### 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.

Original language | English |
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Pages (from-to) | 1762-1768 |

Number of pages | 17 |

Journal | Computers and Mathematics with Applications |

Volume | 58 |

Issue number | 9 |

DOIs | |

State | Published - Nov 2009 |

### Keywords

- Hamiltonian; Augmented cubes

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## Cite this

Lee, C-M., Teng, Y-H., Tan, J-M., & Hsu, L. H. (2009). Embedding Hamiltonian paths in augmented cubes with a required vertex in a fixed position.

*Computers and Mathematics with Applications*,*58*(9), 1762-1768. https://doi.org/10.1016/j.camwa.2009.07.079