What is the maximum number of perfect matchings a genus $g$ balanced $k$-partite graph (number of vertices for each color in all possible $k$-colorings is within a difference of $1$) can have? I am particularly interested in $k=2$.

For planar balanced bipartite graphs (each color has to have same number of vertices assigned) the number of perfect matchings is $2^{O(n)}$ while for genus $\Omega(n^2)$ we can have $2^{\Omega(n\log n)}$. So is maximum number of perfect matchings $2^{O(n\log g)}$ for $k=2$?

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