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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)}.

Joint work with Bucic and Sudakov.