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Asymptotically what is the fraction of balanced bipartite graph on $2n$ vertices with at most $cn^{\beta}$ edges having at most $n^\alpha$ perfect matchings for any fixed $c,\alpha>0$ and fixed $\...

Let $G = (X, Y, E)$ be bipartite graph such that $|X|=|Y|=n$. A matching $M \subseteq E$ is a subset of disjoint edges (i.e., there does not exist a pair of edges $(x, y) \in M$ and $(x', y') \in M$ ...

Given a bipartite graph we know that symmetric difference of any two perfect matchings is union of even cycles. Conversely when is it true that every union of even cycles comes from symmetric ...