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arXiv · 2104.07633

Cycles of many lengths in Hamiltonian graphs

Matija Bucić·Lior Gishboliner·Benny Sudakov
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Abstract

In 1999, Jacobson and Lehel conjectured that for $k \geq 3$, every $k$-regular Hamiltonian graph has cycles of at least linearly many different lengths. This was further strengthened by Verstra\"{e}te, who asked whether the regularity can be replaced with the weaker condition that the minimum degree is at least $3$. Despite attention from various researchers, until now, the best partial result towards both of these conjectures was a $\sqrt{n}$ lower bound on the number of cycle lengths. We resolve these conjectures asymptotically, by showing that the number of cycle lengths is at least $n^{1-o(1)}$.

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Matija Bucić, Lior Gishboliner, Benny Sudakov. 2021-04-15. Cycles of many lengths in Hamiltonian graphs. https://arxiv.org/abs/2104.07633

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