A note on long nontrivial cycle in Hamiltonian graphs

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 $n-O(n^{4\mathrm{/}5})$. In this paper, we improve their bound to $n-O(n^{2\mathrm{/}3})$. 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.