Let $G=(V,E)$ be a graph with an even number of vertices $|V|=2n$. Assume that $G$  is $K_{3,3}$-free i.e. it does not contain a graph isomorphic to [biclique][1]  $K_{3,3}$. A perfect matching of $G$ is a partition of the set of vertices $V$ onto disjoint two-element subsets according to edges $E$.

**Question:**
Are there any known *upper bounds* for the number of perfect matchings of such graphs? 

I am not a specialist in graph theory and after an extensive internet search, I was only able to find that the complexity of computing perfect matchings for such $K_{3,3}$-free graphs is lower than the one in the general [case][2] (it reduces to NC from beeing  #P hard). 

**Motivation**: I stumbled on this problem while studying an unrelated physics problem that concerns contextuality in a restricted model of quantum computing based on Majorana fermions (see for example [this work][3] by Bravyi).


  [1]: https://en.wikipedia.org/wiki/Complete_bipartite_graph
  [2]: https://ecommons.cornell.edu/bitstream/handle/1813/6700/87-860.pdf;jsessionid=1677D94933036CEEAF479CFD9FD7F2EB?sequence=1
  [3]: https://arxiv.org/abs/quant-ph/0511178