Issue 
Wuhan Univ. J. Nat. Sci.
Volume 28, Number 3, June 2023



Page(s)  192  200  
DOI  https://doi.org/10.1051/wujns/2023283192  
Published online  13 July 2023 
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
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$\text{}t$, 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 ($\mathrm{0}\le t<s,\text{}t\equiv r\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{2}$). In this paper, the counting formula of the perfect matchings in $G=\left(n+\mathrm{1,4},t\right)\text{}$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 $G=\left(n+\mathrm{1,4},t\right)$.
Key words: perfect matching / (3,6)fullerene graph / recurrence relation / counting formula
Biography: YANG Rui, female, Ph. D., Associate professor, research direction: graph theory and its application. Email: yangrui@hpu.edu.cn
Fundation item: Supported by National Natural Science Foundation of China (11801148, 11801149 and 11626089) and the Foundation for the Doctor of Henan Polytechnic University (B2014060)
© Wuhan University 2023
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