Mathematics

CLC number: O157.6

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

School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454003, Henan, China

Received: 20 November 2022

Abstract

A $\left(\mathrm{3,6}\right)$-fullerene is a connected cubic plane graph whose faces are only triangles and hexagons, and has the connectivity $\mathrm{2}$ or $\mathrm{3}$. The $\left(\mathrm{3,6}\right)$-fullerenes with connectivity $\mathrm{2}$ are the tubes consisting of $l$ concentric hexagonal layers such that each layer consists of two hexagons, capped on each end by two adjacent triangles, denoted by $T_l\left(l\ge \mathrm{1}\right)$. A $\left(\mathrm{3,6}\right)$-fullerene $T_l$ with $n$ vertices has exactly ${\mathrm{2}}^{\frac{n}{\mathrm{4}}}+\mathrm{1}$ perfect matchings. The structure of a $\left(\mathrm{3,6}\right)$-fullerene $G$ with connectivity $\mathrm{3}$ can be determined by only three parameters $r$, $s$ and$t$, thus we denote it by $G=\left(r,s,t\right)$, where $r$ is the radius (number of rings), $s$ is the size (number of spokes in each layer, $s\ge \mathrm{4}$, $s$ is even), and $t$ is the torsion ($\mathrm{0}\le t<s,t\equiv r\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{2}$). In this paper, the counting formula of the perfect matchings in $G=\left(n+\mathrm{1,4},t\right)$is given, and the number of perfect matchings is obtained. Therefore, the correctness of the conclusion that every bridgeless cubic graph with $p$ vertices has at least ${\mathrm{2}}^{\frac{p}{\mathrm{3656}}}$ perfect matchings proposed by Esperet et al is verified for $\left(\mathrm{3,6}\right)$-fullerene $G=\left(n+\mathrm{1,4},t\right)$.