Given a fixed positive integer k≥2 and a fixed pair of sets of vertices X={x1,x2,⋯,xk} and Y={y1,y2,⋯,yk} in a graph G of sufficiently large order n, the sharp minimum degree condition δ(G)≥(n+k−1)/2 will be shown to imply the existence of a Hamiltonian cycle C such that all of the vertices of X precede the vertices of Y for appropriate initial vertex and orientation of the cycle C. Also, a minimum degree condition along with a connectivity condition will be shown to imply the existence of a Hamiltonian cycle C such that the vertices of X and Y alternate on the cycle C.