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 [European J. Combin., 97 (2021), 103395], 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 nonnegative integer k there is a 5-regular graph on 26 + 6k vertices with 2 ^{k +10} · 3 ^{k +3} Hamiltonian cycles. We prove that for every d ≥ 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.