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

Abstract

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 A Q 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 A Q 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 .