A fully benzenoid system has a unique maximum cardinality resonant set

Abstract

A benzenoid system is a 2-connected plane graph such that its each inner face is a regular hexagon of side length 1. A benzenoid system is Kekulean if it has a perfect matching. Let P be a set of hexagons of a Kekulean benzenoid system B. The set P is called a resonant set of B if the hexagons in P are pair-wise disjoint and the subgraph B?P (obtained by deleting from B the vertices of the hexagons in P) is either empty or has a perfect matching. It was shown (Gutman in Wiss. Z. Thechn. Hochsch. Ilmenau 29:57---65, 1983; Zheng and Chen in Graphs Comb. 1:295---298, 1985) that for every maximum cardinality resonant set P of a Kekulean benzenoid system B, the subgraph B?P is either empty or has a unique perfect matching. A Kekulean benzenoid system B is said to be fully benzenoid if there exists a maximum cardinality resonant set P of B, such that the subgraph B?P is empty. It is shown that a fully benzenoid system has a unique maximum cardinality resonant set, a well-known statement that, so far, has remained without a rigorous proof.