Matchings in vertex-transitive bipartite graphs

Abstract

A theorem of A. Schrijver asserts that a d-regular bipartite graph on 2n vertices has at least $${\left( {\frac{{{{\left( {d - 1} \right)}^{d - 1}}}}{{{d^{d - 2}}}}} \right)^n}$$ 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 $${\left( {\frac{{{{\left( {d - 1} \right)}^{d - 1}}}}{{{d^{d - 2}}}} + c\left( k \right)} \right)^n}$$ perfect matchings.