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Apr 19, 2016 at 1:04 comment added Turbo @JonNoel Seems like it.
Apr 18, 2016 at 23:32 comment added Jon Noel It is Theorem 1 in this paper: combinatorics.org/ojs/index.php/eljc/article/view/v18i1p10. Like I said, this can only give you a lower bound on the ratio that you are interested in and it's very possible that the bound won't be very good!
Apr 17, 2016 at 7:18 comment added Turbo @JonNoel I am unfamiliar with this theorem. Perhaps you could elaborate below.
Apr 17, 2016 at 7:15 comment added Jon Noel Perhaps you could get a lower bound on this quantity using Brègman's Theorem. It would give you a set of degree sequences that are guaranteed to have a small number of perfect matchings (you'd still have to count balanced bipartite graphs with those degree sequences, though).
Apr 17, 2016 at 1:46 history edited Turbo
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Apr 16, 2016 at 20:22 history edited Turbo CC BY-SA 3.0
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Apr 16, 2016 at 20:11 history asked Turbo CC BY-SA 3.0