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arXiv · 2607.01738

A note on long nontrivial cycle in Hamiltonian graphs

Xiaolin Wang·Jiabao Yang·Guangmiao Yu·Ruilin Zheng
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Abstract

Let $G$ be an $n$-vertex graph containing a Hamiltonian cycle and with minimum degree at least $3$. Girão, 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.

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Xiaolin Wang, Jiabao Yang, Guangmiao Yu, Ruilin Zheng. 2026-07-02. A note on long nontrivial cycle in Hamiltonian graphs. https://arxiv.org/abs/2607.01738

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