Skip to main content
SpringerLink
Log in

Hadwiger Numbers of Self-complementary Graphs

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

Abstract

The Hadwiger number of a graph G, denoted by h(G), is the order of the largest complete minor of G. It is known that for any \(n\equiv 0,1 (\text {mod 4})\) and any self-complementary graph G with n vertices, \( \lfloor (n+1)/2 \rfloor \le h(G) \le \lfloor 3n/5\rfloor \). In this article, we revisit the proof of the lower bound, and we prove that for all \(n\equiv 0,1 (\text {mod 4})\) and \( \lfloor (n+1)/2 \rfloor \le h \le \lfloor 3n/5\rfloor \), there exists a self-complementary graph G with n vertices whose Hadwiger number is h. We visit topological properties of self-complementary graphs.

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
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Blain, P., Bowlin, G., Fleming, T., Foisy, J., Hendricks, J., LaCombe, J.: Some results on intrinsically Knotted graphs. J. Knot Theory Ramif. 16, 749–760 (2007)

    Article  MathSciNet  Google Scholar 

  2. Battle, J., Harary, F., Kodama, Y.: Every planar graph with nine points has a non planar complement. Bull. Am. Math. Soc. 68, 569–571 (1962)

    Article  Google Scholar 

  3. Conway, J., Gordon, C.: Knots and links in spatial graphs. J. Graph Theory 7(4), 445–453 (1983)

    Article  MathSciNet  Google Scholar 

  4. de Verdière, C.: Sur un nouvel invariant des graphes et un critère de planaritè. J. Combin. Theory Ser. B 50(1), 11–21 (1990)

    Article  MathSciNet  Google Scholar 

  5. Farrugia, A.: Self-complementary graphs and generalisations: a comprehensive reference manual. Master’s Thesis, University of Malta (1999). Available at http://www.alastairfarrugia.net/sc-graph.html. Accessed 17 Mar 2020

  6. Girse, R., Gillman, R.: Homomorphisms and related contractions of graphs. Int. J. Math. Math. Sci. 11, 95–100 (1988)

    Article  MathSciNet  Google Scholar 

  7. Kostochka, A.V.: A lower bound for the product of the Hadwiger number of a graph and its complement. Komb. Anal. 8, 50–62 (1989). Moscow

    MathSciNet  MATH  Google Scholar 

  8. Kotlov, A., Lovász, L., Vempala, S.: The Colin de Verdière number and sphere representations of a graph. Combinatorica 17(4), 483–521 (1997)

    Article  MathSciNet  Google Scholar 

  9. Kuratowski, K.: Sur le problème des courbes gauches en topologie. Fund. Math. 15, 271–283 (1930)

    Article  Google Scholar 

  10. Kropar, M., Read, R.C.: On the construction of the self-complementary graphs on 12 nodes. J. Graph Theory 3, 111–125 (1979)

    Article  MathSciNet  Google Scholar 

  11. Rao, S.B., Sahoo, U.K.: Complexity results in self-complementary graphs. AIMSCS research report no RR2014-21 (2014)

  12. Ringel, G.: Selbstkomplementäre Graphen. Arch. Math. 14, 354–358 (1963)

    Article  MathSciNet  Google Scholar 

  13. Robertson, N., Seymour, P., Thomas, R.: Linkless embeddings of graphs in 3-space. Bull. Am. Math. Soc. 28(1), 84–89 (1993)

    Article  MathSciNet  Google Scholar 

  14. Sachs, H.: On spatial representations of finite graphs A. In: Hajnal, Lovasz, L., Sós, V.T., (Eds.), Colloq. Math. Soc. János Bolyai, Vol. 37, North-Holland, Amsterdam, pp. 649–662 (1984)

  15. Sachs, H.: Über selbstkomplementäre Graphen. Publ. Math. Debrecen 9, 270–288 (1962)

    MathSciNet  MATH  Google Scholar 

  16. Stiebitz, M.: On Hadwiger’s number—a problem of the Nordhaus–Gaddum type. Discr. Math. 101, 307–317 (1992)

    Article  MathSciNet  Google Scholar 

  17. Wagner, K.: Über eine Eigenschaft der ebenen Komplexe. Math. Ann. 114, 570–590 (1937)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of South Alabama, Mobile, AL, 36688, USA

    Andrei Pavelescu & Elena Pavelescu

Authors
  1. Andrei Pavelescu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Elena Pavelescu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Elena Pavelescu.

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

Pavelescu, A., Pavelescu, E. Hadwiger Numbers of Self-complementary Graphs. Graphs and Combinatorics 36, 865–876 (2020). https://doi.org/10.1007/s00373-020-02159-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-020-02159-8

Keywords

Search

Navigation