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New Bounds on Even Cycle Creating Hamiltonian Paths Using Expander Graphs
Combinatorica ( IF 1.0 ) Pub Date : 2020-03-05 , DOI: 10.1007/s00493-020-4204-z
Gergely Harcos , Daniel Soltész

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:

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。
更新日期:2020-03-05
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