The Asymptotic Lower Matching Conjecture

Let $r\geq 2$, and let $G_n=(V_n,E_n)$, $n=1,2,\ldots$, be a sequence of finite $r$-regular bipartite graphs with $\#V_n\to\infty$. Let $k_n\in[0,\frac{\#V_n}{2}]$ be integers such that

$$\lim_{n\to\infty}\frac{2k_n}{\#V_n}=p\in(0,1].$$

Here $\phi_{G_n}(k_n)$ denotes the number of $k_n$-matchings of $G_n$, and

$$gh_r(p)=\frac{1}{2}\left(p\log r-p\log p-2(1-p)\log(1-p)+(r-p)\log\left(1-\frac{p}{r}\right)\right).$$

The Asymptotic Lower Matching Conjecture. Under these hypotheses,

$$\liminf_{n\to\infty}\frac{\log\phi_{G_n}(k_n)}{\#V_n}\geq gh_r(p).$$

The conjecture is trivial for $r=1$ and is proved for $r=2$; the general case remains open. It gives an asymptotic lower bound for matching numbers in finite regular bipartite graphs.