Order

, Volume 29, Issue 3, pp 499–506 | Cite as

Zero Divisor Graph of a Poset with Respect to an Ideal

Article

Abstract

In this paper, we introduce the zero divisor graph GI(P) of a poset P (with 0) with respect to an ideal I. It is shown that GI(P) is connected with its diameter ≤3, and if GI(P) contains a cycle, then the core K of GI(P) is a union of 3-cycles and 4-cycles. Further, the chromatic number and clique number of GI(P) are shown to be equal. This proves a form of Beck’s conjecture for posets with 0. The complete bipartite zero divisor graphs are characterized.

Keywords

Zero divisor graph Clique number Chromatic number Annihilator Semi-ideal Ideal Prime ideal Semiprime ideal 

Mathematics Subject Classifications (2010)

Primary 05C15; Secondary 06A12 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Anderson, D.D., Naseer, M.: Beck’s coloring of a commutative ring. J. Algebra 159, 500–514 (1993)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Anderson, D.F., Livingstone, P.: The zero divisor graph of a commutative ring. J. Algebra 217, 434–447 (1999)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Beck, I.: Coloring of a commutative ring. J. Algebra 116, 208–226 (1988)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    DeMeyer, F., McKenzie, T., Schneider, K.: The zero divisor graph of a commutative semigroup. Semigroup Forum 65, 206–214 (2002)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Harary, F.: Graph Theory. Narosa, New Delhi (1988)Google Scholar
  6. 6.
    Halaš, R.: Ideals and annihilators in ordered sets. Czech. Math. J. 45, 127–134 (1995)MATHGoogle Scholar
  7. 7.
    Halaš, R., Jukl, M.: On Beck’s coloring of posets. Discrete Math. 309, 4584–4589 (2009)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Halaš, R., Länger, H.: The zero divisor graph of a qoset. Order. doi:10.1007/s11083-009-9120-1
  9. 9.
    Kharat, V.S., Mokbel, K.A.: Semiprime ideals and separation theorems for posets. Order 25, 195–210 (2008)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Maimani, H.R., Pournaki, M.R., Yassemi, S.: Zero divisor graphs with respect to an ideal. Commun. Algebra 34, 923–929 (2006)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Nimbhorkar, S.K., Wasadikar, M.P., DeMeyer, L.: Coloring of semilattices. Ars Comb. 12, 97–104 (2007)MathSciNetGoogle Scholar
  12. 12.
    Redmond, S.P.: An ideal based zero divisor graph of a commutative ring. Commun. Algebra 31, 4425–4423 (2003)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Venkatanarsimhan, P.V.: Semi-ideals in posets. Math. Ann. 185, 338–348 (1970)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PunePuneIndia

Personalised recommendations