Resonant hexagons in fullerene graphs

A fullerene graph is a 3-connected plane cubic graph in which every face is pentagonal or hexagonal. A set of hexagons \(\mathcal {H}\) $\mathcal{H}$ of G is called a resonant pattern if there exists a perfect matching M of G such that exactly three edges of H is contained in M for each member H of \(\mathcal {H}\) $\mathcal{H}$ . In this paper we prove for any natural number k that almost all of the family of k disjoint hexagons are resonant patterns in sufficiently large fullerene graphs.