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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 $\Gamma(R)\neq \emptyset$, 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 $p$, $p\geq 3$, if
$\Gamma (R)$ is a finite complete $p$-partite graph, then
$|\T{Z}(R)|=p^2, |R|=p^3$, and $R$ is isomorphic to exactly one of
the rings $\mathbb{Z}_{p^3}$, $\frac{\mathbb{Z}_p[x,y]}{(xy,
y^2-x)}, \mathrm \ \ \frac{\mathbb{Z}_{p^2}[y]}{(py,y^2-ps)}$,
where $1\leq s < p.$
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