Matchings in Vertex-transitive Bipartite Graphs

A theorem of A. Schrijver asserts that a d–regular bipartite graph on 2n vertices has at least ( (d− 1)d−1 dd−2 )n perfect matchings. L. Gurvits gave an extension of Schrijver’s theorem for matchings of density p. In this paper we give a stronger version of Gurvits’s theorem in the case of vertex-transitive bipartite graphs. This stronger version in particular implies that for every positive integer k, there exists a positive constant c(k) such that if a d-regular vertex-transitive bipartite graph on 2n vertices contains a cycle of length at most k, then it has at least ( (d− 1)d−1 dd−2 + c(k) )n perfect matchings. We also show that if G is d–regular vertex-transitive bipartite graph on 2n vertices and mk(G) denotes the number of matchings of size k, and M(G, t) = 1 +m1(G)t+m2(G)t 2 + · · ·+mn(G)t = n ∏