An Upper Bound for the Clar Number of Fullerene Graphs

A fullerene graph is a three-regular and three-connected plane graph exactly 12 faces of which are pentagons and the remaining faces are hexagons. Let Fn be a fullerene graph with n vertices. The Clar number c(Fn) of Fn is the maximum size of sextet patterns, the sets of disjoint hexagons which are all M-alternating for a perfect matching (or Kekulé structure) M of Fn. A sharp upper bound of Clar number for any fullerene graphs is obtained in this article: $$c(F_n)\leqslant \lfloor \frac {n-12} 6\rfloor$$. Two famous members of fullerenes C60 (Buckministerfullerene) and C70 achieve this upper bound. There exist infinitely many fullerene graphs achieving this upper bound among zigzag and armchair carbon nanotubes.

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