Counting Clar structures of (4, 6)-fullerenes

Abstract

A (4, 6)-fullerene graph G is a plane cubic graph whose faces are squares and hexagons. A resonant set of G is a set of pairwise disjoint faces of G such that the boundaries of such faces are M-alternating cycles for a perfect matching M of G. A resonant set of G is referred to as sextet pattern whenever it only includes hexagonal faces. It was shown that the cardinality of a maximum resonant set of G is equal to the maximum forcing number. In this article, we obtain an expression for the cardinality of a maximum sextet pattern (or Clar formula) of G, which is less than the cardinality of a maximum resonant set of G by one or two. Moreover we mainly characterize all the maximum sextet patterns of G. Via such characterizations we get a formula to count maximum sextet patterns of G only depending on the order of G and the number of fixed subgraphs J1 of G, where the count equals the coefficient of the term with largest degree in its sextet polynomial. Also the number of Clar structures of G is obtained.