arXiv 1407.54092014-07-27math.CO

点传递二部图中的匹配

Matchings in vertex-transitive bipartite graphs

Péter Csikvári

摘要：

A. Schrijver断言$2n$顶点上的$d$ -正则二部图至少有$$\left(\frac{(d-1)^{d-1}}{d^{d-2}}\right)^n$$完美匹配. L. Gurvits给出了Schrijver密度匹配定理的扩展$p$。本文在点传递二部图的情形下给出了Gurvits定理的一个更强的版本。这个更强的版本特别意味着，对于每个正整数$k$，存在一个正常数$c(k)$，使得如果$d$ -$2n$顶点上的正则顶点传递二分图包含长度最多为$k$的圈，则它至少具有$$\left(\frac{(d-1)^{d-1}}{d^{d-2}}+c(k)\right)^n$$完美匹配。证明了如果$(G_i)$是点传递二部图的Benjamini-Schramm收敛图序列，则$$\frac{\ln pm(G_i)}{v(G_i)}$$是收敛的，其中$pm(G)$和$v(G)$分别表示$G$的完美匹配数和顶点数.本文还证明了：如果$G$是$2n$上的$d$ -正则点传递二部图，$m_k(G)$表示$k$的匹配数，$$M(G,t)=1+m_1(G)t+m_2(G)t^2+\dots +m_n(G)t^n=\prod_{k=1}^n(1+γ_k(G)t),$$其中$γ_1(G)\leq \dots \leq γ_n(G)$，则$$γ_k(G)\geq \frac{d^2}{4(d-1)}\frac{k^2}{n^2},$$和$$\frac{m_{n-1}(G)}{m_n(G)}\leq \frac{2}{d}n^2.$$. Kenyon，D. Randall和A.辛克莱尔有$d$ -正则二部图的例子，这些陈述不为真，没有顶点传递的条件。

英文摘要：

A theorem of A. Schrijver asserts that a $d$-regular bipartite graph on $2n$ vertices has at least $$\left(\frac{(d-1)^{d-1}}{d^{d-2}}\right)^n$$ 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 $$\left(\frac{(d-1)^{d-1}}{d^{d-2}}+c(k)\right)^n$$ perfect matchings. We also show that if $(G_i)$ is a Benjamini--Schramm convergent graph sequence of vertex-transitive bipartite graphs, then $$\frac{\ln pm(G_i)}{v(G_i)}$$ is convergent, where $pm(G)$ and $v(G)$ denote the number of perfect matchings and the number of vertices of $G$, respectively. We also show that if $G$ is $d$-regular vertex-transitive bipartite graph on $2n$ vertices and $m_k(G)$ denote the number of matchings of size $k$, and $$M(G,t)=1+m_1(G)t+m_2(G)t^2+\dots +m_n(G)t^n=\prod_{k=1}^n(1+γ_k(G)t),$$ where $γ_1(G)\leq \dots \leq γ_n(G)$, then $$γ_k(G)\geq \frac{d^2}{4(d-1)}\frac{k^2}{n^2},$$ and $$\frac{m_{n-1}(G)}{m_n(G)}\leq \frac{2}{d}n^2.$$ 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.

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