Czechoslovak Mathematical Journal

, Volume 68, Issue 1, pp 19–34 | Cite as

When a line graph associated to annihilating-ideal graph of a lattice is planar or projective

  • Atossa Parsapour
  • Khadijeh Ahmad Javaheri


Let (L,∧, ∨) be a finite lattice with a least element 0. AG(L) is an annihilating-ideal graph of L in which the vertex set is the set of all nontrivial ideals of L, and two distinct vertices I and J are adjacent if and only if IJ = 0. We completely characterize all finite lattices L whose line graph associated to an annihilating-ideal graph, denoted by L(AG(L)), is a planar or projective graph.


annihilating-ideal graph lattice line graph planar graph projective graph 

MSC 2010

05C75 05C10 06B10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    M. Afkhami, S. Bahrami, K. Khashyarmanesh, F. Shahsavar: The annihilating-ideal graph of a lattice. Georgian Math. J. 23 (2016), 1–7.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    D. F. Anderson, M. C. Axtell, J. A. Stickles: Zero-divisor graphs in commutative rings. Commutative Algebra, Noetherian and Non-Noetherian Perspectives (M. Fontana et al., eds.). Springer, New York, 2011, pp. 23–45.Google Scholar
  3. [3]
    D. Archdeacon: A Kuratowski theorem for the projective plane. J. Graph Theory 5 (1981), 243–246.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    I. Beck: Coloring of commutative rings. J. Algebra 116 (1988), 208–226.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    M. Behboodi, Z. Rakeei: The annihilating-ideal graph of commutative rings I. J. Algebra Appl. 10 (2011), 727–739.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    M. Behboodi, Z. Rakeei: The annihilating-ideal graph of commutative rings II. J. Algebra Appl. 10 (2011), 741–753.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    J. A. Bondy, U. S. R. Murty: Graph Theory with Applications, American Elsevier Publishing, New York, 1976.CrossRefzbMATHGoogle Scholar
  8. [8]
    A. Bouchet: Orientable and nonorientable genus of the complete bipartite graph. J. Comb. Theory, Ser. B 24 (1978), 24–33.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    H.-J. Chiang-Hsieh, P.-F. Lee, H.-J. Wang: The embedding of line graphs associated to the zero-divisor graphs of commutative rings. Isr. J. Math. 180 (2010), 193–222.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    B. A. Davey, H. A. Priestley: Introduction to Lattices and Order, Cambridge University Press, Cambridge, 2002.CrossRefzbMATHGoogle Scholar
  11. [11]
    H. H. Glover, J. P. Huneke, C. S. Wang: 103 graphs that are irreducible for the projective plane. J. Comb. Theory, Ser. B 27 (1979), 332–370.Google Scholar
  12. [12]
    C. Godsil, G. Royle: Algebraic Graph Theory. Graduate Texts in Mathematics 207, Springer, New York, 2001.Google Scholar
  13. [13]
    K. Khashyarmanesh, M. R. Khorsandi: Projective total graphs of commutative rings. Rocky Mt. J. Math. 43 (2013), 1207–1213.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    W. S. Massey: Algebraic Topology: An Introduction. Graduate Texts in Mathematics 56, Springer, New York, 1977.Google Scholar
  15. [15]
    J. B. Nation: Notes on Lattice Theory. 1991–2009. Available at http: //www.math. Scholar
  16. [16]
    G. Ringel: Map Color Theorem. Die Grundlehren der mathematischen Wissenschaften 209, Springer, Berlin, 1974.Google Scholar
  17. [17]
    J. Roth, W. Myrvold: Simpler projective plane embedding. Ars Comb. 75 (2005), 135–155.MathSciNetzbMATHGoogle Scholar
  18. [18]
    J. Sedláček: Some properties of interchange graphs. Theory Graphs Appl. Proc. Symp. Smolenice, 1963, Czechoslovak Acad. Sci., Praha, 1964, pp. 145–150.Google Scholar
  19. [19]
    A. T. White: Graphs, Groups and Surfaces. North-Holland Mathematics Studies 8, North-Holland Publishing, Amsterdam-London; American Elsevier Publishing, New York, 1973.Google Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2018

Authors and Affiliations

  1. 1.Department of MathematicsBandar Abbas Branch, Islamic Azad UniversityBandar AbbasIran

Personalised recommendations