Skip to main content
SpringerLink
Log in

Toroidality of Intersection Graphs of Ideals of Commutative Rings

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

Abstract

Let R be a commutative ring with identity and G(R) its intersection graph. In this paper, all toroidal graphs that are intersection graphs are classified. An improvement over the previous results concerning the planarity of these graphs is also presented.

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. Akbari, S., Nikadish, R., Nikmehr, M.J.: Some results on the intersection graphs of ideals of rings. ICTP, Report No. IC/2010/74

  2. Anderson D.F., Badawi A.: The total graph of a commutative ring. J. Algebra. 320, 2706–2719 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  4. Atiyah, M.F., Macdonald, I.G.: Introduction to commutative algebra. Adison-Wesley Publishing Company, Reading, MA (1969)

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

    Article  MATH  MathSciNet  Google Scholar 

  6. Belshoff R., Chapman J.: Planar zero-divisor graph. J. Algebra. 316(1), 471–480 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bloomfield N., Wickham C.: Local rings with genus two zero divisor graph. Comm. Algebra. 38, 2965–2980 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  8. Chakrabarty I., Ghosh S., Mukherjee T.K., Sen M.K.: Intersection graphs of ideals of rings. Discrete Math. 309(17), 5381–5392 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  9. Diestel, R.: Graph Theory. Graduate Texts in Mathematics, 3rd edn, vol. 173, Springer-Verlag, Berlin (2005)

  10. Jafari S.H., Rad N.J.: Planarity of intersection graphs of ideals of rings. Int. El. J. Algebra. 8, 161–166 (2010)

    MATH  Google Scholar 

  11. Kaplansky, I.: Commutative rings, Revised Edition, University of Chicago Press, Chicago, (1974)

  12. Maimani, H.R., Wickham, C., Yassemi, S.: Rings whose total graphs have genus at most one. Rocky Mountain J. Math. (to appear)

  13. Redmond S.P.: On zero-divisor graphs of small finite commutative rings. Discrete Math. 307, 1155–1166 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. Ringel G., Youngs J.W.T.: Solution of the Heawood map-coloring problem. Proc. Nat. Acad. Sei. USA . 60, 438–445 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  15. Wang H.: Zero-divisor graphs of genus one. J. Algebra. 304(2), 666–678 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. Wang H.: Graphs associated to co-maximal ideals of commutative rings. J. Algebra. 320(7), 2917–2933 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  17. Wickham C.: Classification of rings with genus one zero-divisor graphs. Comm. Algebra. 36(2), 1–21 (2008)

    MathSciNet  Google Scholar 

  18. Wickham C.: Rings whose zero-divisor graphs have positive genus. J. Algebra. 321(2), 377–383 (2009)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Civil Engineering, University of Belgrade, Bulevar Kralja Aleksandra 73, 11000, Belgrade, Serbia

    Zoran S. Pucanović

  2. Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000, Belgrade, Serbia

    Zoran Z. Petrović

Authors
  1. Zoran S. Pucanović
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zoran Z. Petrović
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zoran S. Pucanović.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pucanović, Z.S., Petrović, Z.Z. Toroidality of Intersection Graphs of Ideals of Commutative Rings. Graphs and Combinatorics 30, 707–716 (2014). https://doi.org/10.1007/s00373-013-1292-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-013-1292-1

Keywords

Mathematics Subject Classification (2000)

Search

Navigation