Abstract
In this paper, we show that every 3-connected claw-free graph such that every induced cycle of length at least 4 has at most 8 edges contained in a triangle is hamiltonian. This implies that every 3-connected \(\{K_{1,3}\}\cup \{C_i|i\ge 9\}\)-free graph is hamiltonian. We also show that every 3-connected o-heavy graph whose induced cycle of length is at most 8 is hamiltonian and that every 2-connected claw-free graph of longest induced cycle with length at least \(n-2\) is hamiltonian. These results are all best possible. As a byproduct, we show that it is NP-complete to determine the length of a longest induced cycle of a line graph.
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Acknowledgments
The authors are grateful to the editors and anonymous referees for their valuable comments and useful suggestions. This research is supported by Nature Science Funds of China (No. 11171129, No. 1147 1037, No. 61164005) and by Specialized Research Fund for the Doctoral Program of Higher Education (No. 20131101110048).
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Yin, J., Xiong, L. & Zhao, H. Hamiltonian Claw-Free Graphs and o-Heavy Graphs Involving Induced Cycles. Graphs and Combinatorics 32, 823–833 (2016). https://doi.org/10.1007/s00373-015-1605-7
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DOI: https://doi.org/10.1007/s00373-015-1605-7