Reduced zero-divisor graphs of posets

Document Type: Research Paper


North Eastern Hill University


This paper investigates properties of the reduced zero-divisor graph of a poset. We show that a vertex is an annihilator prime ideal if and only if it is adjacent to all other annihilator prime ideals and there are always two annihilator prime ideals which are not adjacent to a non-annihilator prime ideal. We also classify all posets whose reduced zero-divisor graph is planar or toroidal and the number of distinct annihilator prime ideals is four or seven.


Main Subjects

[1] M. Alizadeh, A. K. Das, H. R. Maimani, M. R. Pournaki and S. Yassemi, On the diameter and girth of zero-divisor graphs of posets, Discrete Appl. Math., 160 (2012) 1319–1324.

[2] G. Chartrand and O. R. Oellermann, Applied and Algorithmic Graph Theory, McGraw-Hill, Inc., New York, 1993.

[3] A. K. Das and D. Nongsiang, On Reduced Zero-Divisor Graphs of Posets, J. Discrete Math., 2015 (2015) pp. 7.

[4] B. A. Davey and H. A. Priestley, Introduction to Lattices and Order, second ed., Cambridge University Press, New York, 2002.

[5] R. Halas, Annihilators and ideals in ordered sets, Czechoslovak Math. J., 45 (1995) 127–134.

[6] R. Halas, Relative polars in ordered sets, Czechoslovak Math. J., 50 (2000) 415–429.

[7] R. Halas and M. Jukl, On Beck’s coloring of posets, Discrete Math., 309 (2009) 4584–4589.

[8] E. Neufeld and W. Myrvold, Practical Toroidality Testing, Proceedings of the eighth annual ACM-SIAM symposium on Discrete Algorithms, New Orleans, LA, 1997 574–580.

[9] A. T. White, Graphs, groups and surfaces, North-Holland Mathematics Studies, no. 8., American Elsevier Publishing Co., Inc., New York, 1973.