A note on long nontrivial cycle in Hamiltonian graphs

Xiaolin Wang; Jiabao Yang; Guangmiao Yu; Ruilin Zheng

A note on long nontrivial cycle in Hamiltonian graphs

Combinatorics 2026-07-02 v1

Authors: Xiaolin Wang , Jiabao Yang , Guangmiao Yu , Ruilin Zheng

Abstract

Let $G$ be an $n$ -vertex graph containing a Hamiltonian cycle and with minimum degree at least $3$ . Gir\~{a}o, Kittipassorn and Narayanan (Israel J. Math., 2019) proved that $G$ contains another cycle of length at least $n-O(n^{4/5})$ . In this paper, we improve their bound to $n-O(n^{2/3})$ . Our proof is combined with a constructive method, which is based on a poset result, and a nonconstructive method. And the bound is best possible under these two methods.

Keywords

Hamiltonian graphs long cycles nontrivial cycles poset method extremal graph theory

Cite

@article{arxiv.2607.01738,
  title  = {A note on long nontrivial cycle in Hamiltonian graphs},
  author = {Xiaolin Wang and Jiabao Yang and Guangmiao Yu and Ruilin Zheng},
  journal= {arXiv preprint arXiv:2607.01738},
  year   = {2026}
}

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11 pages