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.

We also show that if G is a d-regular vertex-transitive bipartite graph on 2n vertices and m _k (G) denotes the number of matchings of size k, and

$$M\left( {G,t} \right) = 1 + {m_1}\left( G \right)t + {m_2}\left( G \right){t^2} + \cdot \cdot \cdot + {m_n}\left( G \right){t^n} = \prod\limits_{k = 1}^n {\left( {1 + {\gamma _k}\left( G \right)t} \right),} $$

where γ ₁(G) ≤ ... ≤ γ _n (G), then

$${\gamma _k}\left( G \right) \geqslant \frac{{{d^2}}}{{4\left( {d - 1} \right){n^2}}}\frac{{{k^2}}}{{{n^2}}}$$

and

$$\frac{{{m_{n - 1}}\left( G \right)}}{{{m_n}\left( G \right)}} \leqslant \frac{2}{d}{n^2}$$

. The latter result improves on a previous bound of C. Kenyon, D. Randall and A. Sinclair. There are examples of d-regular bipartite graphs for which these statements fail to be true without the condition of vertex-transitivity.

We also show that if (G _i ) is a Benjamini–Schramm convergent graph sequence of vertex-transitive bipartite graphs, then

$$\frac{{\ln pm\left( {{G_i}} \right)}}{{v\left( {{G_i}} \right)}}$$

is convergent, where pm(G) and v(G) denote the number of perfect matchings and the number of vertices of G, respectively.