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Tight Hamilton cycles in cherry-quasirandom 3-uniform hypergraphs

Part of: Graph theory Extremal combinatorics

Published online by Cambridge University Press:  12 October 2020

Elad Aigner-Horev  and
Gil Levy
Elad Aigner-Horev*
Affiliation:
Department of Mathematics and Computer Science, Ariel University, Israel
Gil Levy
Affiliation:
Department of Mathematics and Computer Science, Ariel University, Israel
*
*Corresponding author. Email: [email protected]
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Abstract

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We employ the absorbing-path method in order to prove two results regarding the emergence of tight Hamilton cycles in the so-called two-path or cherry-quasirandom 3-graphs.

Our first result asserts that for any fixed real α > 0, cherry-quasirandom 3-graphs of sufficiently large order n having minimum 2-degree at least α(n – 2) have a tight Hamilton cycle.

Our second result concerns the minimum 1-degree sufficient for such 3-graphs to have a tight Hamilton cycle. Roughly speaking, we prove that for every d, α > 0 satisfying d + α > 1, any sufficiently large n-vertex such 3-graph H of density d and minimum 1-degree at least $\alpha \left({\matrix{{n - 1} \cr 2 \cr } } \right)$ has a tight Hamilton cycle.

MSC classification

Primary: 05C35: Extremal problems 05C65: Hypergraphs
Secondary: 05C80: Random graphs 05D05: Extremal set theory 05C42: Density (toughness, etc.)
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 30 , Issue 3 , May 2021 , pp. 412 - 443
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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