Skip to main content
SpringerLink
Log in

Edge-Transitive Regular Metacyclic Covers of the Petersen Graph

  • Original Paper
  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

In this paper, a complete classification is given of all the regular metacyclic covers of the Petersen graph, whose fibre-preserving groups act edge-transitively. Moreover, as an application, it is proved that there is no cubic 1-arc-regular graph with order 5m where m is a cube-free integer and coprime to 15.

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

Similar content being viewed by others

References

  1. Bouwer, I.Z. (ed.): The Foster Census. Charles Babbage Research Center, Winnipeg (1988)

    MATH  Google Scholar 

  2. Bosma, W., Cannon, C., Playoust, C.: The Magma algebra system I: the user languages. J. Symb. Comput. 24, 235–265 (2007)

    Article  MathSciNet  Google Scholar 

  3. Conder, M., Dobcsańyi, P.: Trivalent symmetric graphs on up to 768 vertices. J. Comb. Math. Comput. 40, 41–63 (2002)

    MATH  Google Scholar 

  4. Conder, M., Ma, J.C.: Arc-transitive abelian regular covers of cubic graphs. J. Algebra 387, 215–242 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Conder, M., Ma, J.C.: Arc-transitive abelian regular covers of the Heawood graph. J. Algebra 387, 243–267 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fang, X.G., Wang, J., Xu, M.Y.: On 1-arc-regular graphs. Eur. J. Comb. 23, 785–791 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Feng, Y.Q., Kwak, J.H.: Classifying cubic symmetric graphs of order \(10p\) or \(10p^2\). Sci. China 49, 300–319 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Feng, Y.Q., Wang, K.S.: \(s\)-Regular cyclic covers of the three-dimensional hypercube \(Q_3\). Eur. J. Comb. 24, 719–731 (2003)

    Article  MATH  Google Scholar 

  9. Feng, Y.Q., Li, Y.T.: One-regular graphs of square-free order of prime valency. Eur. J. Comb. 32, 261–275 (2011)

    Article  Google Scholar 

  10. Gorenstein, D.: Finite Groups, 2nd edn. AMS Publishing, Providence (1980)

    MATH  Google Scholar 

  11. Gorenstein, D.: Finite Simple Groups. Plenum Press, New York (1982)

    Book  MATH  Google Scholar 

  12. Gross, J.L., Tucker, T.W.: Generating all graph coverings by permutation voltage assignment. Discret. Math. 18, 273–283 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  13. Huppert, B.: Finite Groups. Springer, Berlin (1967)

    Google Scholar 

  14. Kuzman, B.: Arc-transitive elementary abelian covers of the complete graph \({\sf K}_5\). Linear Algebr. Appl. 433, 1909–1921 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kwak, J.H., Oh, J.M.: Arc-transitive elementary abelian covers of the octahedron graph. Linear Algebr. Appl. 429, 2180–2198 (2009)

    Article  MathSciNet  Google Scholar 

  16. Li, C.H., Pan, J.M.: Finite 2-arc-transitive abelian Cayley graphs. Eur. J. Comb. 29, 148–158 (2007)

    Article  MathSciNet  Google Scholar 

  17. Malnič, A., Nedela, R., Škoviera, M.: Lifting graph automorphisms by voltage assignments. Eur. J. Comb. 21, 927–947 (2000)

    Article  MATH  Google Scholar 

  18. Malnič, A., Marušič, D., Potočnik, P.: Elementary abelian covers of graphs. J. Algebr. Comb. 20, 71–97 (2004)

    Article  MATH  Google Scholar 

  19. Malnič, A., Potočnik, P.: Invariant subspaces, duality, and covers of the Petersen graph. Eur. J. Comb. 27, 971–989 (2008)

    Article  Google Scholar 

  20. Malnič, A., Marušič, D., Miklavič, S., Potočnik, P.: Semisymmetric elementary abelian covers of the Möbius–Kantor graph. Discret. Math. 307, 2156–2175 (2007)

    Article  MATH  Google Scholar 

  21. Malnič, A., Marušič, D., Potočnik, P.: On cubic graphs admitting an edge-transitive solvable group. J. Algebr. Comb. 20, 99–113 (2004)

    Article  MATH  Google Scholar 

  22. Oh, J.M.: Arc-transitive elementary abelian covers of the Pappus graphs. Discret. Math. 309, 6590–6611 (2009)

    Article  MATH  Google Scholar 

  23. Pan, J.M., Ding, S.Y., Liu, Y.: Finite arc-regular graphs of prime valency. Sci. China (In Chinese) 44, 307–315 (2014)

    Google Scholar 

  24. Praeger, C.E.: An O’Nan–Scott theorem for finite quasiprimitive permutation groups and an application to 2-arc transitive graphs. J. Lond. Math. Soc. 47, 227–239 (1992)

    MathSciNet  MATH  Google Scholar 

  25. Qiao, S.H., Li, C.H.: The finite groups of cube-free order. J. Algebra 334, 101–108 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Schur, I.: Untersuchen über die Darstellung der endlichen Gruppen durch gebrochenen linearen Substitutionen. Crelle J. 132, 85–137 (1907)

    MathSciNet  MATH  Google Scholar 

  27. Škoviera, M.: A contribution to the theory of voltage graphs. Discret. Math. 61, 281–292 (1986)

    Article  MATH  Google Scholar 

  28. Wang, C.Q., Hao, Y.: Edge-transitive regular \({\sf Z}_n\)-covers of the Heawood graph. Discret. Math. 310, 1752–1758 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. Zhou, J.X., Feng, Y.Q.: Cubic one-regular graphs of order twice a square-free integer. Sci. China 51, 1093–1100 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  30. Zhou, J.X., Feng, Y.Q.: Edge-transitive dihedral or cyclic covers of cubic symmetric graphs of order \(2p\). Combinatorica 34, 115–128 (2014)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors are very grateful to the referees for their valuable remarks.

Author information

Authors and Affiliations

  1. School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, People’s Republic of China

    Jiangmin Pan

  2. School of Mathematics and Statistics, Yunnan University, Kunming, People’s Republic of China

    Jiangmin Pan & Zhaohong Huang

Authors
  1. Jiangmin Pan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhaohong Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jiangmin Pan.

Additional information

This work was partially supported by the National Natural Science Foundation of China (11231008, 11461077).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pan, J., Huang, Z. Edge-Transitive Regular Metacyclic Covers of the Petersen Graph. Graphs and Combinatorics 32, 777–784 (2016). https://doi.org/10.1007/s00373-015-1597-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-015-1597-3

Keywords

Mathematics Subject Classification

Search

Navigation