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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 Z _p³, Z _p[x,y] (xy,y ²−x) , Z _p²[y] (py,y ²−ps) , where 1⩽s<p.