Lower matching conjecture, and a new proof of Schrijver's and Gurvits's theorems

Abstract

Friedland’s Lower Matching Conjecture asserts that if G is a d–regular bipartite graph on v(G) = 2n vertices, and mk(G) denotes the number of matchings of size k, then mk(G) ≥ ( n k )2( d− p d )n(d−p) (dp), where p = k n . When p = 1, this conjecture reduces to a theorem of Schrijver which says that a d–regular bipartite graph on v(G) = 2n vertices has at least ( (d− 1)d−1 dd−2 )n perfect matchings. L. Gurvits proved an asymptotic version of the Lower Matching Conjecture, namely he proved that lnmk(G) v(G) ≥ 1 2 ( p ln ( d p ) + (d− p) ln ( 1− p d ) − 2(1− p) ln(1− p) ) + ov(G)(1). In this paper, we prove the Lower Matching Conjecture. In fact, we will prove a slightly stronger statement which gives an extra cp √ n factor compared to the conjecture if p is separated away from 0 and 1, and is tight up to a constant factor if p is separated away from 1. We will also give a new proof of Gurvits’s and Schrijver’s theorems, and we extend these theorems to (a, b)–biregular bipartite graphs.