Kekuléan benzenoids

Abstract

A Kekule structure for 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 \(K\), i.e. a perfect matching. All fullerenes admit a Kekule structure; however, this is not true for benzenoids. In this paper, we develop methods for deciding whether or not a given benzenoid admits a Kekule structure by constructing Kekule structures that have a high density of benzene rings. The benzene rings of the Kekule structure \(K\) are the faces in \(\Gamma \) that have exactly three edges in \(K\). The Fries number of \(\Gamma \) is the maximum number of benzene rings over all possible Kekule structures for \(\Gamma \) and the set of benzene rings giving the Fries number is called a Fries set. The Clar number is the maximum number of independent benzene rings over all possible Kekule structures for \(\Gamma \) and the set of benzene rings giving the Clar number is called a Clar set. Our method of constructing Kekule structures for benzenoids generally gives good estimates for the Clar and Fries numbers, often the exact values.