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Maximum number of perfect matchings in a graph of genus $g$ balanced $k$-partite graph

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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 $$k=2$$?

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 $$k=2$$.

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

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

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 $$k=2$$.

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