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Drawing 4-Pfaffian graphs on the torus

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Abstract

We say that a graph G is k-Pfaffian if the generating function of its perfect matchings can be expressed as a linear combination of Pfaffians of k matrices corresponding to orientations of G. We prove that 3-Pfaffian graphs are 1-Pfaffian, 5-Pfaffian graphs are 4-Pfaffian and that a graph is 4-Pfaffian if and only if it can be drawn on the torus (possibly with crossings) so that every perfect matching intersects itself an even number of times. We state conjectures and prove partial results for k>5.

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Author notes

  1. Serguei Norine

    Present address: Department of Mathematics, Princeton University, Princeton, NJ, 08540-1000, USA

Authors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia, 30332, USA

    Serguei Norine

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  1. Serguei Norine
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Correspondence to Serguei Norine.

Additional information

The author was supported in part by NSF under Grant No. DMS-0200595 and DMS-0701033.

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Norine, S. Drawing 4-Pfaffian graphs on the torus. Combinatorica 29, 109–119 (2009). https://doi.org/10.1007/s00493-009-2354-0

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  • DOI: https://doi.org/10.1007/s00493-009-2354-0

Mathematics Subject Classification (2000)