Exponential Lower Bounds for the Pfaffian Number of Graphs

Abstract

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 $g\ge1$, there exists a graph of orientable genus at most $g$ whose perfect-matching polynomial requires at least $(8/3)^g$ Pfaffians in any such linear representation. In particular, for every even integer $n\ge6$, there is a graph on $n$ vertices with Pfaffian number at least $(8/3)^{\lfloor n/6\rfloor}$. Moreover, the lower bound is witnessed even by cubic bipartite matching-covered graphs. We prove this by showing that expressing the permanent of an $n\times n$ matrix of distinct variables as a linear combination of determinants obtained by changing signs of its entries requires exponentially many determinants. As a consequence, we improve a recent linear lower bound on the Pfaffian number due to Junchaya, Miranda, and Lucchesi to an exponential lower bound.