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 ll if dAQn(x,y)≤l≤2n−1 between any two distinct vertices x and y of AQnAQn. 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,z;l) from x to z such that dR(x,y,z;l)(x,y)=l for any three distinct vertices x, y, and z of AQnAQn with n≥2n≥2 and for any dAQn(x,y)≤l≤2n−1−dAQn(y,z). Furthermore, there exists a hamiltonian cycle S(x,y;l) such that dS(x,y;l)(x,y)=l for any two distinct vertices x and y and for any dAQn(x,y)≤l≤2n−1.