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An s-Hamiltonian Line Graph Problem

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Abstract

For an integer k  >  0, a graph G is k-triangular if every edge of G lies in at least k distinct 3-cycles of G. In (J Graph Theory 11:399–407 (1987)), Broersma and Veldman proposed an open problem: for a given positive integer k, determine the value s for which the statement “Let G be a k-triangular graph. Then L(G), the line graph of G, is s-hamiltonian if and only L(G) is (s + 2)-connected” is valid. Broersma and Veldman proved in 1987 that the statement above holds for 0 ≤ s ≤ k and asked, specifically, if the statement holds when s = 2k. In this paper, we prove that the statement above holds for 0 ≤ s ≤ max{2k, 6k − 16}.

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Authors and Affiliations

  1. Butler University, Indianapolis, IN, 46208, USA

    Zhi-Hong Chen

  2. West Virginia University, Morgantown, WV, 26506, USA

    Hong-Jian Lai

  3. Hong Kong Baptist University, Hong Kong, China

    Wai-Chee Shiu

  4. School of Information, Renmin University of China, Beijing, People’s Republic of China

    Deying Li

Authors
  1. Zhi-Hong Chen
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  2. Hong-Jian Lai
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  3. Wai-Chee Shiu
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  4. Deying Li
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Correspondence to Zhi-Hong Chen.

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Chen, ZH., Lai, HJ., Shiu, WC. et al. An s-Hamiltonian Line Graph Problem. Graphs and Combinatorics 23, 241–248 (2007). https://doi.org/10.1007/s00373-007-0727-y

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  • DOI: https://doi.org/10.1007/s00373-007-0727-y

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