The classification of hamiltonian generalized Petersen graphs

Abstract

The generalized Petersen graph GP(n, k), n ≥ 2 and 1 ≤ k ≤ n − 1, has vertex-set {u₀, u₁,…, u_{n − 1}, v₀, v₁,…, v_{n − 1}} and edge-set {u_iu_{i + 1}, u_iv_i, v_iv_{i + k} : 0 ≤ i ≤ n − 1 with subscripts reduced modulo n}. In this paper it is proved that GP(n, k) is hamiltonian if and only if it is neither $\text{GP}\text{(n, 2) ≅}\text{GP}\text{(n, n − 2) ≅}\text{GP}\text{(n,}\text{(n − 1)}\text{2}\text{≅}\text{GP}\text{(n,}\text{(n + 1)}\text{2}\text{)}$ when n ≡ 5 (mod 6) nor $\text{GP}\text{(n,}\text{n}\text{2}\text{)}$ when n ≡ 0 (mod 4) and n ≥ 8.