Locating sets of vertices on Hamiltonian cycles

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