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 $\mathcal{P}$ of pair-wise disjoint hexagons of a benzenoid graph B is resonant in B if $B–\mathcal{P}$ contains at least one perfect matching, or if $B–\mathcal{P}$ 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.