SUMMARY
Let R be a commutative ring with identity, and let Z(R) be the set of zero-divisors of R. Let EG(R) be a simple undirect graph associated with R whose vertex set is the set of all nonzero zero-divisors of R and and two distinct vertices x, y in this graph are joined by an edge if and only if AnnR(xy) is an essential ideal. A perfect graph is a graph in which the chromatic number of every induced subgraph equals the size of the largest clique of that subgraph. In this paper, we characterize all rings whose EG(R) is perfect.