Hyper-Hamiltonian generalized Petersen graphs

Assume that n and k are positive integers with n ≥ 2k + 1. A non-hamiltonian graph G is hypo hamiltonian if G − v is hamiltonian for any v ∈ V (G). It is proved that the generalized Petersen graph P (n, k) is hypo hamiltonian if and only if k = 2 and n ≡ 5 (mod 6). Similarly, a hamiltonian graph G is hyper hamiltonian if G−v is hamiltonian for any v ∈ V (G). In this paper, we will give some necessary conditions and some sufficient conditions about the hyper hamiltonian generalized Petersen graphs. In particular, P (n, k) is not hyper hamiltonian if n is even and k is odd. We also prove that P (3k, k) is hyper hamiltonian if and only if k is odd. Moreover, P (n, 3) is hyper hamiltonian if and only if n is odd and P (n, 4) is hyper hamiltonian if and only if n 6= 12. Furthermore, P (n, k) is hyper hamiltonian if k is even with k ≥ 6 and n ≥ 2k + 2 + (4k− 1)(4k + 1), and P (n, k) is hyper hamiltonian if k ≥ 5 is odd and n is odd with n ≥ 6k − 3 + 2k(6k − 2).