The Asymptotic Upper Matching Conjecture

Let $r\geq2$, 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

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

Let $K(r)$ be the countable disjoint union of copies of $K_{r,r}$, and let $h_{K(r)}(p)$ denote its $p$-matching entropy.

The Asymptotic Upper Matching Conjecture. Under these hypotheses,

$$\limsup_{n\to\infty}\frac{\log\phi_{G_n}(k_n)}{\#V_n}\leq h_{K(r)}(p).$$

Equality holds for the sequence $qK_{r,r}$, $q=1,2,\ldots$.

The conjecture is implied by the finite Upper Matching Conjecture and gives the proposed asymptotic upper bound on matching entropy for regular bipartite graphs. The source records the finite conjecture only for $r=2$; the general asymptotic claim remains open.