Kekuléan benzenoids

Abstract

A Kekulé 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 Kekulé structure; however, this is not true for benzenoids. In this paper, we develop methods for deciding whether or not a given benzenoid admits a Kekulé structure by constructing Kekulé structures that have a high density of benzene rings. The benzene rings of the Kekulé 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 Kekulé 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 Kekulé structures for $\Gamma$ and the set of benzene rings giving the Clar number is called a Clar set. Our method of constructing Kekulé structures for benzenoids generally gives good estimates for the Clar and Fries numbers, often the exact values.