  We study how few pairwise distinct longest cycles a regular graph can have
under additional constraints. For each integer $r \geq 5$, we give exponential
improvements for the best asymptotic upper bounds for this invariant under the
additional constraint that the graphs are $r$-regular hamiltonian graphs.
Earlier work showed that a conjecture by Haythorpe on a lower bound for this
invariant is false because of an incorrect constant factor, whereas our results
imply that the conjecture is even asymptotically incorrect. Motivated by a
question of Zamfirescu and work of Chia and Thomassen, we also study this
invariant for non-hamiltonian 2-connected $r$-regular graphs and show that in
this case the invariant can be bounded from above by a constant for all large
enough graphs, even for graphs with arbitrarily large girth.
