An upper bound for the Clar number of fullerene graphs. (English) Zbl 1213.05259

Summary: 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 \(F_n\) be a fullerene graph with \(n\) vertices. The Clar number \(c(F_n)\) of \(F_n\) 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 \(F_n\). A sharp upper bound of Clar number for any fullerene graphs is obtained in this article: \(c(F_n)\leq\lfloor\frac{n-12}{6}\rfloor\). Two famous members of fullerenes \(C_{60}\) (Buckministerfullerene) and \(C_{70}\) achieve this upper bound. There exist infinitely many fullerene graphs achieving this upper bound among zigzag and armchair carbon nanotubes.


05C90 Applications of graph theory
92E10 Molecular structure (graph-theoretic methods, methods of differential topology, etc.)
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