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
Recommended Citation
Belshoff, Richard, and Jeremy Chapman. "Planar zero-divisor graphs." Journal of Algebra 316, no. 1 (2007): 471-480.
Journal Title
Journal of Algebra
