Planar zero-divisor graphs

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.