We know counting perfect matching for bipartite graphs with vertex degree $2$ is in $P$ while counting perfect matching for graphs with vertex degree $3$ is in $\#P$.
We also know there are degree $3$ bipartite graphs that are planar whose perfect matching count is in $P$.
What is known about counting bipartite perfect matching with average degree in $[2,3]$ and max degree $3$?
Is there a cutoff degree $\alpha\in[2,3]$ such that if max degree $3$ holds then counting perfect matching count is in $P$?
Is there any relevant literature?