We say that two graphs on the same vertex set are G -creating if their union (the union of their edges) contains G as a subgraph. Let H n ( G ) be the maximum number of pairwise G -creating Hamiltonian paths of K n . Cohen, Fachini and Körner proved $${n^{\frac{1}{2}n - o\left( n \right)}} \le {H_n}\left( {{C_4}} \right) \le {n^{\frac{3}{4}n + o\left( n \right)}}.$$ In this paper we close the superexponential gap between their lower and upper bounds by proving $${n^{\frac{1}{2}n - \frac{1}{2}\frac{n}{{\log n}} - O\left( 1 \right)}} \le {H_n}\left( {{C_4}} \right) \le {n^{\frac{1}{2}n + o\left( {\frac{n}{{\log n}}} \right)}}.$$ We also improve the previously established upper bounds on {enH} n ({enC} 2 k ) for k >3, and we present a small improvement on the lower bound of Furedi, Kantor, Monti and Sinaimeri on the maximum number of so-called pairwise reversing permutations. One of our main tools is a theorem of Krivelevich, which roughly states that (certain kinds of) good expanders contain many Hamiltonian paths.

中文翻译：

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使用扩展图创建哈密顿路径的偶数循环的新边界

如果它们的并集（它们的边的并集）包含 G 作为子图，我们就说同一顶点集上的两个图是 G 创建的。令 H n ( G ) 是 K n 的成对 G 生成哈密顿路径的最大数量。Cohen、Fachini 和 Körner 证明了 $${n^{\frac{1}{2}n - o\left( n \right)}} \le {H_n}\left( {{C_4}} \right) \le {n^{\frac{3}{4}n + o\left( n \right)}}.$$ 在本文中，我们通过证明 $${n^{\ frac{1}{2}n - \frac{1}{2}\frac{n}{{\log n}} - O\left( 1 \right)}} \le {H_n}\left( {{ C_4}} \right) \le {n^{\frac{1}{2}n + o\left( {\frac{n}{{\log n}}} \right)}}.$$ 我们也改进先前在 {enH} n ({enC} 2 k ) 上建立的上限，k > 3，我们对 Furedi、Kantor 的下限进行了小幅改进，Monti 和 Sinaimeri 关于所谓的成对反转排列的最大数量。我们的主要工具之一是 Krivelevich 定理，它粗略地指出（某些种类的）好的扩展器包含许多哈密顿路径。