2018/08/29 by Libby Taylor, Taylor, Libby
Mathematics · Computer Science · #Graph theory and applications #Advanced Graph Theory Research #Advanced Combinatorial Mathematics
paper · pdf · doi:10.48550/arxiv.1808.09927
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 analogue of Bernardi's results, providing a canonical 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.