Cycles of Many Lengths in Hamiltonian Graphs

Abstract

In 1999, Jacobson and Lehel conjectured that, for 𝑘 ≥ 3, every k-regular Hamiltonian graph has cycles of Θ(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 √ 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) .