1. Introduction

Slater ^[13] and Harary et al. ^[6] introduced the notions of resolvability and locating number in graphs. Chartrand et al. ^[4] introduced the partition dimension of a graph. These concepts have some applications in Chemistry for representing chemical compounds ^[2] or to problems of pattern recognition and image processing, some of which involve the use of hierarchical data structures ^[10].

Kroto et al. ^[9] discovered fullerene molecule and since then, scientists took a great interest in the fullerene graphs. A $(k, 6)$-fullerene graph is a connected cubic plane graph whose faces have sizes $k$ and $6$. There are only three types of fullerene graphs, that is, $(3, 6)$, $(4, 6)$ and $(5, 6)$-fullerene graphs. A $(5, 6)$-fullerene is the usual fullerene as the molecular graph of sphere carbon fullerene. A $(3, 6)$-fullerene graph has cycles of order three and six. The Euler's formula implies that a $(3, 6)$-fullerene graph has exactly four faces of size $3$ and $(n/2) - 2$ hexagons. Similarly $(4, 6)$ and $(5, 6)$-fullerene graphs has cycles of order four and six, and five and six, respectively. The Euler's formula implies that a $(4, 6)$-fullerene graph has exactly six square faces and $(n/2)-4$ hexagons.

Chartrand et al. ^[3] gave useful definitions and results related to the partition dimension of a connected graph. Let $G$ be a connected graph with vertex set $V(G)$ and edge set $E(G)$. If $S$ is a subset of $V(G)$ and $v\in V(G)$ then the distance between $v$ and $S$, denoted by $d(v, S)$, is defined as $d(v, S) = \min\{d(v, x)\mid x\in S\}$. For an ordered $k$-partition $\Pi = \{S_{1}, S_{2}, \dots, S_{k}\}$ of $V(G)$ and a vertex $v$ of $G$, the representation of $v$ with respect to $\Pi$ is defined as the $k$-vector $r(v\mid\Pi) = (d(v, S_{1}), d(v, S_{2}), \dots, d(v, S_{k}))$. The partition $\Pi$ is called a resolving partition if $r(u\mid\Pi)\neq r(v\mid\Pi)$ for each $u, v\in V(G)$, $u\neq v$. The minimum $k$ for which there is a resolving $k$-partition of $V(G)$ is called the partition dimension of $G$ and is denoted by $pd(G)$.

Many authors determined the partition dimension of specific classes of graphs. Rodríguez-Velázquez et al. ^[14,15] find the bounds on the partition dimension of trees and unicyclic graphs. Tomescu et al. ^[16] calculated the partition dimension of a wheel graph and Tomescu ^[17] discussed the metric and partition dimension of a connected graph. Grigorious et al. ^[5] and Javaid et al. ^[7] calculated the partition dimension of some classes of circulant graphs.

The following result is a useful property in determining partition dimension.

Lemma 1.1. ^[3] Let $\Pi$ be a resolving partition of vertex set $V(G)$ of a connected graph $G$ and $u, v\in V(G)$. If $d(u, w) = d(v, w)$ for all $w\in V(G)\setminus\{u, v\}$ then $u$ and $v$ belong to different classes of $\Pi$.

The partition dimension of some families of graphs is given in next lemma.

Lemma 1.2. ^[3] Let $G$ be a connected graph. Then

1. $pd(G) = 2$ if and only if $G = P_{n}$ for $n\geq 2$,

2. $pd(G) = n$ if and only if $G = K_{n}$,

3. $pd(G) = 3$ if $G = C_{n}$ for $n\geq 3$.

Above results are useful in computing the partition dimension of connected graphs. Ashrafi et al. ^[1] studied the topological indices of $(3, 6)$ and $(4, 6)$-fullerene graphs. Moftakhar et al. ^[8] calculated the automorphism group and fixing number of $(3, 6)$ and $(4, 6)$-fullerene graphs. Siddiqui et al. ^[11,12] calculated the metric dimension and partition dimension of nanotubes. In this paper, we calculate the partition dimension of two $(4, 6)$-fullerene graphs. Also we give conjectures on the partition dimension of two $(3, 6)$-fullerene graphs.

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2. Partition dimension of $(4, 6)$-fullerene graphs

In this section, we consider two $(4, 6)$-fullerene graphs $G_1[n]$ and $G_2[n]$ shown in Figure 1 and Figure 2, respectively. It is easily seen that the order of $G_1[n]$ and $G_2[n]$ is $8n$ and $8n+4$, respectively. We calculate the partition dimension of $G_{1}[n]$ and $G_{2}[n]$ graphs.

Theorem 2.1. The partition dimension of fullerene graph $G_{1}[n]$ is $3$.

Proof. Let $\Pi = \{S_{1}, S_{2}, S_{3}\}$, where $S_{1} = \{x_{2n}, x_{2n+1}\}$, $S_{2} = \{y_{2n}\}$ and $S_{3} = V(G_{1}[n])\setminus \{x_{2n}, x_{2n+1}, y_{2n}\}$, be a partition of $V(G_{1}[n])$. We show that $\Pi$ is a resolving partition of $G_{1}[n]$ with minimum cardinality. The representation of each vertex of $G_{1}[n]$ with respect to $\Pi$ is given as follows:

$\begin{eqnarray*} r(x_{2n}\mid\Pi) = (0, 1, 1),&r(x_{2n+1}\mid\Pi) = (0, 2, 1),&r(y_{2n}\mid\Pi) = (1, 0, 1). \end{eqnarray*}$

$\begin{equation*} r(x_{i}\mid\Pi) = \left\{ \begin{array}{l l} (2n-i, 2n-i+1, 0) &{\rm if}~ 1\leq i\leq 2n-1, \\ (i-2n-1, i-2n+1, 0) &{\rm if}~ 2n+2\leq i\leq4n. \end{array} \right. \end{equation*}$

and

$\begin{equation*} r(y_{i}\mid\Pi) = \left\{ \begin{array}{ll} (2n-i+1, 2n-i, 0) &\mbox{if }1\leq i\leq2n-1, \\ (i-2n, i-2n, 0) &\mbox{if }2n+1\leq i\leq4n. \end{array} \right. \end{equation*}$

Therefore, it is easily seen that the representation of each vertex with respect to $\Pi$ is distinct. This shows that $\Pi$ is a resolving partition of $G_{1}[n]$. Thus $pd(G_{1}[n])\leq 3$.

On the other hand, by Lemma 1.2, it follows that $pd(G_{1}[n])\geq3$. Hence $pd(G_{1}[n]) = 3$.

In next theorem, we calculate the partition dimension of $G_2[n]$.

Theorem 2.2. The partition dimension of fullerene graph $G_{2}[n]$ is $3$.

Proof. Let $\Pi = \{S_{1}, S_{2}, S_{3}\}$, where $S_{1} = \{x_{2n+1}, x_{2n+2}\}$, $S_{2} = \{y_{2n+1}\}$ and $S_{3} = V(G_{2}[n])\setminus \{x_{2n+1}, x_{2n+2},$ $y_{2n+1}\}$, be a partition of $V(G_{2}[n])$. We show that $\Pi$ is a resolving partition of $G_{2}[n]$ with minimum cardinality. The representation of each vertex of $G_{2}[n]$ with respect to $\Pi$ is given as follows:

$\begin{eqnarray*} r(x_{2n+1}\mid\Pi) = (0, 1, 1),&r(x_{2n+2}\mid\Pi) = (0, 2, 1),&r(y_{2n+1}\mid\Pi) = (1, 0, 1). \end{eqnarray*}$

$\begin{equation*} r(x_{i}\mid\Pi) = \left\{ \begin{array}{ll} (2n+1-i, 2n+2-i, 0) &\mbox{if }1\leq i\leq2n, \\ (i-2n-2, i-2n, 0) &\mbox{if }2n+3\leq i\leq4n+2. \end{array} \right. \end{equation*}$

and

$\begin{equation*} r(y_{i}\mid\Pi) = \left\{ \begin{array}{ll} (2n+2-i, 2n+1-i, 0) &\mbox{if }1\leq i\leq2n, \\ (i-2n-1, i-2n-1, 0) &\mbox{if }2n+2\leq i\leq4n+2. \end{array} \right. \end{equation*}$

All pairs of vertices can easily be resolved by the partitioning set $\Pi$. Therefore $\Pi$ is a resolving partition of $G_{2}[n]$ and $pd(G_{2}[n])\leq 3$.

On the other hand, by Lemma 1.2, it follows that $pd(G_{2}[n])\geq3$. Hence $pd(G_{2}[n]) = 3$.

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3. Conjectures on partition dimension of two $(3, 6)$-fullerene graphs

In this section, we consider two $(3, 6)$-fullerene graphs $F_3[n]$ and $F_4[n]$ shown in Figure 3 and Figure 4, respectively. We can see that order of $F_3[n]$ and $F_4[n]$ is $16n-32$, $n\geq4$ and $24n$, $n\geq1$, respectively.

Firstly we consider the fullerene graph $F_3[n]$ and give a conjecture on the partition dimension of $F_3[n]$. The set of vertices $V(F_3[n])$, $n\geq5$, is divided into the following sets:

$\begin{equation}\label{qs1} \begin{array}{l l l} X_{1} = \{x_{i}\mid 1\leq i\leq 2n-6\},&X_{2} = \{x_{i}\mid 2n-4\leq i\leq 4n-11\}, &Y_{1} = \{y_{i}\mid 1\leq i\leq 2n-6\}, \\ Y_{2} = \{y_{i}\mid 2n-4\leq i\leq 4n-11\}, &Z_{1} = \{z_{1}, z_{2}, z_{3}\}, &Z_{2} = \{z_{4}, z_{5}, z_{6}\}, \\ A = \{a_{i}\mid 1\leq i\leq 6\}, &B_{1} = \{b_{i}\mid 1\leq i\leq 2n-6\}, &B_{2} = \{b_i\mid 2n-4\leq i\leq 4n-11\}, \\ C_{1} = \{c_{i}\mid 1\leq i\leq 2n-6\}, &C_{2} = \{c_{i}\mid 2n-4\leq i\leq 4n-11\}. \end{array} \end{equation}$

(3.1)

The relations of distances of vertices of $F_3[n]$ are given by:

$\begin{eqnarray}\label{eq21} &d(a_{4}, x) = d(y_{1}, x),&\mbox{ for all }\ x\in X_1, \\ \end{eqnarray}$

(3.2)

$\begin{eqnarray} & d(a_{5}, x) = d(y_{4n-11}, x),&\mbox{ for all }\ x\in X_2, \end{eqnarray}$

(3.3)

$\begin{eqnarray} &d(a_{2}, y) = d(x_{1}, y),&\mbox{ for all }\ y\in Y_1, \end{eqnarray}$

(3.4)

$\begin{eqnarray} &\ \ \ \ \ d(a_{1}, y) = d(x_{4n-11}, y),&\mbox{ for all }\ y\in Y_2, \end{eqnarray}$

(3.5)

$\begin{eqnarray} &d(z_{2}, z) = d(z_{3}, z),&\mbox{ for all }\ z\in Z_2, \end{eqnarray}$

(3.6)

$\begin{eqnarray} &d(z_{5}, z) = d(z_{6}, z),&\mbox{ for all }\ z\in Z_1, \end{eqnarray}$

(3.7)

$\begin{eqnarray} &d(z_{4}, x) = d(z_{6}, x),&\mbox{ for all }\ x\in X_2\cup\{x_{2n-5}\}, \end{eqnarray}$

(3.8)

$\begin{eqnarray} &d(z_{4}, y) = d(z_{5}, y),&\mbox{ for all }\ y\in Y_2\cup\{y_{2n-5}\}, \end{eqnarray}$

(3.9)

$\begin{eqnarray} &d(z_{1}, x) = d(z_{3}, x),&\mbox{ for all }\ x\in X_1\cup\{x_{2n-5}, x_{2n-4}\}, \end{eqnarray}$

(3.10)

$\begin{eqnarray} &d(z_{1}, y) = d(z_{2}, y),&\mbox{ for all }\ y\in Y_1\cup\{y_{2n-5}, y_{2n-4}\}, \end{eqnarray}$

(3.11)

$\begin{eqnarray} &d(a_{1}, x) = d(x_{4n-11}, x),&\mbox{ for all }\ x\in X_1\setminus\{x_1\}, \end{eqnarray}$

(3.12)

$\begin{eqnarray} &d(a_{5}, y) = d(y_{4n-11}, y),&\mbox{ for all }\ y\in Y_1\setminus\{y_1\}, \end{eqnarray}$

(3.13)

$\begin{eqnarray} &d(a_{2}, x) = d(x_{1}, x),&\mbox{ for all }\ x\in X_2\setminus\{x_{2n-4}, x_{4n-11}\}, \end{eqnarray}$

(3.14)

$\begin{eqnarray} &d(a_{4}, y) = d(y_{1}, y),&\mbox{ for all }\ y\in Y_2\setminus\{y_{2n-4}, y_{4n-11}\}, \end{eqnarray}$

(3.15)

$\begin{eqnarray} &d(a_{6}, b) = d(a_{5}, b),&\mbox{ for all }\ b\in B_1\cup B_2\cup \{b_{2n-5}\}\setminus\{b_{1}\}, \end{eqnarray}$

(3.16)

$\begin{eqnarray} &d(a_{6}, c) = d(a_{1}, c),&\mbox{ for all }\ c\in C_1\cup C_2\cup \{c_{2n-5}\}\setminus\{c_{1}\}, \end{eqnarray}$

(3.17)

$\begin{eqnarray} &d(a_{1}, b) = d(x_{2}, b),&\mbox{ for all }\ b\in \{b_1, b_2, b_{4n-12}, b_{4n-11}\}, \end{eqnarray}$

(3.18)

$\begin{eqnarray} &d(a_{5}, c) = d(y_{2}, c),&\mbox{ for all }\ b\in \{c_1, c_2, c_{4n-12}, c_{4n-11}\}. \end{eqnarray}$

(3.19)

The relations of distances of vertices of $C_1\cup\{c_{2n-5}\}$, $C_2\cup\{c_{2n-5}\}$, $B_1\cup\{b_{2n-5}\}$ and $B_2\cup\{b_{2n-5}\}$ are given by:

$d(z_{1}, c) = d(z_{2}, c), \ \ d(z_{1}, c) = d(a_{5}, c), d(z_{2}, c) = d(a_{5}, c) \mbox{ for all }\ c\in C_1\cup\{c_{2n-5}\},$

(3.20)

$d(z_{4}, c) = d(z_{5}, c), \ \ d(z_{4}, c) = d(a_{4}, c), d(z_{5}, c) = d(a_{4}, c) \mbox{ for all }\ c\in C_2\cup\{c_{2n-5}\},$

(3.21)

$d(z_{1}, b) = d(z_{3}, b), \ \ d(z_{1}, b) = d(a_{1}, b), d(z_{3}, b) = d(a_{1}, b) \mbox{ for all }\ b\in B_1\cup\{b_{2n-5}\},$

(3.22)

$d(z_{4}, b) = d(z_{6}, b), \ \ d(z_{4}, b) = d(a_{2}, b), d(z_{6}, b) = d(a_{2}, b) \mbox{ for all }\ b\in B_2\cup\{b_{2n-5}\}.$

(3.23)

The relations of distances of the pair of vertices of $Z_1\cup Z_2$, $A$, $X_1\cup X_2\cup\{x_{2n-5}\}$ and $Y_1\cup Y_2\cup\{y_{2n-5}\}$ are given by:

$d(a_{1}, z) = d(a_{5}, z), d(a_{2}, z) = d(a_{4}, z), \mbox{ for all }\ z\in Z_1\cup Z_2,$

(3.24)

$d(z_{2}, a) = d(z_{3}, a), d(z_{5}, a) = d(z_{6}, a), \mbox{ for all }\ a\in A,$

(3.25)

$d(z_{1}, x) = d(a_{5}, x), d(z_{4}, x) = d(a_{4}, x), \mbox{ for all }\ x\in X_1\cup X_2\cup\{x_{2n-5}\},$

(3.26)

$d(z_{1}, y) = d(a_{1}, y), d(z_{4}, y) = d(a_{2}, y), \mbox{ for all }\ y\in Y_1\cup Y_2\cup\{y_{2n-5}\}.$

(3.27)

The distance between the vertices $b_i\in B_1\cup B_2$ and $c_i\in C_1\cup C_2$ is given as:

$\begin{equation}\label{eq236} d(b_{i}, c_{i}) = \left\{ \begin{array}{ll} 1 &\mbox{for }i\mbox{ is even }, \\ 3 &\mbox{for }i\mbox{ is odd }. \end{array} \right. \end{equation}$

(3.28)

The distance between the vertices $b_i\in B_1\cup B_2$ and $x_i\in X_1\cup X_2$ is given as:

$\begin{equation}\label{eq237} d(x_{i}, b_{i}) = \left\{ \begin{array}{ll} 1 &\mbox{for} i\mbox{ is even }, \\ 3 &\mbox{for }i\mbox{ is odd }. \end{array} \right. \end{equation}$

(3.29)

The distance between the vertices $c_i\in C_1\cup C_2$ and $y_i\in Y_1\cup Y_2$ is given as:

$\begin{equation}\label{eq238} d(y_{i}, c_{i}) = \left\{ \begin{array}{ll} 1 &\mbox{for }i\mbox{ is even }, \\ 3 &\mbox{for }i\mbox{ is odd }. \end{array} \right. \end{equation}$

(3.30)

Lemma 3.1. Let $F_3[n]$ be a fullerene graph shown in Figure 3. Then $3\leq pd(F_{3}[n])\leq4$, where $n\geq5$.

Proof. Let $\{z_{1}, z_{2}, z_{3}\}$ and $\{z_{4}, z_{5}, z_{6}\}$ be the vertices of outer triangles and $\{a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}\}$ be the vertices of outer hexagon of $F_{3}[n]$. Let $\Pi = \{S_{1}, S_{2}, S_{3}, S_{4}\}$, where $S_{1} = \{a_{5}\}$, $S_{2} = \{z_{2}\}$, $S_{3} = \{z_{5}\}$ and $S_{4} = V(F_{3}[n])\setminus \{a_{5}, z_{2}, z_{5}\}$, be a partition of $V(F_{3}[n])$. We show that $\Pi$ is a resolving partition of $F_{3}[n]$ with minimum cardinality. For this we give the representation of each vertex of $F_{3}[n]$ other than $a_{5}, \ z_{2}, \ z_{5}$ with respect to $\Pi$. The representation of vertices of $A$ with respect to $\Pi$ is given by:

$\begin{eqnarray*} r(a_{1}\mid\Pi) = (2, 3, 4, 0),&r(a_{2}\mid\Pi) = (3, 4, 3, 0),&r(a_{3}\mid\Pi) = (2, 5, 2, 0), \\ r(a_{4}\mid\Pi) = (1, 4, 3, 0),&r(a_{6}\mid\Pi) = (1, 2, 5, 0). \end{eqnarray*}$

The representation of vertices of $(Z_1\cup Z_2)\setminus\{z_2, z_5\}$ with respect to $\Pi$ is given by:

$\begin{eqnarray*} r(z_{1}\mid\Pi) = (2, 1, 6, 0),&r(z_{3}\mid\Pi) = (3, 1, 7, 0),&r(z_{4}\mid\Pi) = (3, 6, 1, 0),&r(z_{6}\mid\Pi) = (4, 7, 1, 0). \end{eqnarray*}$

The representation of vertices of $X_1\cup X_2$ with respect to $\Pi$ is given by:

$\begin{equation*} r(x_{i}\mid\Pi) = \left\{ \begin{array}{ll} (3, 2, 5, 0) &\mbox{if }i = 1, \\ (i+2, i+1, i+2, 0) &\mbox{if }2\leq i\leq2n-6, \\ (2n-3, 2n-4, 2n-4, 0) &\mbox{if }i = 2n-5, \\ (4n-i-7, 4n-i-8, 4n-i-9, 0) &\mbox{if }2n-4\leq i\leq4n-12, \\ (4, 5, 2, 0) &\mbox{if }i = 4n-11. \end{array} \right. \end{equation*}$

The representation of vertices of $B_1\cup B_2$ and $C_1\cup C_2$ with respect to $\Pi$ is given by:

$\begin{equation*} r(b_{i}\mid\Pi) = \left\{ \begin{array}{ll} (4, i, i+5, 0) &\mbox{if }i\in \{1, 2\}, \\ (i+2, i, 4n-i-10, 0) &\mbox{if }3\leq i\leq2n-5, \\ (2n-3, 2n-4, 2n-6, 0) &\mbox{if }i = 2n-4, \\ (4n-i-7, 4n-i-7, 4n-i-10, 0) &\mbox{if }2n-3\leq i\leq4n-13, \\ (5, 4n-i-5, 4n-i-10, 0) &\mbox{if }i\in \{4n-12, 4n-11\}. \end{array} \right. \end{equation*}$

$\begin{equation*} r(c_{i}\mid\Pi) = \left\{ \begin{array}{ll} (i+1, i+1, i+5, 0) &\mbox{if }i\in \{1, 2\}, \\ (i+1, i+1, 4n-i-9, 0) &\mbox{if }3\leq i\leq2n-5, \\ (2n-4, 2n-3, 2n-5, 0) &\mbox{if }i = 2n-4, \\ (4n-i-8, 4n-i-6, 4n-i-9, 0) &\mbox{if }2n-3\leq i\leq4n-13, \\ (4n-i-8, 4n-i-5, 4n-i-9, 0) &\mbox{if }i\in \{4n-12, 4n-11\}. \end{array} \right. \end{equation*}$

The representation of vertices of $Y_1\cup Y_2$ with respect to $\Pi$ is given by:

$\begin{equation*} r(y_{i}\mid\Pi) = \left\{ \begin{array}{ll} (1, 3, 5, 0) &\mbox{if }i = 1, \\ (i, i+2, i+3, 0) &\mbox{if }2\leq i\leq2n-6, \\ (2n-5, 2n-3, 2n-3, 0) &\mbox{if }i = 2n-5, \\ (4n-i-9, 4n-i-7, 4n-i-8, 0) &\mbox{if }2n-4\leq i\leq4n-12, \\ (2, 5, 3, 0) &\mbox{if }i = 4n-11. \end{array} \right. \end{equation*}$

It is easily seen that the representation of each vertex with respect to $\Pi$ is distinct. This shows that $\Pi$ is a resolving partition of $F_{3}[n]$. Thus $pd(F_{3}[n])\leq 4$. Also by Lemma 1.2, we have $pd(F_{3}[n])\geq3$.

Suppose that there exists a partition $\widetilde{\Pi}$ of $F_3[n]$, $n\geq5$, such that $|\widetilde{\Pi}| = 3$. Let $\widetilde{\Pi} = \{\widetilde{S}_{1}, \widetilde{S}_{2}, \widetilde{S}_{3}\}$. Consider the following cases:

Case Ⅰ: If two partitioning sets of $\widetilde{\Pi}$ are subsets of either $Z_1$ or $Z_2$ then from (3.6) and (3.7), it is clear that either $r(z_5\mid \widetilde{\Pi}) = r(z_6\mid\widetilde{\Pi})$ or $r(z_2\mid\widetilde{\Pi}) = r(z_3\mid\widetilde{\Pi})$.

Case Ⅱ: If two partitioning sets of $\widetilde{\Pi}$ are subsets of either $A$ or $X_{1}$ or $B_{1}$ then (3.25), (3.2), (3.10) and (3.22) implies that either $r(z_{2}\mid\widetilde{\Pi}) = r(z_{3}\mid\widetilde{\Pi})$ or $r(a_{4}\mid\widetilde{\Pi}) = r(y_{1}\mid\widetilde{\Pi})$ or $r(z_{1}\mid\widetilde{\Pi}) = r(z_{3}\mid\widetilde{\Pi})$.

Case Ⅲ: If two partitioning sets of $\widetilde{\Pi}$ are subsets of either $Y_{1}$ or $C_{1}$ then (3.4), (3.11) and (3.20) implies that either $r(z_{1}\mid\widetilde{\Pi}) = r(z_{2}\mid\widetilde{\Pi})$ or $r(a_{2}\mid\widetilde{\Pi}) = r(x_{1}\mid\widetilde{\Pi})$ or $r(z_{1}\mid\widetilde{\Pi}) = r(a_{5}\mid\widetilde{\Pi})$.

Case Ⅳ: If two partitioning sets of $\widetilde{\Pi}$ are subsets of either $X_{2}$ or $B_{2}$ then from (3.3), (3.8) and (3.23) we obtain either $r(z_{4}\mid\widetilde{\Pi}) = r(z_{6}\mid\widetilde{\Pi})$ or $r(a_{5}\mid\widetilde{\Pi}) = r(y_{4n-11}\mid\widetilde{\Pi})$ or $r(z_{4}\mid\widetilde{\Pi}) = r(a_{2}\mid\widetilde{\Pi})$.

Case Ⅴ: If two partitioning sets of $\widetilde{\Pi}$ are subsets of either $Y_{2}$ or $C_{2}$ then from (3.4), (3.9) and (3.21) we obtain either $r(z_{4}\mid\widetilde{\Pi}) = r(z_{5}\mid\widetilde{\Pi})$ or $r(a_{1}\mid\widetilde{\Pi}) = r(x_{4n-11}\mid\widetilde{\Pi})$.

Case Ⅵ: If two partitioning sets of $\widetilde{\Pi}$ are subsets of either $Z_1\cup Z_2$ or $X_1\cup X_2$ or $Y_1\cup Y_2$ then from (3.24), (3.26) and (3.27), we can easily be seen that either $r(a_{1}\mid\widetilde{\Pi}) = r(a_{5}\mid\widetilde{\Pi})$ or $r(z_{1}\mid\widetilde{\Pi}) = r(a_{5}\mid\widetilde{\Pi})$ or $r(z_{1}\mid\widetilde{\Pi}) = r(a_{1}\mid\widetilde{\Pi})$.

Case Ⅶ: If two partitioning sets of $\widetilde{\Pi}$ are subsets of $B_1\cup B_2$ then from (3.16), (3.18), (3.22) and (3.23) we see that some either $a_{i}$, $a_{j}$ or $a_{i}$, $x_{j}$ or $z_{i}$, $z_{j}$ have same representations with respect to $\widetilde{\Pi}$.

Case Ⅷ: If two partitioning sets of $\widetilde{\Pi}$ are subsets of $C_1\cup C_2$ then from (3.17), and (3.19)-(3.21) we conclude that either $a_{i}$, $a_{j}$ or $a_{i}$, $x_{j}$ or $z_{i}$, $z_{j}$ have same representations with respect to $\widetilde{\Pi}$.

Case Ⅸ: If two partitioning sets of $\widetilde{\Pi}$ are subsets of either $(X_{1}\cup B_{1}\cup \{x_{2n-5}, b_{2n-5}\})$ or $(X_{2}\cup B_{2}\cup \{x_{2n-5}, b_{2n-5}\})$ then from (3.8), (3.10), (3.22) and (3.23) it is clear that either $r(z_{1}\mid\widetilde{\Pi}) = r(z_{3}\mid\widetilde{\Pi})$ or $r(z_{4}\mid\widetilde{\Pi}) = r(z_{6}\mid\widetilde{\Pi})$.

Case Ⅹ: If two partitioning sets of $\widetilde{\Pi}$ are subsets of either $(Y_{1}\cup C_{1}\cup\{y_{2n-5}, c_{2n-5}\})$ or $(Y_{2}\cup C_{2}\cup \{y_{2n-5}, c_{2n-5}\})$ then (3.9), (3.11), (3.20) and (3.21) implies that either $r(z_{1}\mid\widetilde{\Pi}) = r(z_{2}\mid\widetilde{\Pi})$ or $r(z_{4}\mid\widetilde{\Pi}) = r(z_{5}\mid\widetilde{\Pi})$.

Case Ⅺ: Also If two partite sets of $\widetilde{\Pi}$ are subsets of $(C_1\cup C_2\cup B_1\cup B_2)$ then there exists some $x_{i}\in X_1\cup X_2$ and $y_{j}\in Y_1\cup Y_2$ with same representations.

Case Ⅻ: If two partitioning sets of $\widetilde{\Pi}$ are subsets of either $(X_1\cup X_2\cup C_{1}\cup \{c_{2n-5}\})$ or $(X_1\cup X_2\cup C_{2}\cup \{c_{2n-5}\})$ then by (3.20), (3.21)and (3.26) we obtain either $r(z_{1}\mid\widetilde{\Pi}) = r(a_{5}\mid\widetilde{\Pi})$ or $r(z_{4}\mid\widetilde{\Pi}) = r(a_{4}\mid\widetilde{\Pi})$.

Case ⅩⅢ: If two partitioning sets of $\widetilde{\Pi}$ are subsets of either $(Y_1\cup Y_2\cup B_{2}\cup \{b_{2n-5}\})$ or $(Y_1\cup Y_2\cup B_{1}\cup\{c_{2n-5}\})$ then from (3.22), (3.23) and (3.27) either $r(z_{4}\mid\widetilde{\Pi}) = r(a_{2}\mid\widetilde{\Pi})$ or $r(z_{1}\mid\widetilde{\Pi}) = r(a_{1}\mid\widetilde{\Pi})$.

Note that there are total $2047$ possible combinations of subsets of vertex set of $F_3[n]$ shown in (3.1), we guess that no two partite sets of $\widetilde{\Pi}$ can be subsets of combinations of $X_1, \ X_2, \ Y_1, \ Y_2, \ Z_1, \ Z_2, \ A, \ B_1, \ B_2, \ C_1$ and $C_2$. Thus, we have the following conjecture.

Conjecture 3.1. The partition dimension of $F_3[n]$, $n\geq5$, is $4$.

Next, we give the conjecture on the partition dimension of fullerene graph $F_4[n]$. The set of vertices of $F_4[n]$ is divided into the following sets:

$\begin{equation}\label{qs2} \begin{array}{l l l} A = \{a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7}, a_{8}\},&X = \{x_{i}\mid 1\leq i\leq 6n-1\},&B = \{b_{i}\mid 1\leq i\leq 6n-1\}, \\ Y = \{y_{i}\mid 1\leq i\leq 6n-3\},&Z = \{z_{i}\mid 1\leq i\leq 6n-3\}. \end{array} \end{equation}$

(3.31)

The relations of distances of the vertices of $F_4[n]$ are as follows:

$\begin{eqnarray}\label{eq31} &d(x_{1}, a) = d(b_{1}, a),&\mbox{for all }\ a\in A\setminus\{a_7, a_8\}, \end{eqnarray}$

(3.32)

$\begin{eqnarray} &d(x_{6n-1}, a) = d(b_{6n-1}, a),&\mbox{for all }\ a\in A\setminus\{a_3, a_4\}, \end{eqnarray}$

(3.33)

$\begin{eqnarray} &d(a_{2}, x) = d(a_{4}, x),&\mbox{for all }\ x\in X\setminus\{x_1\}, \end{eqnarray}$

(3.34)

$\begin{eqnarray} &d(a_{4}, y) = d(b_{2}, y),&\mbox{for all }\ y\in Y\setminus\{y_1\}, \end{eqnarray}$

(3.35)

$\begin{eqnarray} &d(a_{3}, z) = d(x_{2}, z),&\mbox{for all }\ z\in Z\setminus\{z_1\}, \end{eqnarray}$

(3.36)

$\begin{eqnarray} &d(a_{2}, b) = d(a_{3}, b),&\mbox{for all }\ b\in B\setminus\{b_1\}, \end{eqnarray}$

(3.37)

$\begin{eqnarray} &d(a_{1}, x) = d(b_{1}, x),&\mbox{for all }\ x\in X, \end{eqnarray}$

(3.38)

$\begin{eqnarray} &d(a_{1}, b) = d(x_{1}, b),&\mbox{for all }\ b\in B, \end{eqnarray}$

(3.39)

$\begin{eqnarray} &\ d(a_{7}, z) = d(x_{6n-2}, z),&\mbox{for all }\ z\in Z\setminus\{z_{6n-3}\}, \end{eqnarray}$

(3.40)

$\begin{eqnarray} &\ d(a_{8}, y) = d(b_{6n-2}, y),&\mbox{for all }\ y\in Y\setminus\{y_{6n-3}\}. \end{eqnarray}$

(3.41)

The relations of distances of the vertices of $Z$ and $F_4[n]$ are as follows:

$\begin{eqnarray} d(a_{1}, z) = d(x_{1}, z),&d(a_{4}, z) = d(b_{2}, z),&d(a_{8}, z) = d(b_{6n-2}, z),&d(a_{2}, z) = d(a_{3}, z). \end{eqnarray}$

(3.42)

The relations of distances of the vertices of $Y$ and $F_4[n]$ are as follows:

$\begin{eqnarray} d(a_{1}, y) = d(b_{1}, y),&d(a_{3}, y) = d(x_{2}, y),&d(a_{7}, y) = d(x_{6n-2}, y),&d(a_2, y) = d(a_4, y). \end{eqnarray}$

(3.43)

Lemma 3.2. Let $F_4[n]$ be a fullerene graph shown in Figure 4. Then $3\leq pd(F_{4}[n])\leq4$, where $n\geq1$.

Proof. Let $\Pi = \{S_{1}, S_{2}, S_{3}, S_{4}\}$, where $S_{1} = \{a_{3}\}$, $S_{2} = \{a_{7}\}$, $S_{3} = \{a_{8}\}$ and $S_{4} = V(F_{4}[n])\setminus \{a_{3}, a_{7}, a_{8}\}$, be a partition of $V(F_{4}[n])$. We show that $\Pi$ is a resolving partition of $F_{4}[n]$ with minimum cardinality. The representation of each vertex of $A$ other than $a_{3}, \ a_{7}, \ a_{8}$ with respect to $\Pi$ is given as:

$\begin{eqnarray*} r(a_{1}\mid\Pi) = (2, 6n, 6n, 0),&r(a_{2}\mid\Pi) = (1, 6n-1, 6n-1, 0),&r(a_{4}\mid\Pi) = (1, 6n-1, 6n-2, 0), \\ r(a_{5}\mid\Pi) = (6n, 2, 2, 0),&r(a_{6}\mid\Pi) = (6n-1, 1, 1, 0). \end{eqnarray*}$

The representation of each vertex of $X$ with respect to $\Pi$ is given as:

$\begin{equation*} r(x_{i}\mid\Pi) = \left\{ \begin{array}{ll} (3, 6n-1, 6n, 0) &\mbox{if }i = 1, \\ (i, 6n-i, 6n+1-i, 0) &\mbox{if }2\leq i\leq 6n-2, \\ (6n-1, 3, 3, 0) &\mbox{if }i = 6n-1. \end{array} \right. \end{equation*}$

The representation of each vertex $B$ with respect to $\Pi$ is given as:

$\begin{equation*} r(b_{i}\mid\Pi) = \left\{ \begin{array}{ll} (3, 6n, 6n-1, 0) &\mbox{if }i = 1, \\ (i-1, 6n+1-i, 6n-i, 0) &\mbox{if }2\leq i\leq 6n-2, \\ (6n, 3, 3, 0) &\mbox{if }i = 6n-1. \end{array} \right. \end{equation*}$

The representation of each vertex of $Y$ and $Z$ with respect to $\Pi$ is given as:

$\begin{eqnarray*} r(y_{i}\mid\Pi) = (i, 6n-2-i, 6n-1-i, 0) &\mbox{if }1 \leq i\leq 6n-3, \\ r(z_{i}\mid\Pi) = (i+1, 6n-1-i, 6n-2-i, 0) &\mbox{if }1 \leq i\leq 6n-3. \end{eqnarray*}$

From above representations of vertices with respect to $\Pi$ it can be easily seen that representations are distinct. This implies that $\Pi$ is a resolving partition of $F_{4}[n]$. Thus $pd(F_{4}[n])\leq 4$. Also by Lemma 1.2, we note that $pd(F_{4}[n])\geq3$.

Suppose that there exists partition $\widetilde{\Pi}$ of $F_4[n]$, $n\geq1$, such that $|\widetilde{\Pi}| = 3$. Let $\widetilde{\Pi} = \{\widetilde{S}_{1}, \widetilde{S}_{2}, \widetilde{S}_{3}\}$. Consider the following cases:

Case Ⅰ: If two partitioning sets of $\widetilde{\Pi}$ are subsets of $X$ then by (3.38), we have $r(a_{1}|\widetilde{\Pi}) = r(b_{1}|\widetilde{\Pi})$ and if two partitioning sets of $\widetilde{\Pi}$ are subsets of $Y$ then by (3.43), we have $r(a_{1}\mid\widetilde{\Pi}) = r(b_{1}\mid\widetilde{\Pi})$.

Case Ⅱ: If two partitioning sets of $\widetilde{\Pi}$ are subsets of $A$ except $\{a_{7}, a_{8}\}$ then by (3.32), we have $r(b_{1}\mid\widetilde{\Pi}) = r(x_{1}\mid\widetilde{\Pi})$. If two partitioning sets of $\widetilde{\Pi}$ are subsets of $A$ except $\{a_{3}, a_{4}\}$ then by (3.33), we have and $r(x_{6n-1}\mid\widetilde{\Pi}) = r(b_{6n-1}\mid\widetilde{\Pi})$.

Case Ⅲ: If two partitioning sets of $\widetilde{\Pi}$ are subsets of either $B$ or $Z$ then by (3.39) and (3.42), we have $r(a_{1}\mid\widetilde{\Pi}) = r(x_{1}\mid\widetilde{\Pi})$.

Case Ⅳ: Similarly, from equations (3.38) and (3.43) we observe that if two partitioning sets of $\widetilde{\Pi}$ are subsets of $X\cup Y$ then $r(a_{1}\mid\widetilde{\Pi}) = r(b_{1}\mid\widetilde{\Pi})$.

Case Ⅴ: If two partitioning sets of $\widetilde{\Pi}$ are subsets of $B\cup Z$ then from (3.39) and (3.42), we see that $r(a_{1}\mid\widetilde{\Pi}) = r(x_{1}\mid\widetilde{\Pi})$.

Case Ⅵ: We notice that if two partitioning sets of $\widetilde{\Pi}$ are subsets of $Y\cup Z$ then there exists either some $a_{i}, x_{j}$ or $a_{i}, b_{j}$ with same representations with respect to $\widetilde{\Pi}$.

Note that there are total $31$ possible combinations of subsets of vertex set of $F_4[n]$, shown in (3.31). Thus because of unique structural properties of $F_{4}[n]$, we can observe that no two partitioning sets of $\widetilde{\Pi}$ can be subsets of combinations of $A, \ B, \ X, \ Y$ and $Z$. Thus, we have the following conjecture.

Conjecture 3.2. The partition dimension of $F_4[n]$ is $4$.

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Acknowledgments

This research is supported by the Higher Education Commission of Pakistan under grant No. 20-3067/NRPU /R & D/HEC/12.

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Conflict of interest

All authors declare no conflicts of interest in this paper.

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