When a zero-divisor graph is planar or a complete r-partite graph

Abstract

Let Γ( R) be the zero-divisor graph of a commutative ring R. An interesting question was proposed by Anderson, Frazier, Lauve, and Livingston: For which finite commutative rings R is Γ( R) planar? We give an answer to this question. More precisely, we prove that if R is a local ring with at least 33 elements, and Γ( R)≠∅, then Γ( R) is not planar. We use the set of the associated primes to find the minimal length of a cycle in Γ( R). Also, we determine the rings whose zero-divisor graphs are complete r-partite graphs and show that for any ring R and prime number p, p⩾3, if Γ( R) is a finite complete p-partite graph, then | Z( R)|= p 2, | R|= p 3, and R is isomorphic to exactly one of the rings Z p 3 , Z p[x,y] (xy,y 2−x) , Z p 2 [y] (py,y 2−ps) , where 1⩽ s< p.