#### Proof

We divide the proof in two cases on the basis of parities of *m* and *n*.

**Case I. When ***m* and *n* are of opposite parity

Let *Π* = {*S*_{1}, *S*_{2}, *S*_{3}, *S*_{4}} Where *S*_{1} = {*V*_{1,1}}, *S*_{2} = {*V*_{1,n}}, *S*_{3} = {*V*_{1,2}, *V*_{1,3},...,

*V*_{1,n − 1}, *V*_{2,1}, *V*_{2,2},......,*V*_{2,n},.....,*V*_{m − 2,1}, *V*_{m − 2,2},...,

*V*_{m − 2,n}, *V*_{m − 1,2}, *V*_{m − 1,3},...,*V*_{m − 1,n}} *S*_{4} = {*V*_{m − 1,1}}. We prove that *Π*is a resolving partition for *M*_{m, n}. To find distance vectors we use two parameters *q*, *i* and depending on their different values we divide the entries of distance vectors into four steps.

**Step I**: Distances of *S*_{1} with all vertices of *M*_{m, n}.

In this case for each value of *q* ∈ {1,2,...,*n*} the parameter i varies from 1 to *m* − 1. The entries of different vectors are

$$\begin{array}{}d({S}_{1},{V}_{i,q})=\left\{\begin{array}{l}i+q-2,1\le i\le \frac{1}{2}(m+n-2q+1)\\ m+n-q-i,\frac{1}{2}(m+n-2q+3)\le i\le m-1\end{array}\right.\end{array}$$

**Step II**: Distances of *S*_{2} with all vertices of *M*_{m, n}.

For each value of *q* ∈ {1,2, ..., *n*} the parameter i varies from 1 to m - 1 and we get *d*(*S*_{2}, *V*_{i, q}) = *d*(*S*_{1}, *V*_{i, n + 1 − q}).

**Step III** : Distances of *S*_{3} with all vertices of *M*_{m, n}.

Here for each value of *q* ∈ {1,2,...,*n*} the parameter i varies from 1 to m - 1 and we have

$$\begin{array}{}d({S}_{3},{V}_{i,q})=\left\{\begin{array}{l}1,ifi=1,q=1,q=n\\ 1,ifi=m-1,q=n\\ 0,otherwise\end{array}\right.\end{array}$$

**Step IV**: Distances of *S*_{4} with all vertices of *M*_{m, n}.

Here we have two parts

a) For q = 1 , we have

$$\begin{array}{}d({S}_{4},{V}_{i,q})=\left\{\begin{array}{l}i+n-1,1\le i\le \frac{1}{2}(m-n-1)\\ m-1-i,\frac{1}{2}(m-n+1)\le i\le m-1\end{array}\right.\end{array}$$

b) For each value of *q* ∈ {2,..., *n*} the parameter i varies from 1 to m - 1 and we have *d*(*S*_{4}, *V*_{i, q}) = *d*(*S*_{1}, *V*_{i, n + 2 − q}).

These representations are distinct in at least one coordinate. So *Π* is a resolving partition for *M*_{m, n} so clearly *pd*(*M*_{m, n}) ≤ 4.

#### Example

*Clearly pd*(*M*_{9,4}) ≤ 4 *as the resolving partition for M*_{9,4} is *Π* = {*S*_{1}, *S*_{2}, *S*_{3}, *S*_{4}} *where S*_{1} = {*V*_{1,1}}, *S*_{2} = {*V*_{1,4}}, *S*_{3} = {*V*_{1,2}, *V*_{1,3}, *V*_{2,1}, *V*_{2,2}, *V*_{2,3}, *V*_{2,4},.....

,*V*_{7,1}, *V*_{7,2}, *V*_{7,4}, *V*_{8,2}, *V*_{8,3}, *V*_{8,4}}, *S*_{4} = {*V*_{8,1}}.

*The representations of different vertices of M*_{9,4} *with respect to **Π* are

$$\begin{array}{}{V}_{1,1}(0,3,1,4),{V}_{1,2}(1,2,0,3),{V}_{1,3}(2,1,0,2),{V}_{1,4}(3,0,1,1),\\ {V}_{2,1}(1,4,0,5),{V}_{2,2}(2,3,0,4),{V}_{2,3}(3,2,0,3),{V}_{2,4}(4,1,0,2),\\ {V}_{3,1}(2,5,0,5),{V}_{3,2}(3,4,0,5),{V}_{3,3}(4,3,0,4),{V}_{3,4}(5,2,0,3),\\ {V}_{4,1}(3,5,0,4),{V}_{4,2}(4,5,0,5),{V}_{4,3}(5,4,0,5),{V}_{4,4}(5,3,0,4),\\ {V}_{5,1}(4,4,0,3),{V}_{5,2}(5,5,0,4),{V}_{5,3}(5,5,0,5),{V}_{5,4}(4,4,0,5),\\ {V}_{6,1}(5,3,0,2),{V}_{6,2}(5,4,0,3),{V}_{6,3}(4,5,0,4),{V}_{6,4}(3,5,0,5),\\ {V}_{7,1}(5,2,0,1),{V}_{7,2}(4,3,0,2),{V}_{7,3}(3,4,0,3),{V}_{7,4}(2,5,0,4),\\ {V}_{8,1}(4,1,1,0),{V}_{8,2}(3,2,0,1),{V}_{8,3}(2,3,0,2),{V}_{8,4}(1,4,0,3)\end{array}$$

**Case II: when m and n are of same parity:** We want to prove that *pd*(*M*_{m, n}) ≤ 5 by constructing a general resolving partition of size 5, for *m* − *n* ≥ 4 and *m*, *n* are of same parity.

#### Proof

Let *Π* = {*S*_{1}, *S*_{2}, *S*_{3}, *S*_{4}, *S*_{5}} where *S*_{1} = {*V*_{1,1}}, *S*_{2} = {*V*_{1,n}} , *S*_{3} = {*V*_{1,2}, *V*_{1,3},...,*V*_{1,n − 1}, *V*_{2,1}, *V*_{2,2},......,*V*_{2,n},.....,*V*_{m − 2,1}, *V*_{m − 2,2},...,*V*_{m − 2,n}, *V*_{m − 1,2}, *V*_{m − 1,3},...,*V*_{m − 1,n − 1}}, *S*_{4} = {*V*_{m − 1,1}} , *S*_{5} = {*V*_{m − 1,n}} .

We prove that *Π* is a resolving partition for *M*_{m, n}. To find distance vectors we use two parameters q , i and depending on their different values we divide the entries of distance vectors into five steps.

**Step I**: Distances of *S*_{1} with all vertices of *M*_{m, n}.

In this case for each value of *q* ∈ {1, 2,..., *n*} the parameter i varies from 1 to m - 1. The entries of different vectors are

$$\begin{array}{}d({S}_{1},{V}_{i,q})=\left\{\begin{array}{l}i+q-2,1\le i\le \frac{1}{2}(m+n-2q+2)\\ m+n-q-i,\frac{1}{2}(m+n-2q+4)\le i\le m-1\end{array}\right.\end{array}$$

**Step II**: Distances of *S*_{2} with all vertices of *M*_{m, n}.

For each value of *q* ∈ {1,2,..., *n*} the parameter i varies from 1 to m - 1 and we get *d*(*S*_{2}, *V*_{i, q}) = *d*(*S*_{1}, *V*_{i, n + 1 − q}).

**Step III** : Distances of *S*_{3} with all vertices of *M*_{m, n}.

Here for each value of *q* ∈ {1,2,..., *n*} the parameter i varies from 1 to m - 1 and we have

$$\begin{array}{}d({S}_{3},{V}_{i,q})=\left\{\begin{array}{l}1,ifi=1,q=1,q=n\\ 1,ifi=m-1,q=1,q=n\\ 0,otherwise\end{array}\right.\end{array}$$

**Step IV** : Distances of *S*_{4} with all vertices of *M*_{m, n}. Here we have two parts

a) For q = 1 , we have

$$\begin{array}{}d({S}_{4},{V}_{i,q})=\left\{\begin{array}{l}i+n-1,1\le i\le \frac{1}{2}(m-n)\\ m-1-i,\frac{1}{2}(m-n+2)\le i\le m-1\end{array}\right.\end{array}$$

b) For each value of *q* ∈ {2, ..., *n*} the parameter i varies from 1 to m - 1 and we have *d*(*S*_{4}, *V*_{i, q}) = *d*(*S*_{1}, *V*_{i, n + 2 − q})

**Step V** : Distances of *S*_{5} with all vertices of *M*_{m, n}.

Here for each value of ∈ {1,2,..., *n*} the parameter i varies from 1 to m - 1 and we have *d*(*S*_{5}, *V*_{i, q}) = *d*(*S*_{4}, *V*_{i, n + 1 − q}).

These representations are distinct in at least one coordinate. So *Π* is a resolving partition for *M*_{m, n}. Since there is no 4 resolving partition for *M*_{m, n}, hence *Π* is a minimal resolving partition for *M*_{m, n}. So partition dimension of *M*_{m, n} is 5.

#### Example

*The partition dimension of M*_{9,3} *is 5. The resolving partition for M*_{9,3} *is Π* = {*S*_{1}, *S*_{2}, *S*_{3}, *S*_{4}, *S*_{5}}. *Where*

$$\begin{array}{}{S}_{1}=\{{V}_{1,1}\}\\ {S}_{2}=\{{V}_{1,3}\}\\ {S}_{3}=\{{V}_{1,2},{V}_{2,1},{V}_{2,2},{V}_{2,3},.....,{V}_{7,1},{V}_{7,2},{V}_{7,3},{V}_{8,2},{V}_{8,3}\}\\ {S}_{4}=\{{V}_{8,1}\}\\ {S}_{5}=\{{V}_{8,3}\}\end{array}$$

The representations of different vertices of *M*_{9,3} with respect to *Π* are

$$\begin{array}{}{V}_{1,1}(0,2,1,3,1),{V}_{1,2}(1,1,0,2,2),{V}_{1,3}(2,0,1,1,2),\\ {V}_{2,1}(1,3,0,4,2),{V}_{2,2}(2,2,0,3,3),{V}_{2,3}(3,1,0,2,4),\\ {V}_{3,1}(2,4,0,5,3),{V}_{3,2}(3,3,0,4,4),{V}_{3,3}(4,2,0,3,5),\\ {V}_{4,1}(3,5,0,4,4),{V}_{4,2}(4,4,0,5,5),{V}_{4,3}(5,3,0,4,4),\\ {V}_{5,1}(4,4,0,3,5),{V}_{5,2}(5,5,0,4,4),{V}_{5,3}(4,4,0,5,3),\\ {V}_{6,1}(5,3,0,2,4),{V}_{6,2}(4,4,0,3,3),{V}_{6,3}(3,5,0,4,2),\\ {V}_{7,1}(4,2,0,1,3),{V}_{7,2}(3,3,0,2,2),{V}_{7,3}(2,4,0,3,1),\\ {V}_{8,1}(3,1,1,0,2),{V}_{8,2}(2,2,0,1,1),{V}_{8,3}(1,3,1,2,0,)\end{array}$$

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