What is the number of non-isomorphic $2n$ vertex balanced bipartite graphs of degree at most $d$ and genus $g$?
I am most interested in $d\leq3$ and $g=0$.
Closely related questions are considered in [
Guillaume Chapuy, Éric Fusy, Omer Giménez, Bojan Mohar, and Marc Noy, MR 2745423 Asymptotic enumeration and limit laws for graphs of fixed genus, J. Combin. Theory Ser. A 118 (2011), no. 3, 748--777.]1