Face irregular evaluations of family of grids

Abstract

Let $G=(V,E,F)$ be a connected plane graph. Let $\rho :\alpha (V)\cup \beta (E)\cup \gamma (F)\to \{1,2,\dots ,k\}$ be a k-labeling where $\alpha ,\beta ,\gamma \in \{0,1\}$ and $(\alpha ,\beta ,\gamma )\ne (0,0,0)$. The weight of a face f under ρ is defined as $\mathit{\text{Wt}}(f)=\alpha \sum _{v\sim f}\rho (v)+\beta \sum _{e\sim f}\rho (e)+\gamma \rho (f)$. Then ρ is called a face irregular labeling of type $(\alpha ,\beta ,\gamma )$ if $\mathit{\text{Wt}}(f)\ne \mathit{\text{Wt}}(g)$ for any two faces $f\ne g$. The face irregular strength of G is the smallest integer k such that G admits a face irreg ular k-labeling. In this paper we determine the face irregular strength of the grid graph $P_{n+1}\square P_{m+1}$ under a labeling of type $(\alpha ,\beta ,\gamma )$.

KEYWORDS:

• Face irregular labeling of type $(\alpha ,\beta ,\gamma )$
• face irregular strength
• Cartesian Product of path graphs

1 Introduction

By a graph $G=(V,E,F)$, we mean a simple, finite, undirected plane graph with vertex set V, edge set E and face set F. For a positive integer k, a k-labeling of type $(\alpha ,\beta ,\gamma )$, where $\alpha ,\beta ,\gamma \in \{0,1\}$ is a mapping ρ from the set of graph elements to the set of positive integers $\left\{1,2,3,\dots ,k\right\}$. If the domain of k-labeling of type $(\alpha ,\beta ,\gamma )$ is V(G), E(G), F(G), $V(G)\cup E(G),V(G)\cup F(G),E(G)\cup F(G)$, $V(G)\cup E(G)\cup F(G)$, then we call this a vertex k-labeling of type (1, 0, 0), edge k-labeling of type (0, 1, 0), face k-labeling of type (0, 0, 1), vertex-edge k-labeling of type (1, 1, 0), vertex-face k-labeling of type (1, 0, 1), edge-face k-labeling of type (0, 1, 1), vertex-edge-face k-labeling of type (1, 1, 1) respectively.

Let $G=(V,E,F)$ be a connected plane graph. Let $\rho :\alpha (V)\cup \beta (E)\cup \gamma (F)\to \{1,2,\dots ,k\}$ be a k-labeling where $\alpha ,\beta ,\gamma \in \{0,1\}$ and $(\alpha ,\beta ,\gamma )\ne (0,0,0)$. The weight of a face f under ρ is defined as $$\mathit{\text{Wt}}(f)=\alpha \sum _{v\sim f}\rho (v)+\beta \sum _{e\sim f}\rho (e)+\gamma \rho (f)$$

Then ρ is called a face irregular labeling of type $(\alpha ,\beta ,\gamma )$ if $\mathit{\text{Wt}}(f)\ne \mathit{\text{Wt}}(g)$ for any two faces $f\ne g$. The face irregular strength of G is the smallest integer k such that G admits a face irregular k-labeling. In this paper we determine the face irregular strength of the grid graph $P_{n+1}\square P_{m+1}$ under a labeling of type $(\alpha ,\beta ,\gamma )$

A k-labeling ρ of type $(\alpha ,\beta ,\gamma )$ of the plane graph G will be face irregular k-labeling of type $(\alpha ,\beta ,\gamma )$ of the plane graph G, if for every two different faces f and g of G, we have $$W{t}_{{\rho }_{(\alpha ,\beta ,\gamma )}}(f)\ne W{t}_{{\rho }_{(\alpha ,\beta ,\gamma )}}(g)$$

The minimum number k for which the graph G admits a face irregular k-labeling of type $(\alpha ,\beta ,\gamma )$ is called the face irregularity strength of type $(\alpha ,\beta ,\gamma )$ of the plane graph G and it is denoted by ${\text{fs}}_{(\alpha ,\beta ,\gamma )}(G)$. G. Chartrand, et al. in [10] introduced edge k-labeling phi of a graph G such that $w_{\varphi }(x)\ne w_{\varphi }(y)$ for all vertices $x,y\in V(G)$ for distinct x and y. Such labelings were called irregular assignments and the irregularity strength s(G) of a graph G is known as the minimum k for which G has an irregular assignment using labels at most k and this parameter has attracted attention to [12, 16]. Ahmed, A et al. gives the lower bound for the edge irregular strength and prove it for several families of graphs and generate relation between $\mathit{\text{tes}}(G)$ and $\mathit{\text{es}}(G)$ in [1].

Motivated by this notion Bača et al. in [8] defined a vertex irregular total k-labeling of a graph G to be a total labeling of $G;\lambda :V(G)\cup E(G)\to \{1,2,\dots ,k\}$, such that the total vertex-weights $w{t}_{\lambda }(v)=\lambda (v)+{\Sigma }_{\mathit{\text{vu}}\in E}(G)\lambda (\mathit{\text{vu}})$ are different for all vertices i.e., $w{t}_{\lambda }(v)\ne w{t}_{\lambda }(u)$ where $u\ne v$ and $v,u\in V(G)$. They also defined the total labeling $T:V(G)\cup E(G)\to \{1,2,\dots ,k\}$, such that the total edge-weights $w{t}_{\lambda }(v)=\lambda (v)+{\Sigma }_{\mathit{\text{vu}}\in E}(G)\lambda (\mathit{\text{vu}})$ are different for all edges i.e., $w{t}_{\lambda }(\mathit{\text{uv}})\ne w{t}_{\lambda }(v\prime u\prime )$ for all different edges uv and $u\prime v\prime$, and $\mathit{\text{uv}};u\prime v\prime \in E(G)$. The total edge irregularity strength, $\mathit{\text{tes}}(G)$ is defined as the minimum k for which G has the edge irregular total k-labeling. Some of the interesting results on total vertex irregular strength and total edge irregular strength can be found in [2–4, 9]. For a deep survey on the irregularity strength, one can see [15]. The research papers by Tongi [7], Dimitz et al. [11], Gyárfás [13] and Nierhoff [16] present recent results on irregularity strength. Note, that for some classes of graphs and some values of parameters α, β and γ, the corresponding graph invariant is infinite.

In this article, the cases under consideration are vertex, edge, face, vertex-face and edge-face. We have focussed on finding the exact tight lower bound for the face irregularity strength of plane grid graphs under labeling ρ of type $(\alpha ,\beta ,\gamma )$.

Theorem 1.

[5] Let $G=(V,E,F)$ be a $2-$ connected plane graph with n_i $i-$ sided faces, $i\ge 3$. Let $\alpha ,\beta ,\gamma \in \{0,1\},a=\mathrm{\text{min}}\{i:n_i\ne 0\}$ and $b=\mathrm{\text{max}}\{i:n_i\ne 0\}$. Then the face irregularity strength of type α, β, γ of the plane graph G is $$f{s}_{(\alpha ,\beta ,\gamma )}(G)\ge \lceil \frac{\left(\alpha +\beta \right)a+\gamma +|F(G)|-1}{\left(\alpha +\beta \right)b+\gamma }\rceil$$

Theorem 2.

[5] Let $\alpha ,\beta ,\gamma \in \{0,1\}$. If $n\ge 3$ then $${\text{fs}}_{(\alpha ,\beta ,\gamma )}(C_n)=\{\begin{array}{cc}2,\hfill & \hspace{1em}\text{if}\gamma \text{ = }1,\hfill \\ \infty ,\hfill & \hspace{1em}\text{if}\gamma \text{ = }0.\hfill \end{array}$$

Since the face irregularity strength of type $(\alpha ,\beta ,\gamma )$ for $C_4=P_2\square P_2$ holds.

Theorem 3.

[5] Let $L_{\mathrm{n\text{'}}}{P}_n\square P_2,n\ge 3$, be a ladder and let $\alpha ,\beta ,\gamma \in \{0,1\}$. Then $${\text{fs}}_{(\alpha ,\beta ,\gamma )}(L_n)=\{\begin{array}{cc}n,\hfill & \hspace{1em}\text{if }(\alpha ,\beta ,\gamma )=(0,0,1),\hfill \\ \lceil \frac{n+2}{4}\rceil ,\hfill & \hspace{1em}\text{if }(\alpha ,\beta ,\gamma )=(1,0,0);(0,1,0),\hfill \\ \lceil \frac{n+3}{5}\rceil ,\hfill & \hspace{1em}\text{if }(\alpha ,\beta ,\gamma )=(1,0,1);(0,1,1),\hfill \\ \lceil \frac{n+6}{8}\rceil ,\hfill & \hspace{1em}\text{if }(\alpha ,\beta ,\gamma )=(1,1,0),\hfill \\ \lceil \frac{n+7}{9}\rceil ,\hfill & \hspace{1em}\text{if }(\alpha ,\beta ,\gamma )=(1,1,1)\hfill \end{array}$$

Theorem 4.

[6] Let $m,n\ge 2$ be positive integer and $G_n^m=P_{n+1}\square P_{m+1}$ be grid graph, then $${\text{tfs}}_{(1,1,0)}(G)=\lceil \frac{\mathit{\text{mn}}+7}{8}\rceil$$

2 Face Irregular strength of grid graph $P_{n+1}\square P_{m+1}$

Let $G=P_{n+1}\square P_{m+1}$, where $1\le m\le n$,

$V(P_{n+1}\square P_{m+1})=\{v_i^j:i=1,2,\dots ,n+1,j=1,2,\dots ,m+1\}$ and $$E(P_{n+1}\square P_{m+1})=\{v_i^j{v}_{i+1}^j:i=1,2,\dots ,n,j=1,2,\dots ,m+1\}\cup \{v_i^j{v}_i^{j+1}:i=1,2,\dots ,n+1,j=1,2,\dots ,m\}.$$

Theorem 5.

Let $m\ge 2$ be a positive integers and $G=P_{n+1}\square P_{m+1}$ be grid graph, then $${\text{fs}}_{(1,0,0)}(G)=\lceil \frac{\mathit{\text{mn}}+3}{4}\rceil$$

Proof.

According to (1) it is sufficient to show that ${\text{fs}}_{(1,0,0)}(G)\le \lceil \frac{\mathit{\text{mn}}+3}{4}\rceil$. So we define the face irregular k-labeling ρ of type (1, 0, 0) of $G_n^m$ in the following way $$\rho (v_i^j)=\{\begin{array}{cc}\lceil \frac{j}{4}\rceil +\lfloor \frac{m}{2}\rfloor \lfloor \frac{i+1}{4}\rfloor +\lceil \frac{m}{2}\rceil \lfloor \frac{i}{4}\rfloor ,\hfill & \text{if} i=1,3,5,\dots ,n;n\equiv 1(2) \text{or} i=1,3,5,\dots ,n+1;n\equiv 0(2)\hfill \\ \hfill & \text{and} j=1,3,5,\dots ,m;m\equiv 1(2) \text{or} j=1,3,5,\dots ,m+1;m\equiv 0(2)\hfill \\ \lceil \frac{j}{4}\rceil +\lceil \frac{m}{2}\rceil \lfloor \frac{i+1}{4}\rfloor +\lfloor \frac{m}{2}\rfloor \lfloor \frac{i}{4}\rfloor ,\hfill & \text{if} i=1,3,5,\dots ,n;n\equiv 1(2) \text{or} i=1,3,5,\dots ,n+1;n\equiv 0(2)\hfill \\ \hfill & \text{and} j=2,4,6,\dots ,m+1;m\equiv 1(2)\text{ or } j=2,4,6,\dots ,m;m\equiv 0(2)\hfill \\ \lceil \frac{j+2}{4}\rceil +\lfloor \frac{m}{2}\rfloor \lfloor \frac{i}{4}\rfloor +\lceil \frac{m}{2}\rceil \lfloor \frac{i-1}{4}\rfloor ,\hfill & \text{if} i=2,4,6,\dots ,n+1;n\equiv 1(2) \text{or} i=2,4,6,\dots ,n;n\equiv 0(2)\hfill \\ \hfill & \text{and} j=1,3,5,\dots ,m;m\equiv 1(2) \text{or} j=1,3,5,\dots ,m+1;m\equiv 0(2)\hfill \\ \lceil \frac{j+2}{4}\rceil +\lceil \frac{m}{2}\rceil \lfloor \frac{i}{4}\rfloor +\lfloor \frac{m}{2}\rfloor \lfloor \frac{i-1}{4}\rfloor ,\hfill & \text{if} i=2,4,6,\dots ,n+1;n\equiv 1(2) \text{or} i=2,4,6,\dots ,n;n\equiv 0(2)\hfill \\ \hfill & \text{and} j=2,4,6,\dots ,m+1;m\equiv 1(2)\text{ or} j=2,4,6,\dots ,m;m\equiv 0(2)\hfill \end{array}$$

The weight of the face f under labeling ρ of type (1, 0, 0) is defined as $$W{t}_{\rho (1,0,0)}(f_i^j)=\rho (v_i^j)+\rho (v_i^{j+1})+\rho (v_{i+1}^j)+\rho (v_{i+1}^{j+1})$$

The differences between the weights of the horizontal faces will be calculated as follows: $$\begin{array}{c}\text{For} i=1,3,5,\dots ,n;n\equiv 1(2)\\ \hspace{1em}\text{or} i=1,3,5,\dots ,n+1;n\equiv 0(2)\text{ and}\\ j=1,3,5,\dots ,m;m\equiv 1(2)\\ \hspace{1em}\text{or} j=1,3,5,\dots ,m+1;m\equiv 0(2)\text{ or}\\ j=2,4,6,\dots ,m+1;m\equiv 1(2)\\ \hspace{1em}\text{or} j=2,4,6,\dots ,m;m\equiv 0(2)\end{array}$$

Let, $g=(f_i^{j+1})$ and $h=(f_i^j)$ be two horizontal consecutive faces. Then, $$\begin{array}{c}W{t}_{\rho (1,0,0)}(g)-W{t}_{\rho (1,0,0)}(h)\\ =\left(\rho (v_i^{j+1})+\rho (v_i^{j+2})+\rho (v_{i+1}^{j+1})+\rho (v_{i+1}^{j+2})\right)\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-\left(\rho (v_i^j)+\rho (v_i^{j+1})+\rho (v_{i+1}^j)+\rho (v_{i+1}^{j+1})\right)\\ \hspace{1em}=\lceil \frac{j}{4}\rceil +\lceil \frac{j+1}{4}\rceil +\lceil \frac{j+2}{4}\rceil +\lceil \frac{j+3}{4}\rceil +2m\lfloor \frac{i}{4}\rfloor \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+2m\lfloor \frac{i+1}{4}\rfloor +1-\lceil \frac{j}{4}\rceil -\lceil \frac{j+1}{4}\rceil -\lceil \frac{j+2}{4}\rceil \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-\lceil \frac{j+3}{4}\rceil -2m\lfloor \frac{i}{4}\rfloor -2m\lfloor \frac{i+1}{4}\rfloor \\ \hspace{1em}=1\\ \text{For} i=2,4,6,\dots ,n+1;n\equiv 1(2)\\ \hspace{1em}\text{or} i=2,4,6,\dots ,n;n\equiv 0(2)\text{ and}\\ j=1,3,5,\dots ,m;m\equiv 1(2)\\ \hspace{1em}\text{or} j=1,3,5,\dots ,m+1;m\equiv 0(2)\text{ or}\\ j=2,4,6,\dots ,m+1;m\equiv 1(2)\\ \hspace{1em}\text{or} j=2,4,6,\dots ,m;m\equiv 0(2)\\ W{t}_{\rho (1,0,0)}(g)-W{t}_{\rho (1,0,0)}(h)\\ \hspace{1em}=\left(\rho (v_i^{j+1})+\rho (v_i^{j+2})+\rho (v_{i+1}^{j+1})+\rho (v_{i+1}^{j+2})\right)\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-\left(\rho (v_i^j)+\rho (v_i^{j+1})+\rho (v_{i+1}^j)+\rho (v_{i+1}^{j+1})\right)\\ \hspace{1em}=\lceil \frac{j}{4}\rceil +\lceil \frac{j+1}{4}\rceil +\lceil \frac{j+2}{4}\rceil +\lceil \frac{j+3}{4}\rceil \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+m\lfloor \frac{i}{4}\rfloor +m\lfloor \frac{i+2}{4}\rfloor \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+m\lfloor \frac{i-1}{4}\rfloor +m\lfloor \frac{i+1}{4}\rfloor +1\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-\lceil \frac{j}{4}\rceil -\lceil \frac{j+1}{4}\rceil -\lceil \frac{j+2}{4}\rceil \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-\lceil \frac{j+3}{4}\rceil -m\lfloor \frac{i}{4}\rfloor -m\lfloor \frac{i+2}{4}\rfloor \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-m\lfloor \frac{i-1}{4}\rfloor -m\lfloor \frac{i+1}{4}\rfloor \\ \hspace{1em}=1\end{array}$$

Now we compute differences between the weights of the vertical faces will be calculated as follows: $$\begin{array}{c}\text{For} i=1,3,5,\dots ,n;n\equiv 1(2)\\ \hspace{1em}\text{or} i=1,3,5,\dots ,n+1;n\equiv 0(2)\text{ and}\\ j=1,3,5,\dots ,m;m\equiv 1(2)\\ \hspace{1em}\text{or} j=1,3,5,\dots ,m+1;m\equiv 0(2)\text{ or}\\ j=2,4,6,\dots ,m+1;m\equiv 1(2)\\ \hspace{1em}\text{or} j=2,4,6,\dots ,m;m\equiv 0(2)\end{array}$$

Let, $g=(f_{i+1}^j)$ and $h=(f_i^j)$ be two vertical consecutive faces. Then, $$\begin{array}{c}W{t}_{\rho (1,0,0)}(g)-W{t}_{\rho (1,0,0)}(h)\\ \hspace{1em}=\left(\rho (v_{i+1}^j)+\rho (v_{i+1}^{j+1})+\rho (v_{i+2}^j)+\rho (v_{i+2}^{j+1})\right)\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-\left(\rho (v_i^j)+\rho (v_i^{j+1})+\rho (v_{i+1}^j)+\rho (v_{i+1}^{j+1})\right)\\ \hspace{1em}=m\lfloor \frac{i+3}{4}\rfloor +m\lfloor \frac{i+2}{4}\rfloor -m\lfloor \frac{i+1}{4}\rfloor -m\lfloor \frac{i}{4}\rfloor \\ \hspace{1em}=m\\ \text{For} i=2,4,6,\dots ,n+1;n\equiv 1(2)\\ \hspace{1em}\text{or} i=2,4,6,\dots ,n;n\equiv 0(2)\text{ and}\\ j=1,3,5,\dots ,m;m\equiv 1(2)\\ \hspace{1em}\text{or} j=1,3,5,\dots ,m+1;m\equiv 0(2)\text{ or}\\ j=2,4,6,\dots ,m+1;m\equiv 1(2)\\ \hspace{1em}\text{or} j=2,4,6,\dots ,m;m\equiv 0(2)\\ W{t}_{\rho (1,0,0)}(g)-W{t}_{\rho (1,0,0)}(h)\\ \hspace{1em}=\left(\rho (v_{i+1}^j)+\rho (v_{i+1}^{j+1})+\rho (v_{i+2}^j)+\rho (v_{i+2}^{j+1})\right)\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-\left(\rho (v_i^j)+\rho (v_i^{j+1})+\rho (v_{i+1}^j)+\rho (v_{i+1}^{j+1})\right)\\ \hspace{1em}=m\lfloor \frac{i+2}{4}\rfloor +m\lfloor \frac{i+1}{4}\rfloor -m\lfloor \frac{i}{4}\rfloor -m\lfloor \frac{i-1}{4}\rfloor \\ \hspace{1em}=m\end{array}$$

Hence it follows that the weights of any two faces f and g are distinct. Complete the proof by clearly giving the two required inequalities. □

Theorem 6.

Let $m\ge 2$ be a positive integers and $G=P_{n+1}\square P_{m+1}$ be grid graph, then $${\text{fs}}_{(0,1,0)}(G)=\lceil \frac{3+\mathit{\text{mn}}}{4}\rceil$$

Proof.

We define a face irregular k-labeling ρ of type (0, 1, 0) of G in the following way $$\begin{array}{c}f(v_i^j{v}_i^{j+1})=\{\begin{array}{cc}1+m(\lceil \frac{i}{2}\rceil -1),\hfill & i=1,2,\dots ,2\lceil \frac{k}{m}\rceil \hfill \\ \hfill & \hspace{1em}\text{and }j=1,2,\dots ,m\hfill \\ k,\hfill & i=2\lceil \frac{k}{m}\rceil +1,\dots ,n+1\hfill \\ \hfill & \hspace{1em}\text{and }j=1,2,\dots ,m\hfill \end{array}\\ f(v_i^j{v}_{i+1}^j)=\{\begin{array}{cc}\lceil \frac{j}{2}\rceil ,\hfill & i=1,2,3,\dots ,2\lceil \frac{k}{m}\rceil -1\text{ and}\hfill \\ \hfill & j=1,2,3,\dots ,m+1\hfill \\ \lfloor \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rfloor \lceil \frac{i-2\lceil \frac{k}{m}\rceil +1}{2}\rceil +\lceil \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rceil (i-2\lceil \frac{k}{m}\rceil )+\lceil \frac{j}{2}\rceil ,\hfill & i=2\lceil \frac{k}{m}\rceil ,2\lceil \frac{k}{m}\rceil +1\text{ and}\hfill \\ \hfill & j=1,3,5\dots ,m;m\equiv 1(2)\text{ or}\hfill \\ \hfill & j=1,3,5\dots ,m+1;m\equiv 0(2)\hfill \\ \lceil \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rceil \lceil \frac{i-2\lceil \frac{k}{m}\rceil +1}{2}\rceil +\lfloor \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rfloor (i-2\lceil \frac{k}{m}\rceil )+\lceil \frac{j}{2}\rceil ,\hfill & i=2\lceil \frac{k}{m}\rceil ,2\lceil \frac{k}{m}\rceil +1\text{ and}\hfill \\ \hfill & j=2,4,6\dots ,m+1;m\equiv 1(2)\text{ or}\hfill \\ \hfill & j=2,4,6\dots ,m;m\equiv 0(2)\hfill \\ \lfloor \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rfloor +\lceil \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rceil +\lfloor \frac{m}{2}\rfloor \lceil \frac{i-2\lceil \frac{k}{m}\rceil -1}{2}\rceil \hfill & i=2\lceil \frac{k}{m}\rceil +2,\dots ,n-1\text{ and}\hfill \\ +\lceil \frac{m}{2}\rceil \lceil \frac{i-2\lceil \frac{k}{m}\rceil -2}{2}\rceil +\lceil \frac{j}{2}\rceil ,\hfill & j=1,3,5\dots ,m;m\equiv 1(2)\text{ or}\hfill \\ \hfill & j=1,3,5,\dots ,m+1;m\equiv 0(2)\hfill \\ m\lceil \frac{k}{m}\rceil +1-k+\lceil \frac{m}{2}\rceil \lceil \frac{i-2\lceil \frac{k}{m}\rceil -1}{2}\rceil +\lfloor \frac{m}{2}\rfloor \lceil \frac{i-2\lceil \frac{k}{m}\rceil -2}{2}\rceil +\lceil \frac{j}{2}\rceil ,\hfill & i=2\lceil \frac{k}{m}\rceil +2,\dots ,n-1\text{ and}\hfill \\ \hfill & j=2,4,6,\dots ,m+1;m\equiv 1(2)\text{ or}\hfill \\ \hfill & j=2,4,6,\dots ,m;m\equiv 0(2)\hfill \end{array}\end{array}$$

The weight of the face f under the labeling ρ of type (0, 1, 0) is $$\begin{array}{c}W{t}_{\rho (0,1,0)}(f_i^j)=\rho (v_i^j{v}_i^{j+1})+\rho (v_i^j{v}_{i+1}^j)\\ \hspace{1em}+\rho (v_i^{j+1}{v}_{i+1}^{j+1})+\rho (v_{i+1}^j{v}_{i+1}^{j+1})\end{array}$$

Now we calculate differences between the weights of the horizontal faces as follows:

Let, $g=f_i^{j+1}$ and $h=f_i^j$ be two horizontal consecutive faces.

Then for $i=1,2,\dots ,2\lceil \frac{k}{m}\rceil -1 \text{and} j=1,2,\dots ,m+1$ $$\begin{array}{c}W{t}_{(0,1,0)}(g)-W{t}_{(0,1,0)}(h)\\ \hspace{1em}=\left(\rho (v_i^{j+1}{v}_i^{j+2})+\rho (v_i^{j+1}{v}_{i+1}^{j+1})+\rho (v_i^{j+2}{v}_{i+1}^{j+2})+\rho (v_{i+1}^{j+1}{v}_{i+1}^{j+2})\right)\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-\left(\rho (v_i^j{v}_i^{j+1})+\rho (v_i^j{v}_{i+1}^j)+\rho (v_i^{j+1}{v}_{i+1}^{j+1})+\rho (v_{i+1}^j{v}_{i+1}^{j+1})\right)\\ \hspace{1em}=\left[1+m\left(\lceil \frac{i}{2}\rceil -1\right)\right]+\lceil \frac{j+2}{2}\rceil +\left[1+m\left(\lceil \frac{i+1}{2}\rceil -1\right)\right]\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-\left[1+m\left(\lceil \frac{i}{2}\rceil -1\right)\right]-\lceil \frac{j}{2}\rceil -\left[1+m\left(\lceil \frac{i+1}{2}\rceil -1\right)\right]\\ \hspace{1em}=\lceil \frac{j+2}{2}\rceil -\lceil \frac{j}{2}\rceil \\ \hspace{1em}=1\end{array}$$ $$\begin{array}{c}\text{For} i=2\lceil \frac{k}{m}\rceil ;\\ \hspace{1em}j=1,3,5\dots ,m;m\equiv 1(2)\\ \hspace{1em}\text{or} j=1,3,5\dots ,m+1;m\equiv 0(2),\\ \mathit{\text{and }}j=2,4,6\dots ,m+1;m\equiv 1(2)\\ \hspace{1em}\text{or} j=2,4,6\dots ,m;m\equiv 0(2)\\ W{t}_{(0,1,0)}(g)-W{t}_{(0,1,0)}(h)\\ \hspace{1em}=\left(\rho (v_i^{j+1}{v}_i^{j+2})+\rho (v_i^{j+1}{v}_{i+1}^{j+1})+\rho (v_i^{j+2}{v}_{i+1}^{j+2})+\rho (v_{i+1}^{j+1}{v}_{i+1}^{j+2})\right)\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-\left(\rho (v_i^j{v}_i^{j+1})+\rho (v_i^j{v}_{i+1}^j)+\rho (v_i^{j+1}{v}_{i+1}^{j+1})+\rho (v_{i+1}^j{v}_{i+1}^{j+1})\right)\\ \hspace{1em}=\left[1+m\left(\lceil \frac{i}{2}\rceil -1\right)\right]+\lfloor \frac{m⏧\frac{k}{m}⏧+1-k}{2}\rfloor \lceil \frac{i-2⏧\frac{k}{m}⏧+1}{2}\rceil \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\lceil \frac{m⏧\frac{k}{m}⏧+1-k}{2}\rceil \left(i-2\lceil \frac{k}{m}\rceil \right)+\lceil \frac{j+2}{2}\rceil +k\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-\left[1+m\left(\lceil \frac{i}{2}\rceil -1\right)\right]-\lfloor \frac{m⏧\frac{k}{m}⏧+1-k}{2}\rfloor \lceil \frac{i-2⏧\frac{k}{m}⏧+1}{2}\rceil \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-\lceil \frac{m⏧\frac{k}{m}⏧+1-k}{2}\rceil \left(i-2\lceil \frac{k}{m}\rceil \right)-\lceil \frac{j}{2}\rceil -k\\ \hspace{1em}=\lceil \frac{j+2}{2}\rceil -\lceil \frac{j}{2}\rceil \\ \hspace{1em}=1\end{array}$$ $$\begin{array}{c}\text{For} i=2\lceil \frac{k}{m}\rceil +1;\\ \hspace{1em}j=1,3,5\dots ,m;m\equiv 1(2)\\ \hspace{1em}\text{or} j=1,3,5\dots ,m+1;m\equiv 0(2),\\ \mathit{\text{and}} j=2,4,6\dots ,m+1;m\equiv 1(2)\\ \hspace{1em}\text{or} j=2,4,6\dots ,m;m\equiv 0(2)\\ W{t}_{(0,1,0)}(g)-W{t}_{(0,1,0)}(h)\\ \hspace{1em}=\left(\rho (v_i^{j+1}{v}_i^{j+2})+\rho (v_i^{j+1}{v}_{i+1}^{j+1})+\rho (v_i^{j+2}{v}_{i+1}^{j+2})+\rho (v_{i+1}^{j+1}{v}_{i+1}^{j+2})\right)\\ \hspace{1em}-\left(\rho (v_i^j{v}_i^{j+1})+\rho (v_i^j{v}_{i+1}^j)+\rho (v_i^{j+1}{v}_{i+1}^{j+1})+\rho (v_{i+1}^j{v}_{i+1}^{j+1})\right)\\ \hspace{1em}=k+\lfloor \frac{m⏧\frac{k}{m}⏧+1-k}{2}\rfloor \lceil \frac{i-2⏧\frac{k}{m}⏧+1}{2}\rceil \\ \hspace{1em}+\lceil \frac{m⏧\frac{k}{m}⏧+1-k}{2}\rceil \left(i-2\lceil \frac{k}{m}\rceil \right)\\ \hspace{1em}+\lceil \frac{j+2}{2}\rceil +k-k-\lfloor \frac{m⏧\frac{k}{m}⏧+1-k}{2}\rfloor \lceil \frac{i-2⏧\frac{k}{m}⏧+1}{2}\rceil \\ \hspace{1em}-\lceil \frac{m⏧\frac{k}{m}⏧+1-k}{2}\rceil \left(i-2\lceil \frac{k}{m}\rceil \right)-\lceil \frac{j}{2}\rceil -k\\ \hspace{1em}=\lceil \frac{j+2}{2}\rceil -\lceil \frac{j}{2}\rceil \\ \hspace{1em}=1\end{array}$$ $$\begin{array}{c}\text{For} i=2\lceil \frac{k}{m}\rceil +2,\dots ,n-1;\\ \hspace{1em}j=1,3,5\dots ,m;m\equiv 1(2)\\ \hspace{1em}\text{or} j=1,3,5\dots ,m+1;m\equiv 0(2),\\ \mathit{\text{and}} j=2,4,6\dots ,m+1;m\equiv 1(2)\\ \hspace{1em}\text{or} j=2,4,6\dots ,m;m\equiv 0(2)\\ W{t}_{(0,1,0)}(g)-W{t}_{(0,1,0)}(h)\\ \hspace{1em}=\left(\rho (v_i^{j+1}{v}_i^{j+2})+\rho (v_i^{j+1}{v}_{i+1}^{j+1})+\rho (v_i^{j+2}{v}_{i+1}^{j+2})+\rho (v_{i+1}^{j+1}{v}_{i+1}^{j+2})\right)\\ \hspace{1em}-\left(\rho (v_i^j{v}_i^{j+1})+\rho (v_i^j{v}_{i+1}^j)+\rho (v_i^{j+1}{v}_{i+1}^{j+1})+\rho (v_{i+1}^j{v}_{i+1}^{j+1})\right)\\ \hspace{1em}=\lceil \frac{j+2}{2}\rceil -\lceil \frac{j}{2}\rceil \\ \hspace{1em}=1\end{array}$$

Now we calculate the differences between the weights of the vertically adjacent faces as follows:

Let, $g=f_{i+1}^j$ and $h=f_i^j$ be two vertical consecutive faces.

Then for $i=1,2,\dots ,2\lceil \frac{k}{m}\rceil -2 \text{and} j=1,3,5,\dots ,m+1,m\equiv 1(2)\text{or} j=1,3,5\dots ,m,m\equiv 1(2)$ $$\begin{array}{c}W{t}_{\rho (0,1,0)}(g)-W{t}_{\rho (0,1,0)}(h)\\ \hspace{1em}=\left(\rho (v_{i+1}^j{v}_{i+1}^{j+1})+\rho (v_{i+1}^j{v}_{i+2}^j)+\rho (v_{i+1}^{j+1}{v}_{i+2}^{j+1})+\rho (v_{i+2}^j{v}_{i+2}^{j+1})\right)\\ \hspace{1em}-\left(\rho (v_i^j{v}_i^{j+1})+\rho (v_i^j{v}_{i+1}^j)+\rho (v_i^{j+1}{v}_{i+1}^{j+1})+\rho (v_{i+1}^j{v}_{i+1}^{j+1})\right)\\ \hspace{1em}=\lceil \frac{j}{2}\rceil +\lceil \frac{j+1}{2}\rceil +1+m\left(\lceil \frac{i+2}{2}\rceil -1\right)\\ \hspace{1em}-\lceil \frac{j}{2}\rceil -\lceil \frac{j+1}{2}\rceil -\left(1+m\left(\lceil \frac{i}{2}\rceil -1\right)\right)\\ \hspace{1em}=m\lceil \frac{i+2}{2}\rceil -m\lceil \frac{i}{2}\rceil \\ \hspace{1em}=m\end{array}$$

For, $i=2\lceil \frac{k}{m}\rceil -1 \text{and} j=1,3,5,\dots ,m+1,m\equiv 1(2)\text{or} j=1,3,5\dots ,m,m\equiv 1(2)$ $$\begin{array}{c}W{t}_{\rho (0,1,0)}(g)-W{t}_{\rho (0,1,0)}(h)\\ \hspace{1em}=\left(\rho (v_{i+1}^j{v}_{i+1}^{j+1})+\rho (v_{i+1}^j{v}_{i+2}^j)+\rho (v_{i+1}^{j+1}{v}_{i+2}^{j+1})+\rho (v_{i+2}^j{v}_{i+2}^{j+1})\right)\\ \hspace{1em}-\left(\rho (v_i^j{v}_i^{j+1})+\rho (v_i^j{v}_{i+1}^j)+\rho (v_i^{j+1}{v}_{i+1}^{j+1})+\rho (v_{i+1}^j{v}_{i+1}^{j+1})\right)\\ \hspace{1em}=\lfloor \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rfloor \lfloor \frac{2}{2}\rfloor +\lceil \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rceil \left(1\right)+\lceil \frac{j}{2}\rceil \\ \hspace{1em}+\lceil \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rceil \lceil \frac{2}{2}\rceil \\ \hspace{1em}+\lfloor \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rfloor \left(1\right)+\lceil \frac{j+1}{2}\rceil +k-1-m\left(\lceil \frac{i}{2}\rceil -1\right)\\ \hspace{1em}-\lfloor \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rfloor \lceil \frac{1}{2}\rceil \\ \hspace{1em}-\lceil \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rceil \left(0\right)-\lceil \frac{j}{2}\rceil -\lceil \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rceil \lceil \frac{1}{2}\rceil \\ \hspace{1em}-\lfloor \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rfloor \left(0\right)-\lceil \frac{j+1}{2}\rceil \\ \hspace{1em}=\lfloor \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rfloor +\lceil \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rceil +\lceil \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rceil \\ \hspace{1em}+\lfloor \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rfloor \\ \hspace{1em}+k-1-m\lceil \frac{i}{2}\rceil +m-\lfloor \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rfloor -\lceil \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rceil \\ \hspace{1em}=m\lceil \frac{k}{m}\rceil +1-k+k-1-m\lceil \frac{k}{m}\rceil +m\\ \hspace{1em}=m\end{array}$$

For, $i=2\lceil \frac{k}{m}\rceil \text{and} j=1,3,5,\dots ,m+1,m\equiv 1(2)\text{or} j=1,3,5\dots ,m,m\equiv 1(2)$ $$\begin{array}{c}W{t}_{\rho (0,1,0)}(g)-W{t}_{\rho (0,1,0)}(h)\\ \hspace{1em}=\left(\rho (v_{i+1}^j{v}_{i+1}^{j+1})+\rho (v_{i+1}^j{v}_{i+2}^j)+\rho (v_{i+1}^{j+1}{v}_{i+2}^{j+1})+\rho (v_{i+2}^j{v}_{i+2}^{j+1})\right)\\ \hspace{1em}-\left(\rho (v_i^j{v}_i^{j+1})+\rho (v_i^j{v}_{i+1}^j)+\rho (v_i^{j+1}{v}_{i+1}^{j+1})+\rho (v_{i+1}^j{v}_{i+1}^{j+1})\right)\\ \hspace{1em}=\lfloor \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rfloor \lceil \frac{i-2\lceil \frac{k}{m}\rceil +2}{2}\rceil \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\lceil \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rceil \left(1\right)+\lceil \frac{j}{2}\rceil \\ \hspace{1em}+\lceil \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rceil \lceil \frac{2}{2}\rceil +\lfloor \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rfloor \left(1\right)\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\lceil \frac{j+1}{2}\rceil +k-1-m\left(\lceil \frac{i}{2}\rceil -1\right)\\ \hspace{1em}-\lfloor \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rfloor \lceil \frac{1}{2}\rceil -\lceil \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rceil \left(0\right)\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-\lceil \frac{j}{2}\rceil -\lceil \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rceil \lceil \frac{1}{2}\rceil \end{array}$$ $$\begin{array}{c}\mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-\lfloor \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rfloor \left(0\right)-\lceil \frac{j+1}{2}\rceil \\ \hspace{1em}=\lfloor \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rfloor +\lceil \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rceil \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\lceil \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rceil +\lfloor \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rfloor \\ \hspace{1em}+k-1-m\lceil \frac{i}{2}\rceil +m-\lfloor \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rfloor \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-\lceil \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rceil \\ \hspace{1em}=m\lceil \frac{k}{m}\rceil +1-k+k-1-m\lceil \frac{k}{m}\rceil +m\\ \hspace{1em}=m\end{array}$$

For, $i=2\lceil \frac{k}{m}\rceil +1 \text{and} j=1,3,5,\dots ,m+1,m\equiv 1(2)\text{or} j=1,3,5\dots ,m,m\equiv 1(2)$ $$\begin{array}{c}W{t}_{\rho (0,1,0)}(g)-W{t}_{\rho (0,1,0)}(h)\\ \hspace{1em}=\left(\rho (v_{i+1}^j{v}_{i+1}^{j+1})+\rho (v_{i+1}^j{v}_{i+2}^j)+\rho (v_{i+1}^{j+1}{v}_{i+2}^{j+1})+\rho (v_{i+2}^j{v}_{i+2}^{j+1})\right)\\ \hspace{1em}-\left(\rho (v_i^j{v}_i^{j+1})+\rho (v_i^j{v}_{i+1}^j)+\rho (v_i^{j+1}{v}_{i+1}^{j+1})+\rho (v_{i+1}^j{v}_{i+1}^{j+1})\right)\\ \hspace{1em}=\lfloor \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rfloor +\lceil \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rceil +\lceil \frac{j}{2}\rceil \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\lfloor \frac{m}{2}\rfloor (1)+\lceil \frac{m}{2}\rceil (0)\\ \hspace{1em}+\lfloor \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rfloor (1)+\lceil \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rceil (1)\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\lceil \frac{j+1}{2}\rceil +\lceil \frac{m}{2}\rceil (1)+\lfloor \frac{m}{2}\rfloor (0)\\ \hspace{1em}-\lfloor \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rfloor (1)-\lceil \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rceil (1)\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-\lceil \frac{j}{2}\rceil -\lceil \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rceil (1)\\ \hspace{1em}-\lfloor \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rfloor (1)-\lceil \frac{j+1}{2}\rceil \\ \hspace{1em}=m\end{array}$$

For, $i=2\lceil \frac{k}{m}\rceil +2,\dots ,n-1 \text{and} j=1,3,5,\dots ,m+1,m\equiv 1(2)\text{or} j=1,3,5\dots ,m,m\equiv 1(2)$ $$\begin{array}{c}W{t}_{\rho (0,1,0)}(g)-W{t}_{\rho (0,1,0)}(h)\\ \hspace{1em}=\left(\rho (v_{i+1}^j{v}_{i+1}^{j+1})+\rho (v_{i+1}^j{v}_{i+2}^j)+\rho (v_{i+1}^{j+1}{v}_{i+2}^{j+1})+\rho (v_{i+2}^j{v}_{i+2}^{j+1})\right)\\ -\left(\rho (v_i^j{v}_i^{j+1})+\rho (v_i^j{v}_{i+1}^j)+\rho (v_i^{j+1}{v}_{i+1}^{j+1})+\rho (v_{i+1}^j{v}_{i+1}^{j+1})\right)\\ =2\lfloor \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rfloor +2\lceil \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rceil \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+m\lceil \frac{1-2\lceil \frac{k}{m}\rceil }{2}\rceil +m\lceil \frac{i-2\lceil \frac{k}{m}\rceil -1}{2}\rceil \\ +\lceil \frac{j}{2}\rceil +\lceil \frac{j+1}{2}\rceil -2\lfloor \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rfloor -2\lceil \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rceil \\ -m\lceil \frac{i-2\lceil \frac{k}{m}\rceil -1}{2}\rceil -m\lceil \frac{i-2\lceil \frac{k}{m}\rceil -2}{2}\rceil -\lceil \frac{j}{2}\rceil -\lceil \frac{j+1}{2}\rceil \\ =m\lceil \frac{i-2\lceil \frac{k}{m}\rceil }{2}\rceil -m\lceil \frac{i-2\lceil \frac{k}{m}\rceil }{2}\rceil +m\\ =m\end{array}$$

Hence it follows that the weights of any two faces f and g are distinct. Complete the proof by clearly giving the two required inequalities. □

Theorem 7.

Let $m\ge 2$ be a positive integers and $G=P_{n+1}\square P_{m+1}$ be grid graph, then $${\text{fs}}_{(1,0,1)}(G)=\lceil \frac{4+\mathit{\text{mn}}}{5}\rceil$$

Proof.

We define vertex-face labeling ρ of type (1, 0, 1) of G in the following way $$\begin{array}{c}\rho (v_i^j)=\{\begin{array}{cc}1+\lfloor \frac{m}{2}\rfloor \lfloor \frac{i+1}{4}\rfloor +⏧\frac{m}{2}⏧\lfloor \frac{i-1}{4}\rfloor ,\hfill & \text{if }i=1,2,\dots ,2⏧\frac{2k}{m}⏧\text{ and }j=1,3,\dots ,m+1,m\equiv 0\hspace{1em}\left(\mathrm{\text{mod}}2\right)\hfill \\ \hfill & \text{or }j=1,3,\dots ,m\text{ if }m\equiv 1\hspace{1em}\left(\mathrm{\text{mod}}2\right)\hfill \\ 1+\lceil \frac{m}{2}\rceil \lfloor \frac{i+1}{4}\rfloor +⏧\frac{m}{2}⏧\lfloor \frac{i-1}{4}\rfloor ,\hfill & \text{if }i=1,2,\dots ,2⏧\frac{2k}{m}⏧\text{ and }j=2,4,\dots ,m,m\equiv 0\hspace{1em}\left(\mathrm{\text{mod}}2\right)\hfill \\ \hfill & \text{or }j=2,4,\dots ,m+1,m\equiv 1\hspace{1em}\left(\mathrm{\text{mod}}2\right)\hfill \\ k,\hfill & \text{if }i=2⏧\frac{2k}{m}⏧+1,\dots ,n\text{ and }j=1,2,\dots ,m+1\hfill \end{array}\rho (f_i^j)\\ =\{\begin{array}{cc}j,\hfill & \text{if }i=1,2,\dots ,2⏧\frac{2k}{m}⏧-1\text{ and }j=1,2,3,\dots ,m\hfill \\ \left(i-2⏧\frac{2k}{m}⏧+1\right)\left(2+m⏧\frac{2k}{m}⏧-2k\right)+j,\hfill & \text{if }i=2⏧\frac{2k}{m}⏧,2⏧\frac{2k}{m}⏧+1\text{ and }j=1,2\dots ,m\hfill \\ m(i-1)-4(k-1)+j,\hfill & \text{if }i=2⏧\frac{2k}{m}⏧+2,\dots ,n\text{ and }j=1,2\dots ,m\hfill \end{array}\end{array}$$

The weight under labeling ρ of type (1, 0, 1) can be defined as $$W{t}_{\rho (1,0,1)}(f_i^j)=\rho (v_i^j)+\rho (v_{i+1}^j)+\rho (v_i^{j+1})+\rho (v_{i+1}^{j+1})+\rho (f_i^j)$$

The differences between the weights of every two horizontally adjacent faces will be calculated as follows:

Let, $g=f_i^{j+1}$ and $h=f_i^j$ be two horizontal consecutive faces.

Then for $i=1,2,\dots ,2\lfloor \frac{2k}{m}\rfloor -1$ and $j=1,2,3,\dots ,m$ $$\begin{array}{c}W{t}_{\rho (1,0,1)}(g)-W{t}_{\rho (1,0,1)}(h)\\ \hspace{1em}=\left(\rho (v_i^{j+1})+\rho (v_{i+1}^{j+1})+\rho (v_i^{j+2})+\rho (v_{i+1}^{j+2})+\rho (g)\right)\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-\left(\rho (v_i^j)+\rho (v_{i+1}^j)+\rho (v_i^{j+1})+\rho (v_{i+1}^{j+1})+\rho (h)\right)\\ \hspace{1em}=\left(j+1\right)-j\\ \hspace{1em}=1\end{array}$$

For $i=2\lfloor \frac{2k}{m}\rfloor ,2⏧\frac{2k}{m}⏧+1$ and $j=1,2,3,\dots ,m$ $$\begin{array}{c}W{t}_{\rho (1,0,1)}(g)-W{t}_{\rho (1,0,1)}(h)\\ \hspace{1em}=\left(\rho (v_i^{j+1})+\rho (v_{i+1}^{j+1})+\rho (v_i^{j+2})+\rho (v_{i+1}^{j+2})+\rho (g)\right)\\ \hspace{1em}-\left(\rho (v_i^j)+\rho (v_{i+1}^j)+\rho (v_i^{j+1})+\rho (v_{i+1}^{j+1})+\rho (h)\right)\\ \hspace{1em}=\left(j+1\right)-j\\ \hspace{1em}=1\end{array}$$

For $i=2\lfloor \frac{2k}{m}\rfloor +2,\dots ,n$ and $j=1,2,3,\dots ,m$ $$\begin{array}{c}W{t}_{\rho (1,0,1)}(g)-W{t}_{\rho (1,0,1)}(h)\\ \hspace{1em}=\left(\rho (v_i^{j+1})+\rho (v_{i+1}^{j+1})+\rho (v_i^{j+2})+\rho (v_{i+1}^{j+2})+\rho (g)\right)\\ \hspace{1em}-\left(\rho (v_i^j)+\rho (v_{i+1}^j)+\rho (v_i^{j+1})+\rho (v_{i+1}^{j+1})+\rho (h)\right)\\ \hspace{1em}=\left(j+1\right)-j\\ \hspace{1em}=1\end{array}$$

Now we measure the differences between the weights for every two vertically adjacent faces:

Let, $g=f_{i+1}^j$ and $h=f_i^j$ be two vertical consecutive faces.

Then for $i=1,2,\dots ,2\lfloor \frac{2k}{m}\rfloor -2$ and $j=1,2,3,\dots ,m$ $$\begin{array}{c}W{t}_{\rho (1,0,1)}(g)-W{t}_{\rho (1,0,1)}(h)\\ \hspace{1em}=\left(\rho (v_{i+1}^j)+\rho (v_{i+1}^{j+1})+\rho (v_{i+2}^j)+\rho (v_{i+2}^{j+1})+\rho (g)\right)\\ \hspace{1em}-\left(\rho (v_i^j)+\rho (v_i^{j+1})+\rho (v_{i+1}^j)+\rho (v_{i+1}^{j+1})+\rho (h)\right)\\ \hspace{1em}=\left(2+\left(\lfloor \frac{m}{2}\rfloor +\lceil \frac{m}{2}\rceil \right)\lfloor \frac{i+3}{4}\rfloor +\left(\lceil \frac{m}{2}\rceil +\lfloor \frac{m}{2}\rfloor \right)\lfloor \frac{i+1}{4}\rfloor \right)\\ \hspace{1em}-(2+\left(\lfloor \frac{m}{2}\rfloor +\lceil \frac{m}{2}\rceil \right)\lfloor \frac{i+1}{4}\rfloor +\left(\lceil \frac{m}{2}\rceil +\lfloor \frac{m}{2}\rfloor \right)\lfloor \frac{i-1}{4}\rfloor )\\ \hspace{1em}=m\lfloor \frac{i+3}{4}\rfloor +m\lfloor \frac{i+1}{4}\rfloor -m\lfloor \frac{i+1}{4}\rfloor -m\lfloor \frac{i-1}{4}\rfloor \\ \hspace{1em}=m\left(\lfloor \frac{i+3}{4}\rfloor -\lfloor \frac{i-1}{4}\rfloor \right)\\ \hspace{1em}=m\end{array}$$

For $i=2\lfloor \frac{2k}{m}\rfloor -1$ and $j=1,2,3,\dots ,m$ $$\begin{array}{c}W{t}_{\rho (1,0,1)}(g)-W{t}_{\rho (1,0,1)}(h)\\ \hspace{1em}=\left(\rho (v_{i+1}^j)+\rho (v_{i+1}^{j+1})+\rho (v_{i+2}^j)+\rho (v_{i+2}^{j+1})+\rho (g)\right)\\ \hspace{1em}-\left(\rho (v_i^j)+\rho (v_i^{j+1})+\rho (v_{i+1}^j)+\rho (v_{i+1}^{j+1})+\rho (h)\right)\\ \hspace{1em}=2k-\left(2+m\lfloor \frac{i+1}{4}\rfloor +m\lfloor \frac{i-1}{4}\rfloor \right)\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\left(2+m\lfloor \frac{2k}{m}\rfloor -2k+j-j\right)\\ \hspace{1em}=2k-\left(2+m\lfloor \frac{⏧\frac{2k}{m}⏧}{2}\rfloor +m\lfloor \frac{⏧\frac{2k}{m}⏧-1}{2}\rfloor \right)\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+2+m\lfloor \frac{2k}{m}\rfloor -2k\\ \hspace{1em}=m\lfloor \frac{2k}{m}\rfloor -m\lfloor \frac{⏧\frac{2k}{m}⏧}{2}\rfloor -m\lfloor \frac{⏧\frac{2k}{m}⏧-1}{2}\rfloor \\ \hspace{1em}=m\left(\lfloor \frac{2k}{m}\rfloor -\left(\lfloor \frac{2k}{m}\rfloor -1\right)\right)\\ \hspace{1em}=m\end{array}$$

For $i=2\lfloor \frac{2k}{m}\rfloor$ and $j=1,2,3,\dots ,m$ $$\begin{array}{c}W{t}_{\rho (1,0,1)}(g)-W{t}_{\rho (1,0,1)}(h)\\ \hspace{1em}=\left(\rho (v_{i+1}^j)+\rho (v_{i+1}^{j+1})+\rho (v_{i+2}^j)+\rho (v_{i+2}^{j+1})+\rho (g)\right)\\ \hspace{1em}-\left(\rho (v_i^j)+\rho (v_i^{j+1})+\rho (v_{i+1}^j)+\rho (v_{i+1}^{j+1})+\rho (h)\right)\\ \hspace{1em}=2k-\left(1+\lceil \frac{m}{2}\rceil \lfloor \frac{2\lfloor \frac{2k}{m}\rfloor +1}{4}\rfloor +\lfloor \frac{m}{2}\rfloor \lfloor \frac{2\lfloor \frac{2k}{m}\rfloor -1}{4}\rfloor \right)\\ \hspace{1em}-\left(1+\lceil \frac{m}{2}\rceil \lfloor \frac{2\lfloor \frac{2k}{m}\rfloor +1}{4}\rfloor +\lfloor \frac{m}{2}\rfloor \lfloor \frac{2\lfloor \frac{2k}{m}\rfloor -1}{4}\rfloor \right)\\ \hspace{1em}+2\left(2+m\lfloor \frac{2k}{m}\rfloor -2k\right)-\left(2+m\lfloor \frac{2k}{m}\rfloor -2k\right)\\ \hspace{1em}=m\lfloor \frac{2k}{m}\rfloor -\left(\lfloor \frac{m}{2}\rfloor +\lceil \frac{m}{2}\rceil \right)\lfloor \frac{2\lfloor \frac{2k}{m}\rfloor +1}{4}\rfloor \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-\left(\lceil \frac{m}{2}\rceil +\lfloor \frac{m}{2}\rfloor \right)\lfloor \frac{2\lfloor \frac{2k}{m}\rfloor -1}{4}\rfloor \\ \hspace{1em}=m\lfloor \frac{2k}{m}\rfloor -m\left(\lfloor \frac{2\lfloor \frac{2k}{m}\rfloor +1}{4}\rfloor +\lfloor \frac{2\lfloor \frac{2k}{m}\rfloor -1}{4}\rfloor \right)\\ \hspace{1em}=m\lfloor \frac{2k}{m}\rfloor -m\left(\lfloor \frac{2k}{m}\rfloor -1\right)\\ \hspace{1em}=m\end{array}$$

For $i=2\lfloor \frac{2k}{m}\rfloor +1$ and $j=1,2,3,\dots ,m$ $$\begin{array}{c}W{t}_{\rho (1,0,1)}(g)-W{t}_{\rho (1,0,1)}(h)\\ \hspace{1em}=(\rho (v_{i+1}^j)+\rho (v_{i+1}^{j+1})\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\rho (v_{i+2}^j)+\rho (v_{i+2}^{j+1})+\rho (g))\\ \hspace{1em}-(\rho (v_i^j)+\rho (v_i^{j+1})+\rho (v_{i+1}^j)\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\rho (v_{i+1}^{j+1})+\rho (h))\\ \hspace{1em}=m+2\left(2+m\lfloor \frac{2k}{m}\rfloor -2k\right)\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+j-2\left(2+m\lfloor \frac{2k}{m}\rfloor -2k\right)-j\\ \hspace{1em}=m\end{array}$$

For $i=2\lfloor \frac{2k}{m}\rfloor +2,\dots ,,n$ and $j=1,2,\dots ,m$ $$\begin{array}{c}W{t}_{\rho (1,0,1)}(g)-W{t}_{\rho (1,0,1)}(h)\\ \hspace{1em}=(\rho (v_{i+1}^j)+\rho (v_{i+1}^{j+1})+\rho (v_{i+2}^j)\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\rho (v_{i+2}^{j+1})+\rho (g))\\ \hspace{1em}-(\rho (v_i^j)+\rho (v_i^{j+1})+\rho (v_{i+1}^j)\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\rho (v_{i+1}^{j+1})+\rho (h))\\ \hspace{1em}=\mathit{\text{mi}}-4\left(k-1\right)+j-\left(m\left(i-1\right)-4\left(k-1\right)+j\right)\\ \hspace{1em}=m\end{array}$$

Hence it follows that the weights of any two faces f and g are distinct. Complete the proof by clearly giving the two required inequalities. □

Theorem 8.

Let $m\ge 2$ be a positive integers and $G=P_{n+1}\square P_{m+1}$ be grid graph, then $${\text{fs}}_{(0,1,1)}(G)=\lceil \frac{4+\mathit{\text{mn}}}{5}\rceil$$

Proof.

We define a k-labeling ρ of type (0, 1, 1) of G in the following way

$\begin{array}{c}\rho (v_i^j{v}_i^{j+1})\hspace{1em}=\{\begin{array}{cc}1+m\lfloor \frac{i-1}{2}\rfloor ,\hfill & \text{if }i=1,2,\dots ,2\lceil \frac{k}{m}\rceil \hfill \\ \hfill & \hspace{1em}\text{and }j=1,2,\dots ,m\hfill \\ k,\hfill & \text{if }i=2⏧\frac{k}{m}⏧+1,\dots ,n+1\hfill \\ \hfill & \hspace{1em}\text{and }j=1,2,\dots ,m\hfill \end{array}\\ \rho (v_i^j{v}_{i+1}^j)=\{\begin{array}{cc}1,\hfill & \text{if }i=1,2,\dots ,2\lceil \frac{k}{m}\rceil -1 \text{and}\hfill \\ \hfill & j=1,2,\dots ,m+1\hfill \\ 1+\lfloor \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rfloor \lceil \frac{1}{2}\left(i-2\lceil \frac{k}{m}\rceil +1\right)\rceil \hfill & \text{if }i=2\lceil \frac{k}{m}\rceil ,2\lceil \frac{k}{m}\rceil +1 \text{and}\hfill \\ +\lceil \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rceil \lceil \frac{1}{2}\left(i-2\lceil \frac{k}{m}\rceil \right)\rceil ,\hfill & j=1,3,\dots ,m;m\equiv 1(2)\text{or} j=1,3,\dots ,m+1;m\equiv 0(2)\hfill \\ 1+\lceil \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rceil \lceil \frac{1}{2}\left(i-2\lceil \frac{k}{m}\rceil +1\right)\rceil \hfill & \text{if }i=2\lceil \frac{k}{m}\rceil ,2\lceil \frac{k}{m}\rceil +1 \text{and}\hfill \\ +\lfloor \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rfloor \lceil \frac{1}{2}\left(i-2\lceil \frac{k}{m}\rceil \right)\rceil ,\hfill & j=2,4,\dots ,m+1;m\equiv 1(2)\text{or} j=2,4,\dots ,m;m\equiv 0(2)\hfill \\ 2+m\lceil \frac{k}{m}\rceil -k+\lfloor \frac{m}{2}\rfloor \lceil \frac{1}{2}\left(i-2\lceil \frac{k}{m}\rceil -1\right)\rceil \hfill & \text{if }i=2\lceil \frac{k}{m}\rceil +2,\dots ,2\lceil \frac{k}{m}\rceil +\lceil \frac{k}{\lceil \frac{m}{2}\rceil }\rceil -1 \text{and}\hfill \\ +\lceil \frac{m}{2}\rceil \lceil \frac{1}{2}\left(i-2\lceil \frac{k}{m}\rceil -2\right)\rceil ,\hfill & j=1,3,\dots ,m;m\equiv 1(2)\text{or} j=1,3,\dots ,m+1;m\equiv 0(2)\hfill \\ 2+m\lceil \frac{k}{m}\rceil -k+\lceil \frac{m}{2}\rceil \lceil \frac{1}{2}\left(i-2\lceil \frac{k}{m}\rceil -1\right)\rceil \hfill & \text{if }i=2\lceil \frac{k}{m}\rceil +2,\dots ,2\lceil \frac{k}{m}\rceil +\lceil \frac{k}{\lceil \frac{m}{2}\rceil }\rceil -1 \text{and}\hfill \\ +\lfloor \frac{m}{2}\rfloor \lceil \frac{1}{2}\left(i-2\lceil \frac{k}{m}\rceil -2\right)\rceil ,\hfill & j=2,4,\dots ,m+1;m\equiv 1(2)\text{or} j=2,4,\dots ,m;m\equiv 0(2)\hfill \\ k,\hfill & \text{if }i=2\lceil \frac{k}{m}\rceil +\lceil \frac{k}{\lceil \frac{m}{2}\rceil }\rceil ,\dots ,n \text{and}\hfill \\ \hfill & j=1,2,\dots ,m+1\hfill \end{array}\end{array}$ The face labels are defined as $$\rho (f_i^j)=\{\begin{array}{cc}j,\hfill & \text{if }i=1,2,\dots ,2\lceil \frac{k}{m}\rceil +\lceil \frac{k}{\lceil \frac{m}{2}\rceil }\rceil -1 \text{and} j=1,2,\dots ,m\hfill \\ j+m\left(2\lceil \frac{k}{m}\rceil +\lceil \frac{k}{\lceil \frac{m}{2}\rceil }\rceil -1\right)-4k+4,\hfill & \text{if }i=2\lceil \frac{k}{m}\rceil +\lceil \frac{k}{\lceil \frac{m}{2}\rceil }\rceil \text{and} j=1,2,\dots ,m\hfill \\ j-4k+4+m(i-1),\hfill & \text{if }i=2\lceil \frac{k}{m}\rceil +\lceil \frac{k}{\lceil \frac{m}{2}\rceil }\rceil +1,\dots ,n \text{and} j=1,2,\dots ,m\hfill \end{array}$$

The weights can be defined as $$\begin{array}{c}W{t}_{\rho (0,1,1)}(f_i^j)=\rho (v_i^j{v}_i^{j+1})+\rho (v_i^{j+1}{v}_{i+1}^{j+1})\\ \hspace{1em}+\rho (v_{i+1}^j{v}_{i+1}^{j+1})+\rho (v_i^j{v}_{i+1}^j)+\rho (f_i^j)\end{array}$$

The differences between the horizontal weights for every two adjacent faces are as follows: Let, $g=f_i^{j+1}$ and $h=f_i^j$ be two horizontal consecutive faces.

Then for $i=1,2,\dots n$ and $j=1,2,\dots ,m$ $$\begin{array}{c}W{t}_{\rho (0,1,1)}(g)-W{t}_{\rho (0,1,1)}(h)\\ \hspace{1em}=(\rho (v_i^{j+1}{v}_i^{j+2})+\rho (v_i^{j+2}{v}_{i+1}^{j+2})\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\rho (v_{i+1}^{j+1}{v}_{i+1}^{j+2})+\rho (v_i^{j+1}{v}_{i+1}^{j+1})+\rho (g))\\ \hspace{1em}-(\rho (v_i^j{v}_i^{j+1})+\rho (v_i^{j+1}{v}_{i+1}^{j+1})+\rho (v_{i+1}^j{v}_{i+1}^{j+1})\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\rho (v_i^j{v}_{i+1}^j)+\rho (h))\\ \hspace{1em}=\left(j+1\right)-j\\ \hspace{1em}=1\end{array}$$

Now the differences between the weights of every two adjacent vertical faces of type (0, 1, 1) are as follows:

Let, $g=f_{i+1}^j$ and $h=f_i^j$ be two vertical consecutive faces. Then, $$\begin{array}{c}W{t}_{\rho (0,1,1)}(g)-W{t}_{\rho (0,1,1)}(h)\\ \hspace{1em}=(\rho (v_{i+1}^j{v}_{i+1}^{j+1})+\rho (v_{i+1}^{j+1}{v}_{i+2}^{j+1})\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\rho (v_{i+2}^j{v}_{i+2}^{j+1})+\rho (v_{i+1}^j{v}_{i+2}^j)+\rho (g))\\ \hspace{1em}-(\rho (v_i^j{v}_i^{j+1})+\rho (v_i^{j+1}{v}_{i+1}^{j+1})+\rho (v_{i+1}^j{v}_{i+1}^{j+1})\\ \hspace{1em}+\rho (v_i^j{v}_{i+1}^j)+\rho (h))\end{array}$$

For $i=1,2,\dots ,2\lceil \frac{k}{m}\rceil -2$ and $j=1,2,\dots ,m$ $$\begin{array}{c}W{t}_{\rho (0,1,1)}(g)-W{t}_{\rho (0,1,1)}(h)\\ \hspace{1em}=(\rho (v_{i+1}^j{v}_{i+1}^{j+1})+\rho (v_{i+1}^{j+1}{v}_{i+2}^{j+1})+\rho (v_{i+2}^j{v}_{i+2}^{j+1})\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\rho (v_{i+1}^j{v}_{i+2}^j)+\rho (g))\\ \hspace{1em}-(\rho (v_i^j{v}_i^{j+1})+\rho (v_i^{j+1}{v}_{i+1}^{j+1})+\rho (v_{i+1}^j{v}_{i+1}^{j+1})\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\rho (v_i^j{v}_{i+1}^j)+\rho (h))\\ \hspace{1em}=\left(1+m\lfloor \frac{i+1}{2}\rfloor \right)-\left(1+m\lfloor \frac{i-1}{2}\rfloor \right)\\ \hspace{1em}=m\left(\lfloor \frac{i+1}{2}\rfloor -\lfloor \frac{i-1}{2}\rfloor \right)\\ \hspace{1em}=m\left(1\right)\\ \hspace{1em}=m\end{array}$$

For $i=2\lceil \frac{k}{m}\rceil -1$ and $j=1,2,\dots ,m$ $$\begin{array}{c}W{t}_{\rho (0,1,1)}(g)-W{t}_{\rho (0,1,1)}(h)\\ \hspace{1em}=(\rho (v_{i+1}^j{v}_{i+1}^{j+1})+\rho (v_{i+1}^{j+1}{v}_{i+2}^{j+1})+\rho (v_{i+2}^j{v}_{i+2}^{j+1})\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\rho (v_{i+1}^j{v}_{i+2}^j)+\rho (g))\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-(\rho (v_i^j{v}_i^{j+1})+\rho (v_i^{j+1}{v}_{i+1}^{j+1})+\rho (v_{i+1}^j{v}_{i+1}^{j+1})\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\rho (v_i^j{v}_{i+1}^j)+\rho (h))\\ \hspace{1em}=k+\left(1+\lfloor \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rfloor \right)\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\left(1+\lceil \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rceil \right)\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-\left(1+m\lfloor \frac{2\lceil \frac{k}{m}\rceil -2}{2}\rfloor \right)-\left(1\right)-\left(1\right)\\ \hspace{1em}=k+2+\left(m\lceil \frac{k}{m}\rceil +1-k\right)-3-3\lceil \frac{k}{m}\rceil +m\\ \hspace{1em}=m\end{array}$$

For $i=2\lceil \frac{k}{m}\rceil$ and $j=1,2,\dots ,m$ $$\begin{array}{c}W{t}_{\rho (0,1,1)}(g)-W{t}_{\rho (0,1,1)}(h)\\ \hspace{1em}=(\rho (v_{i+1}^j{v}_{i+1}^{j+1})+\rho (v_{i+1}^{j+1}{v}_{i+2}^{j+1})+\rho (v_{i+2}^j{v}_{i+2}^{j+1})\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\rho (v_{i+1}^j{v}_{i+2}^j)+\rho (g))\\ \hspace{1em}-(\rho (v_i^j{v}_i^{j+1})+\rho (v_i^{j+1}{v}_{i+1}^{j+1})+\rho (v_{i+1}^j{v}_{i+1}^{j+1})\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\rho (v_i^j{v}_{i+1}^j)+\rho (h))\\ \hspace{1em}=k+\left(1+\lfloor \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rfloor +\lceil \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rceil \right)\\ \hspace{1em}+\left(1+\lceil \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rceil +\lfloor \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rfloor \right)\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+j-\left(1+m\lfloor \frac{2\lceil \frac{k}{m}\rceil -1}{2}\rfloor \right)\\ \hspace{1em}-\left(1+\lfloor \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rfloor \right)-\left(1+\lceil \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rceil \right)\\ \hspace{1em}=k+\left(1+m\lceil \frac{k}{m}\rceil +1-k\right)\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\left(1+m\lceil \frac{k}{m}\rceil +1-k\right)-\left(1+m\left(\lceil \frac{k}{m}\rceil -1\right)\right)\\ \hspace{1em}-2-\left(m\lceil \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rceil \right)\\ \hspace{1em}=k+2\left(1+m\lceil \frac{k}{m}\rceil +1-k\right)-4-2m\lceil \frac{k}{m}\rceil +m+k\\ \hspace{1em}=m\end{array}$$

For $i=2\lceil \frac{k}{m}\rceil +1$ and $j=1,2,\dots ,m$ $$\begin{array}{c}W{t}_{\rho (0,1,1)}(g)-W{t}_{\rho (0,1,1)}(h)\\ \hspace{1em}=(\rho (v_{i+1}^j{v}_{i+1}^{j+1})+\rho (v_{i+1}^{j+1}{v}_{i+2}^{j+1})\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\rho (v_{i+2}^j{v}_{i+2}^{j+1})+\rho (v_{i+1}^j{v}_{i+2}^j)+\rho (g))\\ \hspace{1em}-(\rho (v_i^j{v}_i^{j+1})+\rho (v_i^{j+1}{v}_{i+1}^{j+1})\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\rho (v_{i+1}^j{v}_{i+1}^{j+1})+\rho (v_i^j{v}_{i+1}^j)+\rho (h))\\ \hspace{1em}=k+\left(2+m\lceil \frac{k}{m}\rceil -k+\lfloor \frac{m}{2}\rfloor \right)\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\left(2+m\lceil \frac{k}{m}\rceil -k+\lceil \frac{m}{2}\rceil \right)\\ \hspace{1em}-k-\left(1+\lfloor \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rfloor +\lceil \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rceil \right)\\ \hspace{1em}-\left(1+\lceil \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rceil +\lfloor \frac{m\lceil \frac{k}{m}\rceil +1-k}{2}\rfloor \right)\\ \hspace{1em}=k+2\left(2+m\lceil \frac{k}{m}\rceil -k\right)\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+m-k-2\left(1+m\lceil \frac{k}{m}\rceil +1-k\right)\\ \hspace{1em}=m\end{array}$$

For $i=2\lceil \frac{k}{m}\rceil +2,\dots ,2\lceil \frac{k}{m}\rceil +\lceil \frac{k}{\lceil \frac{m}{2}\rceil }\rceil -2$ and $j=1,2,\dots ,m$ $$\begin{array}{c}W{t}_{\rho (0,1,1)}(g)-W{t}_{\rho (0,1,1)}(h)\\ \hspace{1em}=(\rho (v_{i+1}^j{v}_{i+1}^{j+1})+\rho (v_{i+1}^{j+1}{v}_{i+2}^{j+1})\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\rho (v_{i+2}^j{v}_{i+2}^{j+1})+\rho (v_{i+1}^j{v}_{i+2}^j)+\rho (g))\\ \hspace{1em}-(\rho (v_i^j{v}_i^{j+1})+\rho (v_i^{j+1}{v}_{i+1}^{j+1})\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\rho (v_{i+1}^j{v}_{i+1}^{j+1})+\rho (v_i^j{v}_{i+1}^j)+\rho (h))\\ \hspace{1em}=2+m\lceil \frac{k}{m}\rceil -k+\lfloor \frac{m}{2}\rfloor \lceil \frac{1}{2}\left(i-2\lceil \frac{k}{m}\rceil \right)\rceil \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\lceil \frac{m}{2}\rceil \lceil \frac{1}{2}\left(i-2\lceil \frac{k}{m}\rceil -1\right)\rceil \\ \hspace{1em}+2+m\lceil \frac{k}{m}\rceil -k+\lceil \frac{m}{2}\rceil \lceil \frac{1}{2}\left(i-2\lceil \frac{k}{m}\rceil \right)\rceil \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\lfloor \frac{m}{2}\rfloor \lceil \frac{1}{2}\left(i-2\lceil \frac{k}{m}\rceil -1\right)\rceil \\ \hspace{1em}-\{2+m\lceil \frac{k}{m}\rceil -k+\lfloor \frac{m}{2}\rfloor \lceil \frac{1}{2}\left(i-2\lceil \frac{k}{m}\rceil -1\right)\rceil \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\lceil \frac{m}{2}\rceil \lceil \frac{1}{2}\left(i-2\lceil \frac{k}{m}\rceil -2\right)\rceil \}\\ \hspace{1em}-\{2+m\lceil \frac{k}{m}\rceil -k+\lceil \frac{m}{2}\rceil \lceil \frac{1}{2}\left(i-2\lceil \frac{k}{m}\rceil -1\right)\rceil \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\lfloor \frac{m}{2}\rfloor \lceil \frac{1}{2}\left(i-2\lceil \frac{k}{m}\rceil -2\right)\rceil \}\\ \hspace{1em}=m\lceil \frac{1}{2}\left(i-2\lceil \frac{k}{m}\rceil \right)\rceil +m\lceil \frac{1}{2}\left(i-2\lceil \frac{k}{m}\rceil -1\right)\rceil \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-m\lceil \frac{1}{2}\left(i-2\lceil \frac{k}{m}\rceil -1\right)\rceil \\ \hspace{1em}-m\lceil \frac{1}{2}\left(i-2\lceil \frac{k}{m}\rceil -2\right)\rceil \\ \hspace{1em}=m\lceil \frac{1}{2}\left(i-2\lceil \frac{k}{m}\rceil \right)\rceil \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+m\lceil \frac{1}{2}\left(i-2\lceil \frac{k}{m}\rceil -1\right)\rceil \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-m\lceil \frac{1}{2}\left(i-2\lceil \frac{k}{m}\rceil -1\right)\rceil \\ \hspace{1em}-m\lceil \frac{1}{2}\left(i-2\lceil \frac{k}{m}\rceil \right)\rceil +m\\ \hspace{1em}=m\end{array}$$

For $i=2\lceil \frac{k}{m}\rceil +\lceil \frac{k}{\lceil \frac{m}{2}\rceil }\rceil -1$ and $j=1,2,\dots ,m$ $$\begin{array}{c}W{t}_{\rho (0,1,1)}(g)-W{t}_{\rho (0,1,1)}(h)\\ \hspace{1em}=(\rho (v_{i+1}^j{v}_{i+1}^{j+1})+\rho (v_{i+1}^{j+1}{v}_{i+2}^{j+1})\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\rho (v_{i+2}^j{v}_{i+2}^{j+1})+\rho (v_{i+1}^j{v}_{i+2}^j)+(g))\\ \hspace{1em}-(\rho (v_i^j{v}_i^{j+1})+\rho (v_i^{j+1}{v}_{i+1}^{j+1})\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\rho (v_{i+1}^j{v}_{i+1}^{j+1})+\rho (v_i^j{v}_{i+1}^j)+\rho (h))\\ \hspace{1em}=\left(3k\right)+\left(j+m\left(2\lceil \frac{k}{m}\rceil +\lceil \frac{k}{\lceil \frac{m}{2}\rceil }\rceil -1\right)-4k+4\right)-k\\ \hspace{1em}-(4+2m\lceil \frac{k}{m}\rceil -2k+m\lceil \frac{1}{2}\left(i-2\lceil \frac{k}{m}\rceil -1\right)\rceil \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+m\lceil \frac{1}{2}\left(i-2\lceil \frac{k}{m}\rceil -2\right)\rceil )\\ \hspace{1em}=-k+j+m\left(2\lceil \frac{k}{m}\rceil +\lceil \frac{k}{\lceil \frac{m}{2}\rceil }\rceil -1\right)\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+4-k-4-2m\lceil \frac{k}{m}\rceil +2k\\ \hspace{1em}-m\lceil \frac{1}{2}\left(\lceil \frac{k}{\lceil \frac{m}{2}\rceil }\rceil -2\right)\rceil -m\lceil \frac{1}{2}\left(\lceil \frac{k}{\lceil \frac{m}{2}\rceil }\rceil -3\right)\rceil -j\\ \hspace{1em}=m\lceil \frac{k}{\lceil \frac{m}{2}\rceil }\rceil -m-m\lceil \frac{1}{2}\lceil \frac{k}{\lceil \frac{m}{2}\rceil }\rceil \rceil \\ \hspace{1em}+m-m\lceil \frac{1}{2}\left(\lceil \frac{k}{\lceil \frac{m}{2}\rceil }\rceil -1\right)\rceil +m\\ \hspace{1em}=m\lceil \frac{k}{\lceil \frac{m}{2}\rceil }\rceil -m\lceil \frac{1}{2}\lceil \frac{k}{\lceil \frac{m}{2}\rceil }\rceil \rceil \\ \hspace{1em}-m\lceil \frac{1}{2}\left(\lceil \frac{k}{\lceil \frac{m}{2}\rceil }\rceil -1\right)\rceil +m\\ \hspace{1em}=m\lceil \frac{k}{\lceil \frac{m}{2}\rceil }\rceil -m\lceil \frac{k}{\lceil \frac{m}{2}\rceil }\rceil +m\\ \hspace{1em}=m\end{array}$$

For $i=2\lceil \frac{k}{m}\rceil +\lceil \frac{k}{\lceil \frac{m}{2}\rceil }\rceil$ and $j=1,2,\dots ,m$ $$\begin{array}{c}W{t}_{\rho (0,1,1)}(g)-W{t}_{\rho (0,1,1)}(h)\\ \hspace{1em}=(\rho (v_{i+1}^j{v}_{i+1}^{j+1})+\rho (v_{i+1}^{j+1}{v}_{i+2}^{j+1})\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\rho (v_{i+2}^j{v}_{i+2}^{j+1})+\rho (v_{i+1}^j{v}_{i+2}^j)+(g))\\ \hspace{1em}-(\rho (v_i^j{v}_i^{j+1})+\rho (v_i^{j+1}{v}_{i+1}^{j+1})+\rho (v_{i+1}^j{v}_{i+1}^{j+1})\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\rho (v_i^j{v}_{i+1}^j)+\rho (h))\\ \hspace{1em}=j-4k+4+m\left(2\lceil \frac{k}{m}\rceil +\lceil \frac{k}{\lceil \frac{m}{2}\rceil }\rceil \right)\\ \hspace{1em}-\left(j+m\left(2\lceil \frac{k}{m}\rceil +\lceil \frac{k}{\lceil \frac{m}{2}\rceil }\rceil \right)-4k+4\right)\\ \hspace{1em}=m\end{array}$$

For $i=2\lceil \frac{k}{m}\rceil +\lceil \frac{k}{\lceil \frac{m}{2}\rceil }\rceil +1,\dots ,n-1$ and $j=1,2,\dots ,m$ $$\begin{array}{c}W{t}_{\rho (0,1,1)}(g)-W{t}_{\rho (0,1,1)}(h)\\ \hspace{1em}=(\rho (v_{i+1}^j{v}_{i+1}^{j+1})+\rho (v_{i+1}^{j+1}{v}_{i+2}^{j+1})\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\rho (v_{i+2}^j{v}_{i+2}^{j+1})+\rho (v_{i+1}^j{v}_{i+2}^j)+(g))\\ \hspace{1em}-(\rho (v_i^j{v}_i^{j+1})+\rho (v_i^{j+1}{v}_{i+1}^{j+1})\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\rho (v_{i+1}^j{v}_{i+1}^{j+1})+\rho (v_i^j{v}_{i+1}^j)+\rho (h))\\ \hspace{1em}=\left(j-4k+4+\mathit{\text{mi}}\right)-\left(j-4k+4+m\left(i-1\right)\right)\\ \hspace{1em}=\left(\mathit{\text{mi}}-m\left(i-1\right)\right)\\ \hspace{1em}=m\end{array}$$

Hence it follows that the weights of any two faces f and g are distinct. Complete the proof by clearly giving the two required inequalities. □

Theorem 9.

Let $m\ge 2$ be a positive integers and $G=P_{n+1}\square P_{m+1}$ be grid graph, then $${\text{fs}}_{(1,1,1)}(G)=\lceil \frac{7+\mathit{\text{mn}}}{9}\rceil$$

Proof.

Consider, an integer $\lfloor \frac{m+1}{3}\rfloor$ such that $\lfloor \frac{m+1}{3}\rfloor =m-2\lfloor \frac{m+1}{3}\rfloor$. Then the entire face irregular labeling is defined as follow $$\varphi (v_i^j)=\{\begin{array}{cc}1+\lfloor \frac{i-1}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor ,\hfill & i=1,2,3,\dots ,2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil ,\hfill \\ \hfill & \hspace{1em}j=1,2,\dots ,m+1\hfill \\ k,\hfill & i=2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +1,\dots ,n+1\hfill \\ \hfill & \hspace{1em}j=1,2,\dots ,m+1\hfill \end{array}$$

For $j=1,2,\dots ,m$ the face labels are as follows $$\varphi (f_i^j)=\{\begin{array}{c}j,i=1,2,\dots ,2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +1+\lfloor \frac{4k-4-m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil }{\lceil \frac{m}{2}\rceil }\rfloor \hfill \\ j+8+m(2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +\lfloor \frac{4k-m-m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil }{\lceil \frac{m}{2}\rceil }\rfloor +1)-8k,\hfill \\ i=2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +\lfloor \frac{4k-m-m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil }{\lceil \frac{m}{2}\rceil }\rfloor +2\hfill \\ j+8+m(2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +\lfloor \frac{4k-m-m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil }{\lceil \frac{m}{2}\rceil }\rfloor +1)-8k+m(i-2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil -\lfloor \frac{4k-m-m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil }{\lceil \frac{m}{2}\rceil }\rfloor -2),\hfill \\ i=2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +\lfloor \frac{4k-m-m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil }{\lceil \frac{m}{2}\rceil }\rfloor +3,\dots ,n\hfill \end{array}$$ $$\varphi (v_i^j{v}_{i+1}^j)=\{\begin{array}{c}\begin{array}{c}1,i=1,2,\dots ,2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil -1,j=1,2,\dots ,m+1\hfill \\ 1+\lfloor \frac{1}{2}(m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +3\lfloor \frac{m+1}{3}\rfloor -m-3k+3)\rfloor \lceil \frac{1}{2}(i-2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +1)\rceil \hfill \\ +\lceil \frac{1}{2}(m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +3\lfloor \frac{m+1}{3}\rfloor -m-3k+3)\rceil \lceil \frac{1}{2}(i-2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil )\rceil ,\hfill \\ i=2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil ,2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +1,\hfill \\ j=1,3,\dots ,m,m\equiv 1(2)\text{ or }j=1,3,\dots ,m+1,m\equiv 0(2)\hfill \\ 1+\lceil \frac{1}{2}(m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +3\lfloor \frac{m+1}{3}\rfloor -m-3k+3)\rceil \lceil \frac{1}{2}(i-2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +1)\rceil \hfill \\ +\lfloor \frac{1}{2}(m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +3\lfloor \frac{m+1}{3}\rfloor -m-3k+3)\rfloor \lceil \frac{1}{2}(i-2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil )\rceil ,\hfill \end{array}\\ \begin{array}{c}i=2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil ,2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +1,\hfill \\ j=2,4,\dots ,m+1,m\equiv 1(2)\text{ or }j=2,4,\dots ,m,m\equiv 0(2)\hfill \\ 1+\lfloor \frac{m}{2}\rfloor \lceil \frac{1}{2}(i-2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil -1)\rceil +\lceil \frac{m}{2}\rceil \lceil \frac{1}{2}(i-2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil -2)\rceil \hfill \\ +m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +3\lfloor \frac{m+1}{3}\rfloor -m-3k+3,\hfill \\ i=2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +2,\dots ,2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +1+\lfloor \frac{4k-4-m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil }{\lceil \frac{m}{2}\rceil }\rfloor ,\hfill \\ j=1,3,\dots ,m,m\equiv 1(2)\text{ or }j=1,3,\dots ,m+1,m\equiv 0(2)\hfill \end{array}\\ \begin{array}{c}1+\lceil \frac{m}{2}\rceil \lceil \frac{1}{2}(i-2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil -1)\rceil +\lfloor \frac{m}{2}\rfloor \lceil \frac{1}{2}(i-2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil -2)\rceil \hfill \\ +m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +3\lfloor \frac{m+1}{3}\rfloor -m-3k+3,\hfill \\ i=2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +2,\dots ,2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +1+\lfloor \frac{4k-4-m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil }{\lceil \frac{m}{2}\rceil }\rfloor ,\hfill \\ j=2,4,\dots ,m+1,m\equiv 1(2)\text{ or }j=2,4,\dots ,m,m\equiv 0(2)\hfill \\ k,i=2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +\lfloor \frac{4k-4-m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil }{\lceil \frac{m}{2}\rceil }\rfloor +2,\dots ,n,j=1,2,\dots ,m\hfill \end{array}\end{array}$$

For $j=1,2,\dots ,m$ the horizontal edges labels are as follow $$\varphi (v_i^j{v}_i^{j+1})=\{\begin{array}{cc}1+\lfloor \frac{i-1}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor ,\hfill & i=1,2,3,\dots ,2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil \hfill \\ k,\hfill & i=2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +1,\dots ,n\hfill \end{array}$$

The weight of a face is defined as $$W{t}_{(1,1,1)}(f_i^j)=\varphi (v_i^j)+\varphi (v_i^{j+1})+\varphi (v_{i+1}^j)+\varphi (v_{i+1}^{j+1})+\varphi (v_i^j{v}_i^{j+1})+\varphi (v_i^j{v}_{i+1}^j)+\varphi (v_{i+1}^j{v}_{i+1}^{j+1})+\varphi (v_i^{j+1}{v}_{i+1}^{j+1})+\varphi (f_i^j)$$

Now we calculate the difference between the weights for every two adjacent horizontal faces in each row as follow.

Let, $g=f_i^{j+1}$ and $h=f_i^j$ be two horizontal consecutive faces. Then, $$\begin{array}{c}W{t}_{(1,1,1)}(g)-W{t}_{(1,1,1)}(h)\\ \hspace{1em}=\varphi (v_i^{j+1})+\varphi (v_i^{j+2})+\varphi (v_{i+1}^{j+1})\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\varphi (v_{i+1}^{j+2})+\varphi (v_i^{j+1}{v}_i^{j+2})+\varphi (v_i^{j+1}{v}_{i+1}^{j+1})+\varphi (v_{i+1}^{j+1}{v}_{i+1}^{j+2})\\ \hspace{1em}+\varphi (v_i^{j+2}{v}_{i+1}^{j+2})-\varphi (v_i^j)-\varphi (v_i^{j+1})-\varphi (v_{i+1}^j)\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-\varphi (v_{i+1}^{j+1})-\varphi (v_i^j{v}_i^{j+1})-\varphi (v_i^j{v}_{i+1}^j)-\varphi (v_{i+1}^j{v}_{i+1}^{j+1})\\ \hspace{1em}-\varphi (v_i^{j+1}{v}_{i+1}^{j+1})+\varphi (g)-\varphi (h)\\ \hspace{1em}=\varphi (v_i^{j+2})+\varphi (v_{i+1}^{j+2})+\varphi (v_i^{j+1}{v}_i^{j+2})\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\varphi (v_{i+1}^{j+1}{v}_{i+1}^{j+2})+\varphi (v_i^{j+2}{v}_{i+1}^{j+2})+\varphi (g)\\ \hspace{1em}-\varphi (v_i^j)-\varphi (v_{i+1}^j)-\varphi (v_i^j{v}_i^{j+1})-\varphi (v_{i+1}^j{v}_{i+1}^{j+1})\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-\varphi (v_i^j{v}_{i+1}^j)-\varphi (h)\end{array}$$

For $i=1,2,3,\dots ,2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil -2,j=1,2,\dots ,m$ $$\begin{array}{c}W{t}_{(1,1,1)}(g)-W{t}_{(1,1,1)}(h)\\ \hspace{1em}=1+\lfloor \frac{i-1}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor +1+\lfloor \frac{i}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+1+\lfloor \frac{i-1}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor +1+\lfloor \frac{i}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+1+j+1-1-\lfloor \frac{i-1}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-1-\lfloor \frac{i}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor -1-\lfloor \frac{i-1}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-1-\lfloor \frac{i}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor -1-j\end{array}$$ $$W{t}_{(1,1,1)}(g)-W{t}_{(1,1,1)}(h)=1$$

For $i=2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil -1,j=1,2,\dots ,m$ $$\begin{array}{c}W{t}_{(1,1,1)}(g)-W{t}_{(1,1,1)}(h)\\ \hspace{1em}=1+\lfloor \frac{i-1}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor +1+\lfloor \frac{i}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+1+\lfloor \frac{i-1}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor +1+\lfloor \frac{i}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor \\ \hspace{1em}+1+j+1-1-\lfloor \frac{i-1}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-1-\lfloor \frac{i}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor -1-\lfloor \frac{i-1}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor \\ \hspace{1em}-1-\lfloor \frac{i}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor -1-j\\ \hspace{1em}=1\end{array}$$

For $i=2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil ,j=1,2,\dots ,m$ $$\begin{array}{c}W{t}_{(1,1,1)}(g)-W{t}_{(1,1,1)}(h)\\ \hspace{1em}=1+\lfloor \frac{i-1}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor +k+1\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\lfloor \frac{i-1}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor +k+1\\ \hspace{1em}+\lfloor \frac{1}{2}(m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +3\lfloor \frac{m+1}{3}\rfloor -m-3k+3)\rfloor \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}\times \lceil \frac{1}{2}(i-2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +1)\rceil \\ \hspace{1em}+\lceil \frac{1}{2}(m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +3\lfloor \frac{m+1}{3}\rfloor -m-3k+3)\rceil \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}\times \lceil \frac{1}{2}(i-2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil )\rceil \\ \hspace{1em}+j+1-1-\lfloor \frac{i-1}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-k-1-\lfloor \frac{i-1}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor -k-1\\ \hspace{1em}-\lfloor \frac{1}{2}(m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +3\lfloor \frac{m+1}{3}\rfloor -m-3k+3)\rfloor \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}\times \lceil \frac{1}{2}(i-2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +1)\rceil \end{array}$$ $$\begin{array}{c}\hspace{1em}-\lceil \frac{1}{2}(m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +3\lfloor \frac{m+1}{3}\rfloor -m-3k+3)\rceil \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}\times \lceil \frac{1}{2}(i-2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil )\rceil -j\\ \hspace{1em}=1\end{array}$$

For $i=2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +1,j=1,2,\dots ,m$ $$\begin{array}{c}W{t}_{(1,1,1)}(g)-W{t}_{(1,1,1)}(h)\\ \hspace{1em}=k+k+k+k+1+⏧\frac{1}{2}(m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+3\lfloor \frac{m+1}{3}\rfloor -m-3k+3)⏧\lceil \frac{1}{2}(i-2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +1)\rceil \\ \hspace{1em}+\lceil \frac{1}{2}(m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +3\lfloor \frac{m+1}{3}\rfloor -m-3k+3)\rceil \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}\times \lceil \frac{1}{2}(i-2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil )\rceil +j+1-k-k-k\\ \hspace{1em}-k-1-\lfloor \frac{1}{2}(m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +3\lfloor \frac{m+1}{3}\rfloor -m-3k+3)\rfloor \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}\times \lceil \frac{1}{2}(i-2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +1)\rceil \\ \hspace{1em}-\lceil \frac{1}{2}(m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +3\lfloor \frac{m+1}{3}\rfloor -m-3k+3)\rceil \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}\times \lceil \frac{1}{2}(i-2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil )\rceil -j\\ \hspace{1em}=1\end{array}$$

For $i=2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +2,\dots ,2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +1+\lfloor \frac{4k-4-m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil }{\lceil \frac{m}{2}\rceil }\rfloor ,j=1,2,\dots ,m$ $$\begin{array}{c}W{t}_{(1,1,1)}(g)-W{t}_{(1,1,1)}(h)\\ \hspace{1em}=k+k+k+k+1+\lfloor \frac{m}{2}\rfloor \lceil \frac{1}{2}(i-2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil -1)\rceil \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\lceil \frac{m}{2}\rceil \lceil \frac{1}{2}(i-2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil -2)\rceil \\ \hspace{1em}+m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +3\lfloor \frac{m+1}{3}\rfloor -m-3k\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+3+j+1-k-k-k-k-1\\ \hspace{1em}-\lfloor \frac{m}{2}\rfloor \lceil \frac{1}{2}(i-2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil -1)\rceil \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-\lceil \frac{m}{2}\rceil \lceil \frac{1}{2}(i-2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil -2)\rceil \\ \hspace{1em}-m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil -3\lfloor \frac{m+1}{3}\rfloor +m+3k-3-j\\ \hspace{1em}=1\end{array}$$

For $i=2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +\lfloor \frac{4k-4-m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil }{\lceil \frac{m}{2}\rceil }\rfloor +2,j=1,2,\dots ,m$ $$\begin{array}{c}W{t}_{(1,1,1)}(g)-W{t}_{(1,1,1)}(h)\\ \hspace{1em}=k+k+k+k+k+j+1\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+(8+m(2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +\lfloor \frac{4k-m-m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil }{\lceil \frac{m}{2}\rceil }\rfloor +1)-8k)\\ \hspace{1em}-k-k-k-k-k-j\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-(8+m(2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +\lfloor \frac{4k-m-m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil }{\lceil \frac{m}{2}\rceil }\rfloor +1)-8k)\\ \hspace{1em}=1\end{array}$$

For $i=2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +\lfloor \frac{4k-4-m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil }{\lceil \frac{m}{2}\rceil }\rfloor +3,\dots ,n,j=1,2,\dots ,m$ $$\begin{array}{c}W{t}_{(1,1,1)}(g)-W{t}_{(1,1,1)}(h)\\ \hspace{1em}=k+k+k+k+k+j+1+8\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+m(2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +\lfloor \frac{4k-m-m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil }{\lceil \frac{m}{2}\rceil }\rfloor +1)-8k\\ \hspace{1em}+m(i-2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil -\lfloor \frac{4k-m-m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil }{\lceil \frac{m}{2}\rceil }\rfloor -2)\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-k-k-k-k-k-j\\ \hspace{1em}-(8+m(2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +\lfloor \frac{4k-m-m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil }{\lceil \frac{m}{2}\rceil }\rfloor +1)-8k)\\ \hspace{1em}=1\end{array}$$

Now the calculate the difference between the weights for every two adjacent vertical faces as follow.

Let, $g=f_{i+1}^j$ and $h=f_i^j$ be two vertical consecutive faces. $$\begin{array}{c}W{t}_{(1,1,1)}(g)-W{t}_{(1,1,1)}(h)\\ \hspace{1em}=\varphi (v_{i+1}^j)+\varphi (v_{i+1}^{j+1})+\varphi (v_{i+2}^j)+\varphi (v_{i+2}^{j+1})+\varphi (v_{i+1}^j{v}_{i+1}^{j+1})\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\varphi (v_{i+1}^j{v}_{i+2}^j)+\varphi (v_{i+2}^j{v}_{i+2}^{j+1})\\ \hspace{1em}+\varphi (v_{i+1}^{j+1}{v}_{i+2}^{j+1})+\varphi (g)-\varphi (v_i^j)-\varphi (v_i^{j+1})\\ \hspace{1em}-\varphi (v_{i+1}^j)-\varphi (v_{i+1}^{j+1})-\varphi (v_i^j{v}_i^{j+1})-\varphi (v_i^j{v}_{i+1}^j)\\ \hspace{1em}-\varphi (v_{i+1}^j{v}_{i+1}^{j+1})-\varphi (v_i^{j+1}{v}_{i+1}^{j+1})-\varphi (h)\\ \hspace{1em}=\varphi (v_{i+2}^j)+\varphi (v_{i+2}^{j+1})+\varphi (v_{i+1}^j{v}_{i+2}^j)+\varphi (v_{i+2}^j{v}_{i+2}^{j+1})\\ \hspace{1em}+\varphi (v_{i+1}^{j+1}{v}_{i+2}^{j+1})\\ \hspace{1em}+\varphi (g)-\varphi (v_i^j)-\varphi (v_i^{j+1})-\varphi (v_i^j{v}_i^{j+1})-\varphi (v_i^j{v}_{i+1}^j)\\ \hspace{1em}-\varphi (v_i^{j+1}{v}_{i+1}^{j+1})-\varphi (h)\end{array}$$

For $i=1,2,3,\dots ,2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil -2,j=1,2,\dots ,m$ $$\begin{array}{c}W{t}_{(1,1,1)}(g)-W{t}_{(1,1,1)}(h)\\ \hspace{1em}=1+\lfloor \frac{i+1}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor +1+\lfloor \frac{i+1}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+1+1+\lfloor \frac{i+1}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor +1+j\\ \hspace{1em}-1-\lfloor \frac{i-1}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor -1-\lfloor \frac{i-1}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-1-\lfloor \frac{i-1}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor -1-1-j\\ \hspace{1em}=3\lfloor \frac{i+1}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor -3\lfloor \frac{i-1}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor \\ \hspace{1em}=3\lfloor \frac{m+1}{3}\rfloor (\lfloor \frac{i+1}{2}\rfloor -\lfloor \frac{i-1}{2}\rfloor )\end{array}$$

Since $(\lfloor \frac{i+1}{2}\rfloor -\lfloor \frac{i-1}{2}\rfloor )=1$, for every value of i, so we have $$\begin{array}{c}W{t}_{(1,1,1)}(g)-W{t}_{(1,1,1)}(h)=3\lfloor \frac{m+1}{3}\rfloor \\ =m\end{array}$$

For $i=2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil -1,j=1,2,\dots ,m$ $$\begin{array}{c}W{t}_{(1,1,1)}(g)-W{t}_{(1,1,1)}(h)\\ \hspace{1em}=3k+1+1+\lceil \frac{1}{2}(i+1-2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +1)\rceil \\ \hspace{1em}\{\lfloor \frac{(m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +3\lfloor \frac{m+1}{3}\rfloor -m-3k+3)}{2}\rfloor \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\lceil \frac{(m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +3\lfloor \frac{m+1}{3}\rfloor -m-3k+3)}{2}\rceil \}\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+0+j-1-\lfloor \frac{i-1}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor \\ \hspace{1em}-1-\lfloor \frac{i-1}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor -1\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-\lfloor \frac{i-1}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor -1-1-j\\ \hspace{1em}=3k+2+(1)(m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +3\lfloor \frac{m+1}{3}\rfloor -m-3k+3)\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-5-3\lfloor \frac{i-1}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor \\ \hspace{1em}=3k+2+m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +3\lfloor \frac{m+1}{3}\rfloor \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-m-3k+3-5-3\lfloor \frac{i-1}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor \\ \hspace{1em}=m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +3\lfloor \frac{m+1}{3}\rfloor -m-3\lfloor \frac{i-1}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor \\ \hspace{1em}=m\end{array}$$

For $i=2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil ,j=1,2,\dots ,m$ $$\begin{array}{c}W{t}_{(1,1,1)}(g)-W{t}_{(1,1,1)}(h)\\ \hspace{1em}=3k+2+\lceil \frac{1}{2}(i+1-2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +1)\rceil \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}\{\lfloor \frac{(m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +3\lfloor \frac{m+1}{3}\rfloor -m-3k+3)}{2}\rfloor \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\lceil \frac{(m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +3\lfloor \frac{m+1}{3}\rfloor -m-3k+3)}{2}\rceil \}\end{array}$$ $$\begin{array}{c}\mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\lceil \frac{1}{2}(i+1-2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil )\rceil \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}\{\lceil \frac{(m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +3\lfloor \frac{m+1}{3}\rfloor -m-3k+3)}{2}\rceil \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\lfloor \frac{(m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +3\lfloor \frac{m+1}{3}\rfloor -m-3k+3)}{2}\rfloor \}\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+j-1-\lfloor \frac{i-1}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-1-\lfloor \frac{i-1}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor -1-\lfloor \frac{i-1}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor \end{array}$$ $$\begin{array}{c}\mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-2-\lceil \frac{1}{2}(i-2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +1)\rceil \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}\{\lfloor \frac{(m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +3\lfloor \frac{m+1}{3}\rfloor -m-3k+3)}{2}\rfloor \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\lceil \frac{(m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +3\lfloor \frac{m+1}{3}\rfloor -m-3k+3)}{2}\rceil \}\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-\lceil \frac{1}{2}(i-2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil )\rceil \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}\{\lceil \frac{(m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +3\lfloor \frac{m+1}{3}\rfloor -m-3k+3)}{2}\rceil \end{array}$$ $$\begin{array}{c}\mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\lfloor \frac{(m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +3\lfloor \frac{m+1}{3}\rfloor -m-3k+3)}{2}\rfloor \}-j\\ \hspace{1em}=3k+(1)(m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +3\lfloor \frac{m+1}{3}\rfloor -m-3k+3)\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+(1)(m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +3\lfloor \frac{m+1}{3}\rfloor -m-3k+3)\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-3-3\lfloor \frac{i-1}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-(1)(m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +3\lfloor \frac{m+1}{3}\rfloor -m-3k+3)\end{array}$$ $$\begin{array}{c}\hspace{1em}=3k+m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +3\lfloor \frac{m+1}{3}\rfloor \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-m-3k+3+m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +3\lfloor \frac{m+1}{3}\rfloor \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-m-3k+3-3\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-3\lfloor \frac{i-1}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor -m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-3\lfloor \frac{m+1}{3}\rfloor +m+3k-3\\ \hspace{1em}=m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +3\lfloor \frac{m+1}{3}\rfloor -m-3\lfloor \frac{i-1}{2}\rfloor \lfloor \frac{m+1}{3}\rfloor \\ \hspace{1em}=m\end{array}$$

For $i=2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +1,j=1,2,\dots ,m$ $$\begin{array}{c}W{t}_{(1,1,1)}(g)-W{t}_{(1,1,1)}(h)\\ \hspace{1em}=3k+2+\lceil \frac{1}{2}(i+1-2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil -1)\rceil \left\{\lfloor \frac{m}{2}\rfloor +\lceil \frac{m}{2}\rceil \right\}\\ \hspace{1em}+\lceil \frac{1}{2}(i+1-2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil -2)\rceil \left\{\lceil \frac{m}{2}\rceil +\lfloor \frac{m}{2}\rfloor \right\}\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+2m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +6\lfloor \frac{m+1}{3}\rfloor \\ \hspace{1em}-2m-6k+6+j-3k-2-\lceil \frac{1}{2}(i-2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +1)\rceil \end{array}$$ $$\begin{array}{c}\hspace{1em}\{\lfloor \frac{(m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +3\lfloor \frac{m+1}{3}\rfloor -m-3k+3)}{2}\rfloor \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\lceil \frac{(m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +3\lfloor \frac{m+1}{3}\rfloor -m-3k+3)}{2}\rceil \}\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-\lceil \frac{1}{2}(i-2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil )\rceil \\ \hspace{1em}\{\lceil \frac{(m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +3\lfloor \frac{m+1}{3}\rfloor -m-3k+3)}{2}\rceil \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\lfloor \frac{(m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +3\lfloor \frac{m+1}{3}\rfloor -m-3k+3)}{2}\rfloor \}-j\\ \hspace{1em}=3k+2+(1)m+0+2m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+6\lfloor \frac{m+1}{3}\rfloor -2m-6k+6-3k-2\end{array}$$ $$\begin{array}{c}\hspace{1em}-(1)(m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +3\lfloor \frac{m+1}{3}\rfloor -m-3k+3)\\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-(1)(m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +3\lfloor \frac{m+1}{3}\rfloor -m-3k+3)\\ \hspace{1em}=3k+2+m-3k-2\\ \hspace{1em}=m\end{array}$$

For $i=2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +\lfloor \frac{4k-m-m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil }{\lceil \frac{m}{2}\rceil }\rfloor +2,j=1,2,\dots ,m$ $$\begin{array}{c}W{t}_{(1,1,1)}(g)-W{t}_{(1,1,1)}(h)\\ \hspace{1em}=5k+j+(8+m(2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\lfloor \frac{4k-m-m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil }{\lceil \frac{m}{2}\rceil }\rfloor +1)-8k)\\ \hspace{1em}-5k-j-(8+m(2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\lfloor \frac{4k-m-m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil }{\lceil \frac{m}{2}\rceil }\rfloor +1)-8k)\end{array}$$ $$\begin{array}{c}\hspace{1em}=5k+j+(8+m(2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\lfloor \frac{4k-m-m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil }{\lceil \frac{m}{2}\rceil }\rfloor +1+1-1)-8k)-5k\\ \hspace{1em}-j-(8+m(2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\lfloor \frac{4k-m-m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil }{\lceil \frac{m}{2}\rceil }\rfloor +1+1-1)-8k)\end{array}$$ $$\begin{array}{c}\hspace{1em}=(8+m(i+1)-8k-1)-(8+m(i-1)-8k)\\ \hspace{1em}=(8+m(i)-8k)-(8+m(i-1)-8k)\\ \hspace{1em}=8+\mathit{\text{mi}}-8k-8-\mathit{\text{mi}}+m+8k\\ \hspace{1em}=m\end{array}$$

For $i=2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +\lfloor \frac{4k-m-m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil }{\lceil \frac{m}{2}\rceil }\rfloor +3,\dots ,n,j=1,2,\dots ,m$ $$\begin{array}{c}W{t}_{(1,1,1)}(g)-W{t}_{(1,1,1)}(h)\\ \hspace{1em}=5k+j+8+m(2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}+\lfloor \frac{4k-m-m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil }{\lceil \frac{m}{2}\rceil }\rfloor +1)-8k\\ \hspace{1em}+m(i+1-2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil \\ \mathrm{\hspace{1em}}\mathrm{\hspace{1em}}-\lfloor \frac{4k-m-m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil }{\lceil \frac{m}{2}\rceil }\rfloor -2)-5k-j-8\\ \hspace{1em}-m(2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +\lfloor \frac{4k-m-m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil }{\lceil \frac{m}{2}\rceil }\rfloor +1)\end{array}$$ $$\begin{array}{c}\hspace{1em}+8k-m(i-2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil -\lfloor \frac{4k-m-m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil }{\lceil \frac{m}{2}\rceil }\rfloor -2)\\ \hspace{1em}=m(i+1-2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil -\lfloor \frac{4k-m-m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil }{\lceil \frac{m}{2}\rceil }\rfloor -2)\\ \hspace{1em}-m(i-2\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil -\lfloor \frac{4k-m-m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil }{\lceil \frac{m}{2}\rceil }\rfloor -2)\\ \hspace{1em}=\mathit{\text{mi}}+m-2m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil -m\lfloor \frac{4k-m-m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil }{\lceil \frac{m}{2}\rceil }\rfloor -2m\\ \hspace{1em}-\mathit{\text{mi}}+2m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil +m\lfloor \frac{4k-m-m\lceil \frac{k}{\lfloor \frac{m+1}{3}\rfloor }\rceil }{\lceil \frac{m}{2}\rceil }\rfloor +2m\\ =m\end{array}$$

Hence it follows that the weights of any two faces f and g are distinct. Complete the proof by clearly giving the two required inequalities. □

3 Observation

The grid graph $G=P_{n+1}\square P_{m+1}$ contains exactly mn + 1 faces. Consider ρ be face irregular $k-$ labeling of type (0, 0, 1) of a $2-$ connected plane graph G. Since the weight of the face f under a k-labeling ρ of type $(\alpha ,\beta ,\gamma )$, where $\alpha ,\beta ,\gamma \in \{0,1\}$, is defined as $$W{t}_{{\rho }_{(\alpha ,\beta ,\gamma )}}(f)=\alpha \sum _{v\sim f}\rho (v)+\beta \sum _{e\sim f}\rho (e)+\gamma \rho (f)$$

In the case $(\alpha ,\beta )=(0,0)$, the face weight of type (0, 0, 1) is reduced to the label of the respective faces. Since ρ is face irregular k-labeling so face weights must be distinct so that weights of the faces would be label of the respective faces and hence $f{s}_{(0,0,1)}(G)=\mathit{\text{mn}}+1$.

4 Conclusion

In this paper, we have applied new characteristic of face irregular evaluations of type $(\alpha ,\beta ,\gamma )$ for grid graphs for parameters, where grid graph G with n rows and m columns is the graph of cartesian product of path graphs $P_{n+1}\square P_{m+1}$. To determine the face irregularity strength of type $(\alpha ,\beta ,\gamma )$ of planar graphs is a very complicated task. We investigated the tight lower bound for the face irregular evaluations for all possible combinations for parameters $\alpha ,\beta ,\gamma \in \{0,1\}$ are explained in this research for G.