Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Thursday, October 14, 2021, 12:15 pm

Duration: 30 minutes

Location: CAB G51 and Zoom: conference room

Speaker: Lior Gishboliner

Cycles of many lengths in Hamiltonian graphs

In 1999, Jacobson and Lehel conjectured that for $k \geq 3$, every $k$-regular Hamiltonian graph on $n$ vertices has cycles of $\Theta(n)$ many different lengths. This was further strengthened by Verstraete, 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 Matija Bucic and Benny Sudakov.

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