Clar and Fries numbers for benzenoids

A Kekule structure of a benzenoid or a fullerene \(\Gamma \) is a set of edges \(K\) such that each vertex of \(\Gamma \) is incident with exactly one edge in \(K\). The set of faces in \(\Gamma \) that have exactly three edges in \(K\) are called the benzene faces of \(K\). The Fries number of \(\Gamma \) is the maximum number of benzene faces over all possible Kekule structures for \(\Gamma \). The Clar number is the maximum number of independent benzene faces over all possible Kekule structures for \(\Gamma \). It is often assumed, but never proved, that some set of independent benzene faces giving the Clar number is a subset of a set of benzene faces giving the Fries number. In Hartung (The Clar structure of fullerenes, Ph.D. Dissertation. Syracuse University, 2012) it is shown that this assumption is false for a large class of fullerenes. In this paper, we prove that this assumption is valid for a large a class of benzenoids.