The Number of Perfect Matchings in (3,6)-Fullerene

Abstract

A [see formula in PDF]-fullerene is a connected cubic plane graph whose faces are only triangles and hexagons, and has the connectivity [see formula in PDF] or [see formula in PDF]. The [see formula in PDF]-fullerenes with connectivity [see formula in PDF] are the tubes consisting of [see formula in PDF] concentric hexagonal layers such that each layer consists of two hexagons, capped on each end by two adjacent triangles, denoted by [see formula in PDF]. A [see formula in PDF]-fullerene [see formula in PDF] with [see formula in PDF] vertices has exactly [see formula in PDF] perfect matchings. The structure of a [see formula in PDF]-fullerene [see formula in PDF] with connectivity [see formula in PDF] can be determined by only three parameters [see formula in PDF], [see formula in PDF] and[see formula in PDF], thus we denote it by [see formula in PDF], where [see formula in PDF] is the radius (number of rings), [see formula in PDF] is the size (number of spokes in each layer, [see formula in PDF], [see formula in PDF] is even), and [see formula in PDF] is the torsion ([see formula in PDF]). In this paper, the counting formula of the perfect matchings in [see formula in PDF]is given, and the number of perfect matchings is obtained. Therefore, the correctness of the conclusion that every bridgeless cubic graph with [see formula in PDF] vertices has at least [see formula in PDF] perfect matchings proposed by Esperet et al is verified for [see formula in PDF]-fullerene [see formula in PDF].