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 in...

Let G be a connected graph with vertex set V (G) = {v1, v2, · · · , vν} , which may have multiple edges but have no loops, and 2 ≤ dG(vi) ≤ 3 for i = 1, 2, · · · , ν, where dG(v) denotes the degree of vertex v of G. We show that if G has an even number of edges, then the number of perfect matchings of the line graph of G equals 2, where n is the number of 3-degree vertices of G. As a corollary,...

Aharoni and Berger conjectured that every bipartite graph which is the union of n matchings of size n + 1 contains a rainbow matching of size n. This conjecture is a generalization of several old conjectures of Ryser, Brualdi, and Stein about transversals in Latin squares. When the matchings are all edge-disjoint and perfect, an approximate version of this conjecture follows from a theorem of H...