Regular graphs with few longest cycles

Motivated by work of Haythorpe, Thomassen and the author showed that there exists a positive constant $c$ such that there is an infinite family of 4-regular 4-connected graphs, each containing exactly $c$ hamiltonian cycles. We complement this by proving that the same conclusion holds for planar 4-regular 3-connected graphs, although it does not hold for planar 4-regular 4-connected graphs by a result of Brinkmann and Van Cleemput, and that it holds for 4-regular graphs of connectivity~2 with the constant $144 < c$, which we believe to be minimal among all hamiltonian 4-regular graphs of sufficiently large order. We then disprove a conjecture of Haythorpe by showing that for every non-negative integer $k$ there is a 5-regular graph on $26 + 6k$ vertices with $2^{k+10} \cdot 3^{k+3}$ hamiltonian cycles. We prove that for every $d \ge 3$ there is an infinite family of hamiltonian 3-connected graphs with minimum degree $d$, with a bounded number of hamiltonian cycles. It is shown that if a 3-regular graph $G$ has a unique longest cycle $C$, at least two components of $G - E(C)$ have an odd number of vertices on $C$, and that there exist 3-regular graphs with exactly two such components.