Abstract
Abstract
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
$n^{1-o(1)}$
.
Publisher
Cambridge University Press (CUP)
Subject
Computational Mathematics,Discrete Mathematics and Combinatorics,Geometry and Topology,Mathematical Physics,Statistics and Probability,Algebra and Number Theory,Theoretical Computer Science,Analysis
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