Embedding hamiltonian paths in hypercubes with a required vertex in a fixed position

Assume that n is a positive integer with n⩾2. It is proved that between any two different vertices x and y of Qn there exists a path Pl(x,y) of length l for any l with h(x,y)⩽l⩽n2−1 and 2|(l−h(x,y)). We expect such path Pl(x,y) can be further extended by including the vertices not in Pl(x,y) into a hamiltonian path from x to a fixed vertex z or a hamiltonian cycle. In this paper, we prove that for any two vertices x and z from different partite set of n-dimensional hypercube Qn, for any vertex y∈V(Qn)−{x,z}, and for any integer l with h(x,y)⩽l⩽n2−1−h(y,z) and 2|(l−h(x,y)), there exists a hamiltonian path R(x,y,z;l) from x to z such that dR(x,y,z;l)(x,y)=l. Moreover, for any two distinct vertices x and y of Qn and for any integer l with h(x,y)⩽l⩽2n−1 and 2|(l−h(x,y)), there exists a hamiltonian cycle S(x,y;l) such that dS(x,y;l)(x,y)=l.