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Hamiltonicity of hypercubes with a constraint of required and faulty edges

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Abstract

Let R and F be two disjoint edge sets in an n-dimensional hypercube Q n . We give two constructing methods to build a Hamiltonian cycle or path that includes all the edges of R but excludes all of F. Besides, considering every vertex of Q n incident to at most n−2 edges of F, we show that a Hamiltonian cycle exists if (A) |R|+2|F|≤2n−3 when |R|≥2, or (B) |R|+2|F|≤4n−9 when |R|≤1. Both bounds are tight. The analogous property for Hamiltonian paths is also given.

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Authors and Affiliations

  1. Department of Computer Science and Information Engineering, Providence University, Taichung County, Taiwan

    Lih-Hsing Hsu

  2. General Education Center, China University of Technology, Hsinchu, Taiwan

    Shu-Chung Liu

  3. Institute of Mathematics, Academia Sinica, Taipei, Taiwan

    Yeong-Nan Yeh

Authors
  1. Lih-Hsing Hsu
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  2. Shu-Chung Liu
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  3. Yeong-Nan Yeh
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Corresponding author

Correspondence to Shu-Chung Liu.

Additional information

Dedicated to Professor Frank K. Hwang on the occasion of his 65th birthday.

Lih-Hsing Hsu’s research project is partially supported by NSC 95-2221-E-233-002.

Shu-Chung Liu’s research project is partially supported by NSC 90-2115-M-163-003 and 95-2115-M-163-002.

Yeong-Nan Yeh’s research project is partially supported by NSC 95-2115-M-001-009.

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Hsu, LH., Liu, SC. & Yeh, YN. Hamiltonicity of hypercubes with a constraint of required and faulty edges. J Comb Optim 14, 197–204 (2007). https://doi.org/10.1007/s10878-007-9059-3

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  • DOI: https://doi.org/10.1007/s10878-007-9059-3

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