Skip to main content
SpringerLink
Log in

A New Zero-divisor Graph Contradicting Beck’s Conjecture, and the Classification for a Family of Polynomial Quotients

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

Abstract

We classify all possible zero-divisor graphs of a particular family of quotients of \(\mathbf{Z}_4[x,y,w,z]\). As the 90 quotients vary, we obtain a total of 7 graphs, corresponding to seven isomorphism classes, and one of these graphs provides a new example which contradicts Beck’s conjecture on the chromatic number of a zero-divisor graph. The algebraic analysis is strongly supported by the combinatorial setting, as already shown in a previous paper, where the graph-theoretical tools were presented and successfully applied to \(\mathbf{Z}_4[x,y,z]\)—therefore, the just smaller case—in order to get a deeper knowledge of the classical counterexample to Beck’s conjecture.

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

Similar content being viewed by others

References

  1. Anderson, D.D., Naseer, M.: Beck’s coloring of a commutative ring. J. Algebra 159, 500–514 (1993)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Atiyah, M.F., Macdonald, I.G.: Introduction to Commutative Algebra. Addison-Wesley, Menlo Park (1969)

    MATH  Google Scholar 

  4. Axtell, M., Coykendall, J., Stickles, J.: Zero-divisor graphs of polynomials and power series over commutative rings. Comm. Algebra 33, 2043–2050 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Axtell, M., Stickles, J., Warfel, J.: Zero-divisor graphs of direct products of commutative rings. Houston J. Math. 32, 985–994 (2006). (electronic)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Bryant, V.: Aspects of Combinatorics: a Wide-ranging Introduction. Cambridge University Press, Cambridge (1993)

    MATH  Google Scholar 

  8. DeMeyer, F., McKenzie, T., Schneider, K.: The zero-divisor graph of a commutative semigroup. Semigroup Forum 65, 206–214 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. DeMeyer, L., D’Sa, M., Epstein, I., Geiser, A., Smith, K.: Semigroups and the zero divisor graph. Bull. Inst. Combin. Appl. 57, 60–70 (2009)

    MathSciNet  MATH  Google Scholar 

  10. Vietri, A.: A combinatorial analysis of zero-divisor graphs on certain polynomial rings. Comm. Algebra 41, 2040–2047 (2013)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The author is grateful to the anonymous referees, for their valuable comments and suggestions.

Author information

Authors and Affiliations

  1. Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Sapienza Università di Roma, via A. Scarpa 16, 00161, Rome, Italy

    Andrea Vietri

Authors
  1. Andrea Vietri
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrea Vietri.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vietri, A. A New Zero-divisor Graph Contradicting Beck’s Conjecture, and the Classification for a Family of Polynomial Quotients. Graphs and Combinatorics 31, 2413–2423 (2015). https://doi.org/10.1007/s00373-014-1501-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-014-1501-6

Keywords

Mathematics Subject Classification

Search

Navigation