Abstract
A fullerene is a 3-regular plane graph with only pentagonal and hexagonal faces. The Fries and Clar number of a fullerene are two related parameters, and the Clar number is less understood. We introduce the Clar Structure of a fullerene, a decomposition designed to compute the Clar number for classes of fullerenes. We also settle an open question with a counterexample: we prove that the Clar and Fries number of a fullerene cannot always be obtained with the same Kekule structure.
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