Hyper-Hamiltonian generalized Petersen graphs

Abstract

Assume that n and k are positive integers with n ≥ 2 k + 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 for 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 ( 3 k , 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 ≠ 12 . Furthermore, P ( n , k ) is hyper-Hamiltonian if k is even with k ≥ 6 and n ≥ 2 k + 2 + ( 4 k − 1 ) ( 4 k + 1 ) , and P ( n , k ) is hyper-Hamiltonian if k ≥ 5 is odd and n is odd with n ≥ 6 k − 3 + 2 k ( 6 k − 2 ) .