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

Let Gamma (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 Gamma(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 Gamma(R) not equal theta, then Gamma(R) is not planar. We use the set of the associated primes to find the minimal length of a cycle in Gamma(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 greater than or equal to 3, if Gamma(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), ((xy,y2-x))/(Zp[x,y]), ((py,y2-ps))/(Zp2[y]), where 1 less than or equal to s < p. (C) 2003 Published by Elsevier Inc.