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

It is proved that there exists a path Pl.x; y/ of length l if dAQn .x; y/ l 2n 􀀀 1 between any two distinct vertices x and y of AQn. Obviously, we expect that such a 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 there exists a hamiltonian path R.x; y; zI l/ from x to z such that dR.x;y;zIl/.x; y/ D l for any three distinct vertices x, y, and z of AQn with n 2 and for any dAQn .x; y/ l 2n 􀀀 1 􀀀 dAQn .y; z/. Furthermore, there exists a hamiltonian cycle S.x; yI l/ such that dS.x;yIl/.x; y/ D l for any two distinct vertices x and y and for any dAQn .x; y/ l 2n􀀀1.