### Abstract

A bipartite graph is bipancyclic if it contains a cycle of every even length from 4 to vertical bar V(G)vertical bar inclusive. A hamiltonian bipartite graph G is bipanpositionable if, for any two different vertices x and y, there exists a hamiltonian cycle C of G such that d(c)(x, y) = k for any integer k with d(G)(x, y) <= k <= vertical bar V(G)vertical bar/2 and (k - d(G)(x, y)) being even. A bipartite graph G is k-cycle bipanpositionable if, for any two different vertices x and y, there exists a cycle of G with d(C)(x, y) = l and vertical bar V(C)vertical bar = k for any integer l with d(G)(x, y) <= l <= k/2 and (l - d(G)(x, y)) being even. A bipartite graph G is bipanpositionable bipancyclic if G is k-cycle bipanpositionable for every even integer k, 4 <= k <= vertical bar V(G)vertical bar. We prove that the hypercube Q(n) is bipanpositionable bipancyclic for n >= 2. (C) 2009 Elsevier Ltd. All rights reserved.

Original language | English |
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Pages (from-to) | 1722-1724 |

Journal | Computers and Mathematics with Applications |

Volume | 58 |

Issue number | 9 |

DOIs | |

State | Published - Nov 2009 |

### Keywords

- Bipanpositionable; Bipancyclic; Hypercube; Hamiltonian

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## Cite this

Shih, Y-K., Lin, C. K., Tan, J-M., & Hsu, L. H. (2009). The bipanpositionable bipancyclic property of the hypercube.

*Computers and Mathematics with Applications*,*58*(9), 1722-1724. https://doi.org/10.1016/j.camwa.2009.07.087