On partition dimension of fullerene graphs

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Abstract

Let $G = (V(G), E(G))$ be a connected graph and $\Pi = \{S_{1}, S_2, \dots, S_{k}\}$ be a $k$-partition of $V(G)$. The representation $r(v|\Pi)$ of a vertex $v$ with respect to $\Pi$ is the vector $(d(v, S_{1}), d(v, S_2), \dots, d(v, S_{k}))$, where $d(v, S_{i}) = \min\{d(v, s_{i})\mid s_{i}\in S_{i}\}$. The partition $\Pi$ is called a resolving partition of $G$ if $r(u|\Pi)\neq r(v|\Pi)$ for all distinct $u, v\in V(G)$. The partition dimension of $G$, denoted by $pd(G)$, is the cardinality of a minimum resolving partition of $G$. In this paper, we calculate the partition dimension of two $(4, 6)$-fullerene graphs. We also give conjectures on the partition dimension of two $(3, 6)$-fullerene graphs.

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