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, among all graphs of orientable genus at most $g$, the maximum possible Pfaffian number is at least $(8/3)^g$. This lower bound holds even for connected matching-covered graphs. We also obtain exponential lower bounds for the Pfaffian number of complete bipartite graphs, and hence for even complete graphs, improving asymptotically on a recent linear lower bound of Junchaya, Lucchesi, and Miranda.

翻译：Galluccio-Loebl和Tesler表明，嵌入在亏格为$g$的可定向曲面上的图的完美匹配多项式可以表示为最多$4^g$个Pfaffians的线性组合。我们证明，一般而言，需要指数多个Pfaffians。更精确地说，在所有可定向亏格不超过$g$的图中，最大可能的Pfaffian数至少为$(8/3)^g$。该下界甚至对连通匹配覆盖图也成立。我们还获得了完全二分图（进而对偶完全图）的Pfaffian数的指数下界，渐近地改进了Junchaya、Lucchesi和Miranda近期的一个线性下界。