KEKULE PATTERNS AND CLAR PATTERNS IN BIPARTITE PLANE GRAPHS

Let G be a finite bipartite plane graph. In a chemical context, a set of pairwise disjoint edges that cover all vertices of G (i.e., a perfe ct matching of G) is called a Kekule pattern of G, and a set of pairwi se disjoint cells of G such that the deletion of all vertices incident to these cells results in a graph that has a Kekule pattern, or is em pty, is called a Clar pattern of G. Let k(G) and c(G) denote the numbe r of Kekule patterns and of Clar patterns of G, respectively. It is sh own that k(G) is not smaller than c(G) and that equality holds if G is an outerplane graph. This result generalizes a well known proposition of the theory of benzenoid hydrocarbons; the proof uses a new idea.