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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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  • $\begingroup$ genus alone is not a big obstacle, it appears. e.g. fullerens (particular kind of planar graphs) can have exponentially many perfect matchings : arxiv.org/abs/0801.1438 $\endgroup$
    – Dima Pasechnik
    May 28, 2019 at 7:03
  • $\begingroup$ @DimaPasechnik You dont get the point. Balanced bipartite graphs can have $2^{\Omega(n\log n)}$ perfect matchings however genus needs to be high and if you have planarity you are constrained by $2^{O(n)}$ perfect matchings. $\endgroup$
    – Turbo
    May 28, 2019 at 7:08
  • $\begingroup$ planar 4-gonal $2k\times 2k$ grids can have exponentially many perfect matchnigs, too - one covers such a grid by $2\times 2$ squares, each containing a 4-gon, so you can switch each of them in 2 different ways. $\endgroup$
    – Dima Pasechnik
    May 28, 2019 at 7:09
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    $\begingroup$ Forgot the reference but check mathoverflow.net/questions/273765/…. $\endgroup$
    – Turbo
    May 28, 2019 at 7:22
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    $\begingroup$ The (bipartite) hypercube on $N=2^n$ vertices has genus $1+(n-4)2^{n-3} \approx C N \log(N)$, and it has roughly $2^{c N \log(\log(N))}$ perfect matchings. So there’s something with not huge genus having superexponentialy many perfect matchings. $\endgroup$
    – Pat Devlin
    May 29, 2019 at 1:59

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Yes, but not for a great reason.

Fact: genus of a graph is bounded below by $|E|/6 - n/2+1$.

Case 1: suppose $|E| \leq 10 n$. Then the number of perfect matchings is at most ${ {|E|} \choose {n/2}} \leq 2^{|E|} \leq 2^{10n}.$

Case 2: suppose $|E| \geq 10n$ so that (by fact) the genus satisfies $g \geq n$. We know every graph has at most $2^{O(n \log(n))}$ perfect matchings, and we’re done since $g\geq n$.

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  • $\begingroup$ Yes... If $g=\log(n)$, then there are at most $10n$ edges, so at most $2^{10n}$ perfect matchings [see case 1]. $\endgroup$
    – Pat Devlin
    May 29, 2019 at 15:38
  • $\begingroup$ Great. In which case the genus is greater than $n$ [see case 2]. $\endgroup$
    – Pat Devlin
    May 29, 2019 at 15:40
  • $\begingroup$ Are you hoping for a construction or a lower bound? If the genus is too small, there are not many edges at all (at most linearly many). And if there are only $cn$ edges, the number of perfect matchings is at most $2^{cn}$, which is in fact less than $2^{cn \log(g)}$. This is a better upper bound than what you were hoping for. So there is no lower bound matching the upper bound hoped for in OP. $\endgroup$
    – Pat Devlin
    May 29, 2019 at 15:47
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    $\begingroup$ (Commenting on “all or nothing” comment) Yea. Seems like it. The upper bounds of $2^{E}$ and $n!$ are always smaller than $2^{n \log(g)}$. I think one of the morals here is that “logs and tolerable multiplicative factors in the exponent cover a multitude of sin.” $\endgroup$
    – Pat Devlin
    May 29, 2019 at 16:16
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    $\begingroup$ Page $18$ cs.columbia.edu/~cs6204/files/Lec2-Min&MaxGenus.pdf refers. $\endgroup$
    – Turbo
    May 29, 2019 at 16:24

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