2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/68187It is known that if G is a connected simple graph, then G^3 is Hamiltonian (in fact, Hamilton-connected). A simple graph is k-ordered Hamiltonian if for any sequence v_1, v_2, ..., v_k of k vertices there is a Hamiltonian cycle containing these vertices in the given order. In this paper, we prove that G^(3k/2 + 1) is k-ordered Hamiltonian for a connected graph G on at least k vertices. We further show that if G is connected, then G^4 is 4-ordered Hamiltonian and that if G is Hamiltonian, then G^3 is 5-ordered Hamiltonian. We also give bounds on the smallest power p_k such that G^p_k is k-ordered Hamiltonian for G=P_n and G=C_n.18 pages, 8 figures; submitted to J. Graph TheoryCombinatorics05C45; 05C12; 05C38text