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The Sandpile Group of Polygon Rings and Twisted Polygon Rings

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Abstract

Let \(C_{k_1}, \ldots , C_{k_n}\) be cycles with \(k_i\ge 2\) vertices (\(1\le i\le n\)). By attaching these n cycles together in a linear order, we obtain a graph called a polygon chain. By attaching these n cycles together in a cyclic order, we obtain a graph, which is called a polygon ring if it can be embedded on the plane; and called a twisted polygon ring if it can be embedded on the Möbius band. It is known that the sandpile group of a polygon chain is always cyclic. Furthermore, there exist edge generators. In this paper, we not only show that the sandpile group of any (twisted) polygon ring can be generated by at most three edges, but also give an explicit relation matrix among these edges. So we obtain a uniform method to compute the sandpile group of arbitrary (twisted) polygon rings, as well as the number of spanning trees of (twisted) polygon rings. As an application, we compute the sandpile groups of several infinite families of polygon rings, including some that have been done before by ad hoc methods, such as, generalized wheel graphs, ladders and Möbius ladders.

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All data generated or analysed during this study are included in this published article.

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Acknowledgements

We thank the referee for his helpful comments, which provide the content of Remark 6.

Funding

This work was done while the H. Y. Chen visited The Simon Fraser University. The hospitality of the hosting institution is greatly acknowledged. The visit was funded by the Fujian Provincial Education Department. H. Y. Chen was supported by the National Natural Science Foundation of China (Grant numbers 11771181, 12071180). B. Mohar was supported in part by the NSERC Discovery Grant R611450 (Canada), by the Canada Research Chairs program, and by the Research Project J1-8130 of ARRS (Slovenia). On leave from IMFM, Department of Mathematics, University of Ljubljana.

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Authors and Affiliations

  1. The School of Sciences, Jimei University, Xiamen, Fujian, China

    Haiyan Chen

  2. Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada

    Bojan Mohar

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  1. Haiyan Chen
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Correspondence to Haiyan Chen.

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Chen, H., Mohar, B. The Sandpile Group of Polygon Rings and Twisted Polygon Rings. Graphs and Combinatorics 38, 113 (2022). https://doi.org/10.1007/s00373-022-02514-x

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