A lower bound on the number of hamiltonian cycles

Thomassen (J. Combin. Theory Ser. B 72 (1998) 104–109) showed that any r-regular hamiltonian graph, r⩾300, has a second hamiltonian cycle. Refining his methods we prove: Let G be a hamiltonian graph, Δ and δ be its maximum and minimum degree, respectively. Then for any real number k⩾1 there exists Δ(k) so that if Δ⩾Δ(k) then G has at least δ−⌊Δ/k⌋+2 hamiltonian cycles. In particular, if k⩾Δ/δ and Δ⩾Δ(k) then G has a second hamiltonian cycle. A simple method for calculating an upper bound on Δ(k) is given. For example, Δ(1)⩽73,Δ(1.1)⩽93,Δ(2)⩽382,Δ(50)⩽545 800. In addition, it is shown that this bounds are nearly best possible if one confines himself to methods introduced by Thomassen.