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arXiv · 2606.16114

The number of cycles of a given length in dense hamiltonian graphs: proving Hilton's conjecture

Chengli Li·Leyou Xu·Bo Zhou
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Abstract

A classical theorem of Sheehan in 1977 states that every hamiltonian graph $G$ of order $n$ satisfying $e(G)>\left\lfloor \frac{n^2}{4}\right\rfloor+1$ contains at least two cycles of every length $\ell$, $3\le \ell\le n$. In the same paper, Sheehan recorded a conjecture of Hilton, which strengthens this conclusion by asserting that such a graph contains at least $n-\ell+2$ cycles of length $\ell$ for each $3\le \ell\le n$. We prove Hilton's conjecture for all hamiltonian graphs of order at least $440$.

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Chengli Li, Leyou Xu, Bo Zhou. 2026-06-15. The number of cycles of a given length in dense hamiltonian graphs: proving Hilton's conjecture. https://arxiv.org/abs/2606.16114

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