Skip to main content
SpringerLink
Log in

Hamiltonian cycles of power graph of abelian groups

  • Published:
Afrika Matematika Aims and scope Submit manuscript

Abstract

In this article we discuss the question of existence of Hamiltonian cycles in the undirected power graph of a group, where power graph is defined as a graph with the group as the vertex set and edges between two distinct elements whenever one is a power of the other. We describe a new structural description of power graphs through vertex weighted directed graphs. We develop the theory of weighted Hamiltonian paths in a weighted graph. We solve the Hamiltonian question completely for power graphs of a class of finite abelian groups, namely \(({\mathbb {Z}}_p)^n \times ({\mathbb {Z}}_q)^m\) where p, q are distinct primes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Aalipour, G., Akbari, S., Cameron, P.J., Nikandish, R., Shaveisi, F.: On the structure of the power graph and the enhanced power graph of a group. Electron. J. Comb. 24(3), Paper 3.16, 18 (2017)

    MathSciNet  MATH  Google Scholar 

  2. Abawajy, J., Kelarev, A., Chowdhury, M.: Power graphs: a survey. Electron. J. Graph Theory Appl. (EJGTA) 1(2), 125–147 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Anderson, D.F., Livingston, P.S.: The zero-divisor graph of a commutative ring. J. Algebra 217(2), 434–447 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Beck, I.: Coloring of commutative rings. J. Algebra 116(1), 208–226 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  5. Beineke, L.W., Wilson, R.J. (eds.): Topics in Algebraic Graph Theory. Encyclopedia of Mathematics and its Applications, vol. 102. Cambridge University Press, Cambridge (2004)

    Google Scholar 

  6. Burton, D.M.: Elementary Number Theory, 2nd edn. W. C. Brown Publishers, Dubuque (1989)

    Google Scholar 

  7. Cameron, P.J.: The power graph of a finite group. II. J. Group Theory 13(6), 779–783 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cameron, P.J., Ghosh, S.: The power graph of a finite group. Discrete Math. 311(13), 1220–1222 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chakrabarty, I., Ghosh, S., Sen, M.K.: Undirected power graphs of semigroups. Semigroup Forum 78(3), 410–426 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Curran, S.J., Gallian, J.A.: Hamiltonian cycles and paths in cayley graphs and digraphs—a survey. Discrete Math. 156(1–3), 1–18 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. Curtin, B., Yousefi-Azari, G.P.H.: On the punctured power graph of a finite group. Aust. J. Comb. 62(1), 1–7 (2015)

    MathSciNet  MATH  Google Scholar 

  12. Dehn, M.: Papers on Group Theory and Topology. Springer, New York (1987). (translated from the German and with introductions and an appendix by John Stillwell, With an appendix by Otto Schreier)

    Book  MATH  Google Scholar 

  13. DeMeyer, F., DeMeyer, L.: Zero divisor graphs of semigroups. J. Algebra 283(1), 190–198 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Doostabadi, A., Erfanian, A., Farrokhi, M.D.G.: On power graphs of finite groups with forbidden induced subgraphs. Indag. Math. (N.S.) 25(3), 525–533 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Feng, M., Ma, X., Wang, K.: The structure and metric dimension of the power graph of a finite group. Eur. J. Comb. 43, 82–97 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Godsil, C., Royle, G.F.: Algebraic Graph Theory, vol. 207. Springer, New York (2013)

    MATH  Google Scholar 

  17. Gould, R.J.: Advances on the hamiltonian problem—a survey. Graphs Comb. 19(1), 7–52 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Harary, F.: Graph Theory. Addison-Wesley Publishing Co., Reading (1969)

    Book  MATH  Google Scholar 

  19. Kelarev, A.V., Quinn, S.J.: Directed graphs and combinatorial properties of semigroups. J. Algebra 251(1), 16–26 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ma, X., Feng, M., Wang, K.: The power index of a graph. Graphs Comb. 33(5), 1381–1391 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ma, X., Feng, M., Wang, K.: The strong metric dimension of the power graph of a finite group. Discrete Appl. Math. 239, 159–164 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ma, X., Fu, R., Lu, X.: On the independence number of the power graph of a finite group. Indag. Math. (N.S.) 29(2), 794–806 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mirzargar, M., Ashrafi, A., Nadjafi-Arani, M.: On the power graph of a finite group. Filomat 26(6), 1201–1208 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Moghaddamfar, A., Rahbariyan, S., Shi, W.: Certain properties of the power graph associated with a finite group. J. Algebra Appl. 13(07), 1450040 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. Witte, D., Gallian, J.A.: A survey: Hamiltonian cycles in Cayley graphs. Discrete Math. 51(3), 293–304 (1984)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. BITS Pilani K K Birla Goa Campus Zuarinagar, Goa, India

    Himadri Mukherjee

Authors
  1. Himadri Mukherjee
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Himadri Mukherjee.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mukherjee, H. Hamiltonian cycles of power graph of abelian groups. Afr. Mat. 30, 1025–1040 (2019). https://doi.org/10.1007/s13370-019-00699-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13370-019-00699-8

Keywords

Mathematics Subject Classification

Search

Navigation