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Pancyclicity of 4-Connected \(\{K_{1,3},Z_8\}\)-Free Graphs

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Abstract

A graph G is said to be pancyclic if G contains cycles of lengths from 3 to |V(G)|. For a positive integer i, we use \(Z_i\) to denote the graph obtained by identifying an endpoint of the path \(P_{i+1}\) with a vertex of a triangle. In this paper, we show that every 4-connected claw-free \(Z_8\)-free graph is either pancyclic or is the line graph of the Petersen graph. This implies that every 4-connected claw-free \(Z_6\)-free graph is pancyclic, and every 5-connected claw-free \(Z_8\)-free graph is pancyclic.

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References

  1. Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications. Elsevier, New York (1976)

    Book  MATH  Google Scholar 

  2. Ferrara, M., Gould, R., Gehrke, S., Magnant, C., Pfender, F.: Pancyclicity of 4-connected claw, generalized net-free graphs (submitted) (2010)

  3. Ferrara, M., Morris, T., Wenger, P.: Pancyclicity of 4-connected, claw-free, \(P_{10}\)-free graphs. J. Graph Theory 71(4), 435–447 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Gould, R., Łuczak, T., Pfender, F.: Pancyclicity of 3-connected graphs: pairs of forbidden subgraphs. J. Graph Theory 47(3), 183–202 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Lai, H.-J., Xiong, L., Yan, H., Yan, J.: Every 3-connected claw-free \(Z_8\)-free graph is Hamiltonian. J. Graph Theory 64(1), 1–11 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Matthews, M.M., Sumner, D.P.: Hamiltonian results in \(K_{1,3}\)-free graphs. J. Graph Theory 8(1), 139–146 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ryjác̆ek, Z.: On a closure concept in claw-free graphs. J. Combin. Theory Ser. B 70(2), 217–224 (1997)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, West Virginia University, Morgantown, WV, 26506, USA

    Hong-Jian Lai

  2. College of Mathematics and System Sciences, Xinjiang University, Ürümqi, 830046, Xinjiang, People’s Republic of China

    Hong-Jian Lai

  3. Department of Mathematics, Millersville University of Pennsylvania, Millersville, PA, 17551, USA

    Mingquan Zhan

  4. Department of Mathematics, Penn State Worthington Scranton, Dunmore, PA, 18512, USA

    Taoye Zhang

  5. Department of Mathematics, Kutztown University of Pennsylvania, Kutztown, PA, 19530, USA

    Ju Zhou

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  1. Hong-Jian Lai
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  2. Mingquan Zhan
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  3. Taoye Zhang
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  4. Ju Zhou
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Correspondence to Mingquan Zhan.

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Lai, HJ., Zhan, M., Zhang, T. et al. Pancyclicity of 4-Connected \(\{K_{1,3},Z_8\}\)-Free Graphs. Graphs and Combinatorics 35, 67–89 (2019). https://doi.org/10.1007/s00373-018-1987-4

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  • DOI: https://doi.org/10.1007/s00373-018-1987-4

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