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

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)=p2, R=p3, and R is isomorphic to exactly one of the rings , , , where 1 s