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The Journal of Supercomputing

, Volume 54, Issue 2, pp 239–251 | Cite as

Mutually independent Hamiltonian cycles in dual-cubes

  • Yuan-Kang Shih
  • Hui-Chun Chuang
  • Shin-Shin KaoEmail author
  • Jimmy J. M. Tan
Article

Abstract

The hypercube family Q n is one of the most well-known interconnection networks in parallel computers. With Q n , dual-cube networks, denoted by DC n , was introduced and shown to be a (n+1)-regular, vertex symmetric graph with some fault-tolerant Hamiltonian properties. In addition, DC n ’s are shown to be superior to Q n ’s in many aspects. In this article, we will prove that the n-dimensional dual-cube DC n contains n+1 mutually independent Hamiltonian cycles for n≥2. More specifically, let v i V(DC n ) for 0≤i≤|V(DC n )|−1 and let \(\langle v_{0},v_{1},\ldots ,v_{|V(\mathit{DC}_{n})|-1},v_{0}\rangle\) be a Hamiltonian cycle of DC n . We prove that DC n contains n+1 Hamiltonian cycles of the form \(\langle v_{0},v_{1}^{k},\ldots,v_{|V(\mathit{DC}_{n})|-1}^{k},v_{0}\rangle\) for 0≤kn, in which v i k v i k whenever kk′. The result is optimal since each vertex of DC n has only n+1 neighbors.

Keywords

Hypercube Dual-cube Hamiltonian cycle Hamiltonian connected Mutually independent 

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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Yuan-Kang Shih
    • 1
  • Hui-Chun Chuang
    • 2
  • Shin-Shin Kao
    • 2
    Email author
  • Jimmy J. M. Tan
    • 1
  1. 1.Department of Computer ScienceNational Chiao Tung UniversityHsinchuTaiwan, China
  2. 2.Department of Applied MathematicsChung-Yuan Christian UniversityChung-LiTaiwan, China

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