Geodesic
Cycles of many lengths in Hamiltonian graphs
Forum of Mathematics, Sigma, Tome 10 (2022)

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In 1999, Jacobson and Lehel conjectured that, for $k \geq 3$, every k-regular Hamiltonian graph has cycles of $\Theta (n)$ many different lengths. This was further strengthened by Verstraëte, 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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     author = {Matija Buci\'c and Lior Gishboliner and Benny Sudakov},
     title = {Cycles of many lengths in {Hamiltonian} graphs},
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Matija Bucić; Lior Gishboliner; Benny Sudakov. Cycles of many lengths in Hamiltonian graphs. Forum of Mathematics, Sigma, Tome 10 (2022). doi: 10.1017/fms.2022.42

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