Exponential Lower Bounds for the Pfaffian Number of Graphs

TL;DR

This paper proves that the number of Pfaffians needed to represent the perfect-matching polynomial of certain graphs grows exponentially with the genus, establishing new lower bounds.

Contribution

It establishes exponential lower bounds on the Pfaffian number for graphs embedded in surfaces of genus g, improving previous linear bounds.

Findings

Maximum Pfaffian number for graphs of genus g is at least (8/3)^g.

Exponential lower bounds are shown for complete bipartite graphs.

Results hold even for connected matching-covered 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, 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.