New Bounds on Even Cycle Creating Hamiltonian Paths Using Expander Graphs

Abstract

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 Hn(G) be the maximum number of pairwise G-creating Hamiltonian paths of Kn. 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}2k) 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.

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Correspondence to Daniel Soltész.

Additional information

First author supported by NKFIH (National Research, Development and Innovation Office) grant K 119528, and by the MTA Rényi Intézet Lendület Automorphic Research Group. Second author supported by NKFIH (National Research, Development and Innovation Office) grants K 108947, K 120706, KH 126853, KH 130371.

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Harcos, G., Soltész, D. New Bounds on Even Cycle Creating Hamiltonian Paths Using Expander Graphs. Combinatorica 40, 435–454 (2020). https://doi.org/10.1007/s00493-020-4204-z

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Mathematics Subject Classification (2010)

  • 05C35
  • 05D99