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Cycle Lengths of Hamiltonian \(P_\ell \)-free Graphs

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Abstract

For an integer \(\ell \) at least three, we prove that every Hamiltonian \(P_\ell \)-free graph \(G\) on \(n>\ell \) vertices has cycles of at least \(\frac{2}{\ell }n-1\) different lengths. For small values of \(\ell \), we can improve the bound as follows. If \(4\le \ell \le 7\), then \(G\) has cycles of at least \(\frac{1}{2}n-1\) different lengths, and if \(\ell \) is \(4\) or \(5\) and \(n\) is odd, then \(G\) has cycles of at least \(n-\ell +2\) different lengths.

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Acknowledgments

We were supported by the DFG project “Cycle Spectra of Graphs” RA873/5-1.

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Correspondence to Dieter Rautenbach.

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Meierling, D., Rautenbach, D. Cycle Lengths of Hamiltonian \(P_\ell \)-free Graphs. Graphs and Combinatorics 31, 2335–2345 (2015). https://doi.org/10.1007/s00373-014-1494-1

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  • DOI: https://doi.org/10.1007/s00373-014-1494-1

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