Asymptotics of the Upper Matching Conjecture

We give upper bounds for the number Φℓ(G) of matchings of size ℓ in (i) bipartite graphs G=(X∪Y,E) with specified degrees dx (x∈X), and (ii) general graphs G=(V,E) with all degrees specified. In particular, for d-regular, N-vertex graphs, our bound is best possible up to an error factor of the form exp[od(1)N], where od(1)→0 as d→∞. This represents the best progress to date on the “Upper Matching Conjecture” of Friedland, Krop, Lundow and Markström.Some further possibilities are also suggested.