Fullerene graphs have exponentially many perfect matchings

Abstract

A fullerene graph is a planar cubic 3-connected graph with only pentagonal and hexagonal faces. We show that fullerene graphs have exponentially many perfect matchings.Comment: 7 pages, 3 figure

Program Fullerene: A software package for constructing and analyzing structures of regular fullerenes

“…However, the assignment of double bonds (Kekul e structures) to fullerenes is equivalent in graph theory to finding perfect matchings, [18] of which fullerene graphs have exponentially many. [19] While the exponential theoretical lower bound for the number of perfect matchings in fullerenes only kicks in at C 380 , the rapid growth in their numbers starts much sooner: The IPR fullerene I h -C 60 (I h -C 70 ) has as many as 12,500 (52,168) Kekul e structures, of which 158 (2780) are nonisomorphic. [20,21] As this number grows, searching through all Kekul e structures for the most stable one becomes impractical, and force fields for fullerenes should, therefore, not be designed with double bonds in mind.…”

Exponentially many perfect matchings in cubic graphs

“…We refer the reader to [11] for an exposition of this connection and of an elegant proof of Gurvits generalizing Schrijver's result. For fullerene graphs, a class of planar cubic graphs for which the conjecture relates to molecular stability and aromaticity of fullerene molecules, the problem was settled by Kardoš, Král', Miškuf and Sereni [9]. Chudnovsky and Seymour recently proved the conjecture for all cubic bridgeless planar graphs [1].…”

Long cycles in fullerene graphs

“…Proof. In [11], it was proven that for every fullerene graph G on n vertices there is a decomposition of its edges into three perfect matchings M 1 , M 2 , M 3 , such that at least one of them, say M 1 , has at least n−380 61 resonant hexagons. Then G ′ = M 2 ∪ M 3 is a 2-factor of the graph G having at least n−380 61 shifts.…”