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Exponential Lower Bounds for the Pfaffian Number of Graphs
Galluccio--Loebl and Tesler showed that the perfect-matching polynomial of a graph embedded in an orientable surface of genus $g$ can be written as a linear combination of at most $4^g$ Pfaffians. We show that, in general, exponentially many Pfaffians are necessary. More precisely, for every $g\ge1$
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