12+1+m+13i2+j+22−1−m+13i−12−1−m+13i2−1−m+13i−12−j2−1−m+13i2=j+22−j2=1,for all values ofj,Fori=2km+1/3andj=1,2,…,m+1,Wt1,1,0fij+1−Wt1,1,0fij=ϕvij+1+ϕvij+2+ϕvi+1j+1+ϕvi+1j+2+ϕvij+1vij+2+ϕvij+1vi+1j+1+ϕvi+1j+1vi+1j+2+ϕvij+2vi+1j+2−ϕvij−ϕvij+1−ϕvi+1j−ϕvi+1j+1−ϕvijvij+1−ϕvijvi+1j−ϕvi+1jvi+1j+1−ϕvij+1vi+1j+1,Wt1,1,0fij+1−Wt1,1,0fij=1+m+13i−12+k+1+m+13i−12+k+j+22+12m−3k+3+3m+13km+1/3−112i+1−2km+1/3+1+12m−3k+3+3m+13km+1/3−112i+1−2km+1/3+1−1−m+13i−12−k−1−m+13i−12−j2−12m−3k+3+3m+13km+1/3−112i−2km+1/3+1−12m−3k+3+3m+13km+1/3−112i−2km+1/3+1−k=j+22+12m−3k+3+3m+13km+1/3−11+0−j2−12m−3k+3+3m+13km+1/3−11−0=j+22−j2=1,for allj=1,2,…,m+1,Fori=2km+1/3+1andj=1,3,…,m;m≡1mod2orj=1,3,…,m+1;m≡0mod2,Wt1,1,0fij+1−Wt1,1,0fij=ϕvij+1+ϕvij+2+ϕvi+1j+1+ϕvi+1j+2+ϕvij+1vij+2+ϕvij+1vi+1j+1+ϕvi+1j+1vi+1j+2+ϕvij+2vi+1j+2−ϕvij−ϕvij+1−ϕvi+1j−ϕvi+1j+1−ϕvijvij+1−ϕvijvi+1j−ϕvi+1jvi+1j+1−ϕvij+1vi+1j+1,Wt1,1,0fij+1−Wt1,1,0fij=k+k+k+k+j+22+12m−3k+3+3m+13km+1/3−112i−2km+1/3+1+12m−3k+3+3m+13km+1/3−112i−2km+1/3+1−k−k−k−k−j2−12m−3k+3+3m+13km+1/3−112i−2km+1/3+1−12m−3k+3+3m+13km+1/3−112i−2km+1/3+1=j+22−j2=1,for every odd value ofj,Fori=2km+1/3+1andj=2,4,…,m+1;m≡1mod2orj=2,4,…,m;m≡0mod2,Wt1,1,0fij+1−Wt1,1,0fij=ϕvij+1+ϕvij+2+ϕvi+1j+1+ϕvi+1j+2+ϕvij+1vij+2+ϕvij+1vi+1j+1+ϕvi+1j+1vi+1j+2+ϕvij+2vi+1j+2−ϕvij−ϕvij+1−ϕvi+1j−ϕvi+1j+1−ϕvijvij+1−ϕvijvi+1j−ϕvi+1jvi+1j+1−ϕvij+1vi+1j+1,Wt1,1,0fij+1−Wt1,1,0fij=k+k+k+k+j+22+12m−3k+3+3m+13km+1/3−112i−2km+1/3+1+12m−3k+3+3m+13km+1/3−112i−2km+1/3+1−k−k−k−k−j2−12m−3k+3+3m+13km+1/3−112i−2km+1/3+1−12m−3k+3+3m+13km+1/3−112i−2km+1/3+1=j+22−j2=1,for every even value ofj,Fori=2km+1/3+2,…,n;j=1,3,…,m;m≡1mod2ORj=1,3,…,m+1;m≡0mod2,Wt1,1,0fij+1−Wt1,1,0fij=ϕvij+1+ϕvij+2+ϕvi+1j+1+ϕvi+1j+2+ϕvij+1vij+2+ϕvij+1vi+1j+1+ϕvi+1j+1vi+1j+2+ϕvij+2vi+1j+2−ϕvij−ϕvij+1−ϕvi+1j−ϕvi+1j+1−ϕvijvij+1−ϕvijvi+1j−ϕvi+1jvi+1j+1−ϕvij+1vi+1j+1,Wt1,1,0fij+1−Wt1,1,0fij=k+k+k+k+j+22+m−3k+3+3m+13km+1/3−1+m212i−2km+1/3−1+m212i−2km+1/3−1−k−k−k−k−j2−m−3k+3+3m+13km+1/3−1−m212i−2km+1/3−1−m212i−2km+1/3−1=j+22−j2=1,for all odd values ofj,Fori=2km+1/3+2,…,n;j=2,4,…,m+1;m≡1mod2ORj=2,4,…,m;m≡0mod2,Wt1,1,0fij+1−Wt1,1,0fij=ϕvij+1+ϕvij+2+ϕvi+1j+1+ϕvi+1j+2+ϕvij+1vij+2+ϕvij+1vi+1j+1+ϕvi+1j+1vi+1j+2+ϕvij+2vi+1j+2−ϕvij−ϕvij+1−ϕvi+1j−ϕvi+1j+1−ϕvijvij+1−ϕvijvi+1j−ϕvi+1jvi+1j+1−ϕvij+1vi+1j+1,Wt1,1,0fij+1−Wt1,1,0fij=k+k+k+k+j+22+m−3k+3+3m+13km+1/3−1+m212i−2km+1/3−1+m212i−2km+1/3−1−k−k−k−k−j2−m−3k+3+3m+13km+1/3−1−m212i−2km+1/3−1−m212i−2km+1/3−1=j+22−j2=1,for all even values ofj.Vertical differences in weights among different intervals ofi and j can be calculated as follows:(17)Fori=1,2,…,2km+1/3−2andj=1,2,…,m+1,Wt1,1,0fi+1j−Wt1,1,0fij=ϕvi+1j+ϕvi+1j+1+ϕvi+2j+ϕvi+2j+1+ϕvi+1jvi+1j+1+ϕvi+1jvi+2j+ϕvi+2jvi+2j+1+ϕvi+1j+1vi+2j+1−ϕvij−ϕvij+1−ϕvi+1j−ϕvi+1j+1−ϕvijvij+1−ϕvijvi+1j−ϕvi+1jvi+1j+1−ϕvij+1vi+1j+1,Wt1,1,0fi+1j−Wt1,1,0fij=1+m+13i+12+1+m+13i+12+j2+j+12+1+m+13i+12−1−m+13i−12−1−m+13i−12−1−m+13i−12−j2−j+12=3m+13i+12−3m+13i−12=3m+13i+12−i−12=3m+131=m,Fori=2km+1/3−1;j=1,2,…,m+1,Wt1,1,0fi+1j−Wt1,1,0fij=ϕvi+1j+ϕvi+1j+1+ϕvi+2j+ϕvi+2j+1+ϕvi+1jvi+1j+1+ϕvi+1jvi+2j+ϕvi+2jvi+2j+1+ϕvi+1j+1vi+2j+1−ϕvij−ϕvij+1−ϕvi+1j−ϕvi+1j+1−ϕvijvij+1−ϕvijvi+1j−ϕvi+1jvi+1j+1−ϕvij+1vi+1j+1,Wt1,1,0fi+1j−Wt1,1,0fij=k+k+k+j2+j+12+12i+1−2km+1/3+112m−3k+3+3m+13km+1/3−1+12m−3k+3+3m+13km+1/3−1+12i+1−2km+1/3+112m−3k+3+3m+13km+1/3−1+12m−3k+3+3m+13km+1/3−1−1−m+13i−12−1−m+13i−12−1−m+13i−12−j2−j+12=3k+12i+1−2km+1/3+1,m−3k+3+3m+1/3k/m+1/3−12+m−3k+3+3m+1/3k/m+1/3−12+12i+1−2k/m+1/3+1m−3k+3+3m+1/3k/m+1/3−12+m−3k+3+3m+1/3k/m+1/3−12−3−3m+13i−12=3k+1m−3k+3+3m+13km+1/3−1+0−3−3m+13i−12=3k+m−3k+3+3m+13km+1/3−3m+13−3−3m+13i−12=m+3m+13km+1/3−1−i−12=m+3m+13km+1/3−1−km+1/3−1=m,Fori=2km+1/3;j=1,2,…,m+1,Wt1,1,0fi+1j−Wt1,1,0fij=ϕvi+1j+ϕvi+1j+1+ϕvi+2j+ϕvi+2j+1+ϕvi+1jvi+1j+1+ϕvi+1jvi+2j+ϕvi+2jvi+2j+1+ϕvi+1j+1vi+2j+1−ϕvij−ϕvij+1−ϕvi+1j−ϕvi+1j+1−ϕvijvij+1−ϕvijvi+1j−ϕvi+1jvi+1j+1−ϕvij+1vi+1j+1,Wt1,1,0fi+1j−Wt1,1,0fij=k+k+k+j2+j+12+12i+1−2km+1/3+112m−3k+3+3m+13km+1/3−1+12m−3k+3+3m+13km+1/3−1+12i+1−2km+1/3+112m−3k+3+3m+13km+1/3−1+12m−3k+3+3m+13km+1/3−1−1−m+13i−12−1−m+13i−12−1−m+13i−12−j2−j+12−12i−2km+1/3+112m−3k+3+3m+13km+1/3−1+12m−3k+3+3m+13km+1/3−1,−12i−2km+1/3+112m−3k+3+3m+13km+1/3−1+12m−3k+3+3m+13km+1/3−1=3k+12m−3k+3+3m+13km+1/3−1+12m−3k+3+3m+13km+1/3−1+12m−3k+3+3m+13km+1/3−1+12m−3k+3+3m+13km+1/3−1−3−3m+13i−12−12m−3k+3+3m+13km+1/3−1+12m−3k+3+3m+13km+1/3−1=3k+m−3k+3+3m+1/3k/m+1/3−12+m−3k+3+3m+1/3k/m+1/3−12−3−3m+13i−12=3k+m−3k+3+3m+13km+1/3−1−3−3m+13i−12=m+3m+13km+1/3−1−i−12=m+3m+13km+1/3−1−i−12=m,Fori=2km+1/3+1;j=1,2,…,m+1,Wt1,1,0fi+1j−Wt1,1,0fij=ϕvi+1j+ϕvi+1j+1+ϕvi+2j+ϕvi+2j+1+ϕvi+1jvi+1j+1+ϕvi+1jvi+2j+ϕvi+2jvi+2j+1+ϕvi+1j+1vi+2j+1−ϕvij−ϕvij+1−ϕvi+1j−ϕvi+1j+1−ϕvijvij+1−ϕvijvi+1j−ϕvi+1jvi+1j+1−ϕvij+1vi+1j+1,Wt1,1,0fi+1j−Wt1,1,0fij=k+k+k+j2+j+12+2m−3k+3+3m+13km+1/3−1+12i+1−2km+1/3−1m2+m2+12i+1−2km+1/3−1m2+m2−k−k−k−j2−j+12−12i−2km+1/3+1m−3k+3+3m+1/3k/m+1/3−12+m−3k+3+3m+1/3k/m+1/3−12−12i−2km+1/3+1m−3k+3+3m+1/3k/m+1/3−12+m−3k+3+3m+1/3k/m+1/3−12=2m−3k+3+3m+13km+1/3−1+1m−1m−3k+3+3m+13km+1/3−1−1m−3k+3+3m+13km+1/3−1=m,Fori=2km+1/3+2,…,n;j=1,2,…,m+1,Wt1,1,0fi+1j−Wt1,1,0fij=ϕvi+1j+ϕvi+1j+1+ϕvi+2j+ϕvi+2j+1+ϕvi+1jvi+1j+1+ϕvi+1jvi+2j+ϕvi+2jvi+2j+1+ϕvi+1j+1vi+2j+1−ϕvij−ϕvij+1−ϕvi+1j−ϕvi+1j+1−ϕvijvij+1−ϕvijvi+1j−ϕvi+1jvi+1j+1−ϕvij+1vi+1j+1,Wt1,1,0fi+1j−Wt1,1,0fij=k+k+k+j2+j+12+2m−3k+3+3m+13km+1/3−1+12i+1−2km+1/3−1m2+m2+12i+1−2km+1/3−1m2+m2−k−k−k−j2−j+12−2m−3k+3+3m+13km+1/3−1−12i−2km+1/3−1m2+m2−12i−2km+1/3−1m2+m2=m12i+1−2km+1/3−1+m12i+1−2km+1/3−1−m12i−2km+1/3−1−m12i−2km+1/3−1=mi+1−2k/m+1/3−12+i+1−2k/m+1/3−12−mi−2k/m+1/3−12+i−2k/m+1/3−12=mi+1−2km+1/3−1−mi−2km+1/3−1=mi+1−2km+1/3−1−i+2km+1/3+1=m.Example 1.
The total face irregularity strength of grid graphG63, under a k-labeling of type 1,1,0 is 4.Proof.
The graph under consideration isG63=P7□P4. Figure 2 is a 4-labeling of type 1,1,0 for the grid graph G63, and it will help us in calculating total face irregularity strength in different intervals of the grid graph.
Here,k=⌈18+7/8⌉=4, ⌊m+1/3⌋=1, ⌈k/⌊m+1/3⌋⌉=4, and m−2⌊m+1/3⌋=1
In order to show thattfs1,1,0G63=4, 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:(18)Fori=1,2,3,4,5,6andj=1,2,3,4,Wt1,1,0fij+1−Wt1,1,0fij=ϕvij+1+ϕvij+2+ϕvi+1j+1+ϕvi+1j+2+ϕvij+1vij+2+ϕvij+1vi+1j+1+ϕvi+1j+1vi+1j+2+ϕvij+2vi+1j+2−ϕvij−ϕvij+1−ϕvi+1j−ϕvi+1j+1−ϕvijvij+1−ϕvijvi+1j−ϕvi+1jvi+1j+1−ϕvij+1vi+1j+1=j+22−j2=1,for every value ofj,Fori=7andj=1,2,…,m+1,Wt1,1,0fij+1−Wt1,1,0fij=ϕvij+1+ϕvij+2+ϕvi+1j+1+ϕvi+1j+2+ϕvij+1vij+2+ϕvij+1vi+1j+1+ϕvi+1j+1vi+1j+2+ϕvij+2vi+1j+2−ϕvij−ϕvij+1−ϕvi+1j−ϕvi+1j+1−ϕvijvij+1−ϕvijvi+1j−ϕvi+1jvi+1j+1−ϕvij+1vi+1j+1=j+22−j2=1,for all values ofj.
Vertical differences in face weights can be calculated as follows:(19)Fori=1,2,3,4,5,6andj=1,2,3,4,Wt1,1,0fi+1j−Wt1,1,0fij=ϕvi+1j+ϕvi+1j+1+ϕvi+2j+ϕvi+2j+1+ϕvi+1jvi+1j+1+ϕvi+1jvi+2j+ϕvi+2jvi+2j+1+ϕvi+1j+1vi+2j+1−ϕvij−ϕvij+1−ϕvi+1j−ϕvi+1j+1−ϕvijvij+1−ϕvijvi+1j−ϕvi+1jvi+1j+1−ϕvij+1vi+1j+1=3m+13=3,Fori=7;j=1,2,3,4,Wt1,1,0fi+1j−Wt1,1,0fij=ϕvi+1j+ϕvi+1j+1+ϕvi+2j+ϕvi+2j+1+ϕvi+1jvi+1j+1+ϕvi+1jvi+2j+ϕvi+2jvi+2j+1+ϕvi+1j+1vi+2j+1−ϕvij−ϕvij+1−ϕvi+1j−ϕvi+1j+1−ϕvijvij+1−ϕvijvi+1j−ϕvi+1jvi+1j+1−ϕvij+1vi+1j+1=m+3m+13km+1/3−1−km+1/3−1=3.
It shows that all the differences of horizontal faces are equal to one and all the differences of vertical faces are equal tom. Hence, total face irregularity strength of grid graph G63 is 4.Figure 2
Total face irregular 4-labeling of the (1, 1, 0) of grid graphG63.Theorem 4.
. LetWn be a wheel graph with n+1 vertices, where n≥3. Then, under a total k-labeling of type 1,1,0, we have(20)tfsWn=n+45.Proof.
LetWn 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,(21)EWn=2n,FWn=n+1.
As we see that a wheel graph has3−sided internal faces and external face, so by using Theorem 2, we have(22)tfsWn≥n+45.
In Figure3, v is the vertex in the center of wheel graph Wn which is connecting to all the vertices vi for 1≤i≤n. Similarly, for 1≤i≤n−1, the edges of the wheel graph can be constructed as EWn=vvi,vivi+1,vvn,v1vn. Also for 1≤i≤n−1, there will be exterior face, the nth interior face can be written as fWn=vv1vnv, and other all 3-sided interior faces can be written as fWn=vvivi+1v. Let us define a total k-labeling ϕ:V∪E⟶1,2,3,…,⌈n+4/5⌉.
In Figure4, we consider a finite wheel graph W3 which is labeled under a 2-labeling of type 1,1,0. So, for 1≤i≤3, we have(23)ϕv=ϕv2=ϕvvi=ϕv1v2=1,ϕv1=ϕv3=ϕv1v3=ϕv2v3=2.
Weight of exterior face will be(24)Wtfexterior=10.
Weight of interior faces will be(25)Wtfi=i+6.
Now, let us talk about the graphs exceptW3 for which we define the labeling as ϕv=1:(i)
For1≤i≤⌈n/2⌉+1, we have ϕvi=⌈2i/5⌉(ii)
For⌈n/2⌉+2≤i≤n, we have ϕvi=⌈2n−i+1/5⌉+1(iii)
For1≤i≤⌈n/2⌉+1, we have ϕvvi=⌈2i−1/5⌉(iv)
For⌈n/2⌉+2≤i≤n, we have ϕvvi=⌈2n−i+1/5⌉+1(v)
Fori=1, we have ϕv1vn=1(vi)
For2≤i≤⌈n/2⌉, we have ϕvivi+1=⌈2i−2/5⌉(vii)
Fori=⌈n/2⌉+1, where n≡mmod6 in which m=2,3,4,5, we have ϕvivi+1=⌈2i−2/5⌉(viii)
Fori=⌈n/2⌉+1 where n≡mmod6 in which m∈0,1, we have ϕvivi+1=⌈n+4/5⌉(xi)
Fori=⌈n/2⌉+2≤i≤n−1, we have ϕvivi+1=⌈2n−i−2/5⌉+1
For weights of the wheel graphWn, we proceed as follows:(i)
Fori=1, we have Wtfi=6(ii)
For2≤i≤⌈n+1/2⌉, we have Wtfi=2i+3(iii)
For⌈n+1/2⌉+1≤i≤n, we have Wtfi=2n−i+8(iv)
For external weight, we will useWtfexterior≥2n+8
We can easily observe that under thek-labeling ϕ of type 1,1,0, the minimum k for which the wheel graph admits total face irregular strength is ⌈n+4/5⌉. Hence,(26)tfsWn=n+45.Figure 3
Wheel graphWn.Figure 4
2-labeling of a wheel graphW3.


## 3. Conclusion

We investigated total face irregularity strength of generalized plane grid graphsGnm and wheel graphs Wn under a graph k-labeling of type α,β,γ where α,β∈0,1. 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 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,γ.



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