Improved asymptotic upper bounds for the minimum number of pairwise distinct longest cycles in regular graphs

We study how few pairwise distinct longest cycles a regular graph can have under additional constraints. For each integer $r\ge 5$, 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.