AIMS Mathematics, 2018, 3(3): 343-352. doi: 10.3934/Math.2018.3.343

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On partition dimension of fullerene graphs

School of Natural Sciences, National University of Sciences and Technology, H-12 Islamabad,Pakistan

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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