A 1-factor in an n -vertex graph G is a collection of $$\frac{n}{2}$$ vertex-disjoint edges and a 1-factorization of G is a partition of its edges into edge-disjoint 1-factors. Clearly, a 1-factorization of G cannot exist unless n is even and G is regular (that is, all vertices are of the same degree). The problem of finding 1-factorizations in graphs goes back to a paper of Kirkman in 1847 and has been extensively studied since then. Deciding whether a graph has a 1-factorization is usually a very difficult question. For example, it took more than 60 years and an impressive tour de force of Csaba, Kühn, Lo, Osthus and Treglown to prove an old conjecture of Dirac from the 1950s, which says that every d -regular graph on n vertices contains a 1-factorization, provided that n is even and $$d \geqslant 2\left[ {\frac{n}{4}} \right] - 1$$ . In this paper we address the natural question of estimating F ( n , d ), the number of 1-factorizations in d -regular graphs on an even number of vertices, provided that $$d \geqslant \left[ {\frac{n}{2}} \right] + \varepsilon n$$ . Improving upon a recent result of Ferber and Jain, which itself improved upon a result of Cameron from the 1970s, we show that $$F\left( {n,\,d} \right) \geqslant {\left( {\left( {1 + o\left( 1 \right)} \right)\frac{d}{{{e^2}}}} \right)^{nd/2}}$$ , which is asymptotically best possible.

中文翻译：

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正则高阶图的 1-Factorizations 数

n 顶点图 G 中的 1 因子是 $$\frac{n}{2}$$ 顶点不相交边的集合，G 的 1 因子分解是将其边划分为边不相交 1-因素。显然，除非 n 是偶数且 G 是正则的（即所有顶点的度数相同），否则 G 的 1-因式分解不可能存在。在图中寻找 1 分解的问题可以追溯到 1847 年 Kirkman 的一篇论文，并从那时起得到了广泛的研究。确定一个图是否具有 1 因子分解通常是一个非常困难的问题。例如，Csaba、Kühn、Lo、Osthus 和 Treglown 用了 60 多年的时间证明了 1950 年代狄拉克的一个古老猜想，即在 n 个顶点上的每个 d 正则图都包含一个 1 - 分解，前提是 n 是偶数且 $$d \geqslant 2\left[ {\frac{n}{4}} \right] - 1$$ 。在本文中，我们解决了估计 F ( n , d ) 的自然问题，即偶数个顶点上的 d -正则图中的 1 因式分解的数量，前提是 $$d \geqslant \left[ {\frac{n }{2}} \right] + \varepsilon n$$ 。改进 Ferber 和 Jain 的最新结果，该结果本身改进了 1970 年代 Cameron 的结果，我们表明 $$F\left( {n,\,d} \right) \geqslant {\left( {\left ( {1 + o\left( 1 \right)} \right)\frac{d}{{{e^2}}}} \right)^{nd/2}}$$ ，这是渐近最好的。