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Generalized connectivity of some total graphs

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Abstract

We study the generalized k-connectivity κk(G) as introduced by Hager in 1985, as well as the more recently introduced generalized k-edge-connectivity λk(G). We determine the exact value of κk (G) and λk (G) for the line graphs and total graphs of trees, unicyclic graphs, and also for complete graphs for the case k = 3.

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References

  1. L. W. Beineke, R. J. Wilson (eds.): Topics in Structural Graph Theory. Encyclopedia of Mathematics and its Applications 147. Cambrige University Press, Cambridge, 2013.

    MATH  Google Scholar 

  2. F. T. Boesch, S. Chen: A generalization of line connectivity and optimally invulnerable graphs. SIAM J. Appl. Math. 34 (1978), 657–665.

    Article  MathSciNet  Google Scholar 

  3. J. A. Bondy, U. S. R. Murty: Graph Theory. Graduate Texts in Mathematics 244. Springer, Berlin, 2008.

    Book  Google Scholar 

  4. G. Chartrand, S. F. Kappor, L. Lesniak, D. R. Lick: Generalized connectivity in graphs. Bull. Bombay Math. Colloq. 2 (1984), 1–6.

    Google Scholar 

  5. G. Chartrand, F. Okamoto, P. Zhang: Rainbow trees in graphs and generalized connectivity. Networks 55 (2010), 360–367.

    MathSciNet  MATH  Google Scholar 

  6. G. Chartrand, M. J. Stewart: The connectivity of line-graphs. Math. Ann. 182 (1969), 170–174.

    Article  MathSciNet  Google Scholar 

  7. R. Gu, X. Li, Y. Shi: The generalized 3-connectivity of random graphs. Acta Math. Sin., Chin. Ser. 57 (2014), 321–330. (In Chinese.)

    MATH  Google Scholar 

  8. M. Hager: Pendant tree-connectivity. J. Comb. Theory, Ser. B 38 (1985), 179–189.

    Article  MathSciNet  Google Scholar 

  9. T. Hamada, T. Nonaka, I. Yoshimura: On the connectivity of total graphs. Math. Ann. 196 (1972), 30–38.

    Article  MathSciNet  Google Scholar 

  10. M. Kriesell: Edge-disjoint trees containing some given vertices in a graph. J. Comb. Theory, Ser. B 88 (2003), 53–65.

    Article  MathSciNet  Google Scholar 

  11. M. Kriesell: Edge disjoint Steiner trees in graphs without large bridges. J. Graph Theory 62 (2009), 188–198.

    Article  MathSciNet  Google Scholar 

  12. S. Li, X. Li: Note on the hardness of generalized connectivity. J. Comb. Optim. 24 (2012), 389–396.

    Article  MathSciNet  Google Scholar 

  13. S. Li, X. Li, Y. Shi: The minimal size of a graph with generalized connectivity κ3 = 2. Australas. J. Comb. 51 (2011), 209–220.

    MathSciNet  MATH  Google Scholar 

  14. S. Li, X. Li, W. Zhou: Sharp bounds for the generalized connectivity κ3(G). Discrete Math. 310 (2010), 2147–2163.

    Article  MathSciNet  Google Scholar 

  15. S. Li, W. Li, Y. Shi, H. Sun: On minimally 2-connected graphs with generalized connectivity κ3 = 2. J. Comb. Optim. 34 (2017), 141–164.

    Article  MathSciNet  Google Scholar 

  16. S. Li, Y. Shi, J. Tu: The generalized 3-connectivity of Cayley graphs on symmetric groups generated by trees and cycles. Graphs Comb. 33 (2017), 1195–1209.

    Article  MathSciNet  Google Scholar 

  17. X. Li, Y. Mao: Generalized Connectivity of Graphs. SpringerBriefs in Mathematics. Springer, Cham, 2016.

    Book  Google Scholar 

  18. X. Li, Y. Mao, Y. Sun: On the generalized (edge-)connectivity of graphs. Australas. J. Comb. 58 (2014), 304–319.

    MathSciNet  MATH  Google Scholar 

  19. C. S. J. A. Nash-Williams: Edge-disjoint spanning trees of finite graphs. J. Lond. Math. Soc. 36 (1961), 445–450.

    Article  MathSciNet  Google Scholar 

  20. F. Okamoto, P. Zhang: The tree connectivity of regular complete bipartite graphs. J. Comb. Math. Comb. Comput. 74 (2010), 279–293.

    MathSciNet  MATH  Google Scholar 

  21. W. T. Tutte: On the problem of decomposing a graph into n connected factors. J. Lond. Math. Soc. 36 (1961), 221–230.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, Qinghai Nationalities University, No. 3 Bayizhonglu, Chengdong, Xining, Qinghai, 810007, P. R. China

    Yinkui Li

  2. School of Mathematics and Statistics, Qinghai Normal University, No. 38 W. Wusi Avenue, Xining, Qinghai, 810008, P. R. China

    Yaping Mao

  3. School of Science, China Jiliang University, No. 258, Xueyuan Street, Hangzhou, Zhejiang, 310018, P. R. China

    Zhao Wang

  4. School of Science, Xianing University of Architecture and Technology, Xianing, No. 13, Yanta Road, Shaanxi, 710055, P. R. China

    Zongtian Wei

Authors
  1. Yinkui Li
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  2. Yaping Mao
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  3. Zhao Wang
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  4. Zongtian Wei
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Corresponding author

Correspondence to Yinkui Li.

Additional information

Supported by the National Science Foundation of China (No. 11661066) and the Science Found of Qinghai Province (No. 2017-ZJ-701) and the Science Found of Qinghai Nationalities University (2019XJG10, 2020XJGH14).

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Li, Y., Mao, Y., Wang, Z. et al. Generalized connectivity of some total graphs. Czech Math J 71, 623–640 (2021). https://doi.org/10.21136/CMJ.2021.0287-19

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  • DOI: https://doi.org/10.21136/CMJ.2021.0287-19

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