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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Bauer, D., Schmeichel, E.F.: Hamiltonian degree conditions which imply a graph is pancyclic. J. Combin. Theory (B) 48, 111–116 (1990)
Bondy, J.A.: Pancyclic graphs. J. Combin. Theory (B) 11, 80–84 (1971)
Bondy, J.A.: Longest paths and cycles in graphs of high degree. Res. Rep. No. CORR 80–16, Univ. Waterloo, Waterloo, ON (1980)
Chvátal, V.: On Hamiltons ideals. J. Combin. Theory (B) 12, 163–168 (1972)
Fan, G.: New sufficient conditions for cycles in graphs. J. Combin. Theory (B) 37, 221–227 (1984)
Milans, K.G., Pfender, F., Rautenbach, D., Regen, F., West, D.B.: Cycle spectra of Hamiltonian graphs. J. Combin. Theory (B) 102, 869–874 (2012)
Marczyk, A., Woźniak, M.: Cycles in hamiltonian graphs of prescribed maximum degree. Disc. Math. 266, 321–326 (2003)
Müttel, J., Rautenbach, D., Regen, F., Sasse, T.: On the cycle spectrum of cubic Hamiltonian graphs. Graphs Comb. 29, 1067–1076 (2013)
Ore, O.: Note on Hamilton circuits. Am. Math. Monthly 67, 55 (1960)
Schmeichel, E.F., Hakimi, S.L.: A cycle structure theorem for Hamiltonian graphs. J. Combin. Theory (B) 45, 99–107 (1988)
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We were supported by the DFG project “Cycle Spectra of Graphs” RA873/5-1.
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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