Embedding Hamiltonian paths in augmented cubes with a required vertex in a fixed position

It is proved that there exists a path P"l(x,y) of length l if d"A"Q"""n(x,y)@?l@?2^n-1 between any two distinct vertices x and y of AQ"n. Obviously, we expect that such a path P"l(x,y) can be further extended by including the vertices not in P"l(x,y) into a hamiltonian path from x to a fixed vertex z or a hamiltonian cycle. In this paper, we prove that there exists a hamiltonian path R(x,y,z;l) from x to z such that d"R"("x","y","z";"l")(x,y)=l for any three distinct vertices x, y, and z of AQ"n with n>=2 and for any d"A"Q"""n(x,y)@?l@?2^n-1-d"A"Q"""n(y,z). Furthermore, there exists a hamiltonian cycle S(x,y;l) such that d"S"("x","y";"l")(x,y)=l for any two distinct vertices x and y and for any d"A"Q"""n(x,y)@?l@?2^n^-^1.