Perfect Zero-Divisor Graphs of the Ring of Gaussian Integers Modulo pn

Abstract The zero-divisor graph of a ring R is a graph whose vertex set is the set of nonzero zero-divisors of R where two vertices u and υ are connected by an edge if and only if uυ = 0. In 6, Smith studied the perfectness of the zero-divisor graph of the ring ℤ n . By definition, a perfect graph is a graph G for which every induced subgraph of G has chromatic number equal to its clique number. In this paper, we extend the work of Smith to the zero-divisor graphs of the ring ℤ p n i , where p is a prime in ℤ, n is a positive integer and i is an imaginary unit in the ring ℂ of complex numbers.