Planar zero-divisor graphs

Abstract

This paper answers the question of Anderson, Frazier, Lauve, and Livingston: for which finite commutative rings R is the zero-divisor graph Γ (R) planar? We build upon and extend work of Akbari, Maimani, and Yassemi, who proved that if R is any local ring with more than 32 elements, and R is not a field, then Γ (R) is not planar. They left open the question: "Is it true that, for any local ring R of cardinality 32, which is not a field, Γ (R) is not planar?" In this paper we answer this question in the affirmative. We prove that if R is any local ring with more than 27 elements, and R is not a field, then Γ (R) is not planar. Moreover, we determine all finite commutative local rings whose zero-divisor graph is planar. © 2007 Elsevier Inc. All rights reserved.

Department(s)

Mathematics

Document Type

Article

DOI

https://doi.org/10.1016/j.jalgebra.2007.01.049

Keywords

Finite commutative ring, Planar graph, Zero-divisor graph

Publication Date

10-1-2007

Journal Title

Journal of Algebra

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