A 1‐factorization of a graph G is a collection of edge‐disjoint perfect matchings whose union is E (G ). In this paper, we prove that for any ϵ >0, an (n ,d ,λ )‐graph G admits a 1‐factorization provided that n is even, C ₀ ≤ d  ≤ n −1 (where C ₀=C ₀(ϵ ) is a constant depending only on ϵ ), and λ  ≤ d ^1−ϵ . In particular, since (as is well known) a typical random d ‐regular graph G _{n ,d} is such a graph, we obtain the existence of a 1‐factorization in a typical G _{n ,d} for all C ₀ ≤ d  ≤ n −1, thereby extending to all possible values of d results obtained by Janson, and independently by Molloy, Robalewska, Robinson, and Wormald for fixed d . Moreover, we also obtain a lower bound for the number of distinct 1‐factorizations of such graphs G , which is better by a factor of 2^{nd /2} than the previously best known lower bounds, even in the simplest case where G is the complete graph.

中文翻译：

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1-伪随机图的因子化

图G的1因式分解是边不相交的完全匹配的集合，其并集为E（G）。在本文中，我们证明了对任何ε > 0，一个（Ñ，d，λ） -图ģ承认1-因式分解提供Ñ为偶数时，c ^ ₀  ≤  d  ≤  Ñ -1（其中，C ^ ₀ = C ^ ₀（ε）是仅依赖于一个常数ε），和λ  ≤  d ^{1- ε}。特别地，由于（如众所周知的那样）一个典型的随机d -regular图表ģ _Ñ，d是这样的图中，我们获得的1因式分解的存在在一个典型的ģ _Ñ，d为所有Ç ₀  ≤  d  ≤  Ñ -1，从而扩展到詹森以及对于固定d的Molloy，Robalewska，Robinson和Wormald独立获得的d结果的所有可能值。此外，我们还获得了此类图G的不同1分解次数的下界，其下限为2 ^{nd / 2}更好即使在最简单的情况下G是完整图形的情况下，也比以前最著名的下限要好。