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Strongly Spanning Trailable Graphs with Small Circumference and Hamilton-Connected Claw-Free Graphs

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Abstract

A graph G is strongly spanning trailable if for any \(e_1=u_1v_1, e_2=u_2v_2\in E(G)\) (possibly \(e_1=e_2\)), \(G(e_1, e_2)\), which is obtained from G by replacing \(e_1\) by a path \(u_1v_{e_1}v_1\) and by replacing \(e_2\) by a path \(u_2v_{e_2}v_2\), has a spanning \((v_{e_1}, v_{e_2})\)-trail. A graph G is Hamilton-connected if there is a spanning path between any two vertices of V(G). In this paper, we first show that every 2-connected 3-edge-connected graph with circumference at most 8 is strongly spanning trailable with an exception of order 8. As applications, we prove that every 3-connected \(\{K_{1, 3}, N_{1, 2, 4}\}\)-free graph is Hamilton-connected and every 3-connected \(\{K_{1, 3}, P_{10}\}\)-free graph is Hamilton-connected with a well-defined exception. The last two results extend the results in Hu and Zhang (Graphs Comb 32: 685–705, 2016) and Bian et al. (Graphs Comb 30: 1099–1122, 2014) respectively.

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References

  1. Bian, Q., Gould, R.J., Horn, P., Janiszewski, S., Fleur, S., Wrayno, P.: 3-connected {K1, 3, P9}-free graphs are Hamiltonian-connected. Graphs Comb. 30, 1099–1122 (2014)

    Article  Google Scholar 

  2. Bondy, J.A., Murty, U.S.R.: Graph Theory. Springer, Berlin (2008)

    Book  Google Scholar 

  3. Broersma, H., Faudree, R.J., Huck, A., Trommel, H., Veldman, H.J.: Forbidden subgraphs that imply Hamiltonian-connectedness. J. Graph Theory 40, 104–119 (2002)

    Article  MathSciNet  Google Scholar 

  4. Brousek, J., Ryjáček, Z., Favaron, O.: Forbidden subgraphs, Hamiltonicity and closure in claw-free graphs. Discrete Math. 196, 29–50 (1999)

    Article  MathSciNet  Google Scholar 

  5. Catlin, P.A.: A reduction method to find spanning Eulerian subgraphs. J. Graph Theory 12, 29–44 (1988)

    Article  MathSciNet  Google Scholar 

  6. Catlin, P.A.: Supereulerian graphs, collapsible graphs and four-cycles. Congressus Numerantium 58, 233–246 (1988)

    MathSciNet  Google Scholar 

  7. Catlin, P.A., Han, Z.Y., Lai, H.-J.: Graphs without spanning closed trals. Discrete Math. 160, 81–91 (1996)

    Article  MathSciNet  Google Scholar 

  8. Chen, G., Gould, R.J.: Hamiltonian connected graphs involving forbidden subgraphs. Bull. Inst. Combin. Appl. 29, 25–32 (2000)

    MathSciNet  Google Scholar 

  9. Faudree, J.R., Faudree, R.J., Ryjáček, Z., Vrána, P.: On forbidden pairs implying Hamiltionconnectedness. J. Graph Theory 72, 327–345 (2013)

    Article  MathSciNet  Google Scholar 

  10. Faudree, R.J., Gould, R.J.: Characterizing forbidden pairs for Hamiltonian properties. Discrete Math. 173, 45–60 (1997)

    Article  MathSciNet  Google Scholar 

  11. Fujita, S., Kawarabayashi, K., Lucchesi, C.L., Ota, K., Plummer, M., Saito, A.: A pair of forbidden subgraphs and perfect matchings. J. Combin. Theory Ser. B 96, 315–324 (2006)

    Article  MathSciNet  Google Scholar 

  12. Hu, Z., Zhang, S.: Every 3-connected $\{K_{1, 3}, N_{1, 2, 3}\}$-free graph is Hamilton-connected. Graphs Comb. 32, 685–705 (2016)

    Article  MathSciNet  Google Scholar 

  13. Kužel, R., Ryjáček, Z., Teska, J., Vrána, P.: Closure, clique covering and degree conditions for Hamilton-connectedness in claw-free graphs. Discrete Math. 312, 2177–2189 (2012)

    Article  MathSciNet  Google Scholar 

  14. Lai, H.-J., Shao, Y., Yu, G., Zhan, M.: Hamiltonian connectedness in 3-connected line graphs. Discrete Math. 157, 152–173 (2009)

    MathSciNet  Google Scholar 

  15. Lai H.-J., Wang K., Xie X., Zhan M.: Catlins reduced graphs with small orders (Submitted)

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

    Article  MathSciNet  Google Scholar 

  17. Li, D., Lai, H.-J., Zhan, M.: Eulerian subgraphs and Hamilton-connected line graphs. Discrete Appl. Math. 145, 422–428 (2005)

    Article  MathSciNet  Google Scholar 

  18. Li, P., Wang, K., Zhan, M., Lai, H.-J.: Strongly spanning trailable graphs with short longest paths. ARS Comb. 137, 3–39 (2018)

    MathSciNet  Google Scholar 

  19. Liu, D., Lai, H.-J., Chen, Z.H.: Reinforcing the number of disjoint spanning trees. ARS Comb. 93, 113–127 (2009)

    MathSciNet  Google Scholar 

  20. Liu, X., Lin, H., Xiong, L.: Forbidden subgraphs and weak locally connected graphs. Graphs Comb. 34, 1671–1690 (2018)

    Article  MathSciNet  Google Scholar 

  21. Liu, J., Yu, A., Wang, K., Lai, H.-J.: Degree sum and Hamiltonian-connected line graphs. Discrete Math. 341, 1363–3179 (2018)

    Article  MathSciNet  Google Scholar 

  22. Ma, X., Lai, H.-J., Xiong, W., Wu, B., An, X.: Supereulerian graphs with small circumference and 3-connected Hamiltonian claw-free graphs. Discrete Math. 202, 111–130 (2016)

    Article  MathSciNet  Google Scholar 

  23. Shao, Y.: Claw-free graphs and line graphs. Ph.D Thesis, West Virginia University (2005)

  24. Shepherd, F.B.: Hamiltonicity in claw-free graphs. J. Combin. Theory Ser. B 53, 173–194 (1991)

    Article  MathSciNet  Google Scholar 

  25. Wang, K.: Supereulerian properties in graphs and Hamiltonian properties in line graphs. Ph.D Thesis, West Virginia University (2015)

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Acknowledgements

The authors thank the referees very much for their carefully reading. The work is supported by the Natural Science Funds of China (nos: 11871099 and 11671037).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, People’s Republic of China

    Xia Liu

  2. School of Mathematics and Statistics, Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing, 100081, People’s Republic of China

    Liming Xiong

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

    Hong-Jian Lai

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  1. Xia Liu
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  2. Liming Xiong
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  3. Hong-Jian Lai
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Correspondence to Liming Xiong.

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Liu, X., Xiong, L. & Lai, HJ. Strongly Spanning Trailable Graphs with Small Circumference and Hamilton-Connected Claw-Free Graphs. Graphs and Combinatorics 37, 65–85 (2021). https://doi.org/10.1007/s00373-020-02236-y

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