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Kasteleyn cokernels and perfect matchings on planar bipartite graphs

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Abstract

The determinant method of Kasteleyn gives a method of computing the number of perfect matchings of a planar bipartite graph. In addition, results of Bernardi exhibit a bijection between spanning trees of a planar bipartite graph and elements of its Jacobian. In this paper, we explore an adaptation of Bernardi’s results, providing a simply transitive group action of the Kasteleyn cokernel of a planar bipartite graph on its set of perfect matchings, when the planar bipartite graph in question is of the form \(G^+\), as defined by Kenyon, Propp and Wilson.

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Acknowledgements

The author would like to thank Matt Baker for suggesting the problem and for many helpful conversations. She would also like to thank Marcel Celaya and an anonymous referee for helpful feedback on an earlier version of the paper.

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  1. Stanford University, Stanford, USA

    Libby Taylor

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  1. Libby Taylor
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Correspondence to Libby Taylor.

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Taylor, L. Kasteleyn cokernels and perfect matchings on planar bipartite graphs. J Algebr Comb 57, 727–737 (2023). https://doi.org/10.1007/s10801-022-01186-3

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  • DOI: https://doi.org/10.1007/s10801-022-01186-3

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