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(AQn) (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(AQn) (x, y)<= l <= 2(n) - 1 - d(AQn) (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(AQn) (x, y) <= l <= 2(n-1). (C) 2009 Elsevier Ltd. All rights reserved.