On the role of hypercubes in the resonance graphs of benzenoid graphs

Abstract

The resonance graph R ( B ) of a benzenoid graph B has the perfect matchings of B as vertices, two perfect matchings being adjacent if their symmetric difference forms the edge set of a hexagon of B. A family P of pair-wise disjoint hexagons of a benzenoid graph B is resonant in B if B – P contains at least one perfect matching, or if B – P is empty. It is proven that there exists a surjective map f from the set of hypercubes of R ( B ) onto the resonant sets of B such that a k-dimensional hypercube is mapped into a resonant set of cardinality k.