The zero-divisor graph of a ring R, denoted &Gamma(R), is the graph whose vertex set is the collection of zero-divisors in R, with edges between two distinct vertices u and v if and only if uv=0. In this paper, we restrict our attention to &Gamma(Zn), the zero-divisor graph of the ring of integers modulo n. Specifically, we determine all values of n for which &Gamma(Zn) is perfect. Our classification depends on the prime factorization of n, with relatively simple prime factorizations corresponding to perfect graphs. In fact, for values of n with at most two distinct prime factors, &Gamma(Zn) is perfect; for n with at least 5 distinct prime factors, &Gamma(Zn) is not perfect; and for n with either three or four distinct prime factors, &Gamma(Zn) is perfect only in the cases where n=paqr and n=pqrs for distinct primes p, q, r, s and positive integer a. In proving our results, we make heavy use of the Strong Perfect Graph Theorem.
Mary Marshall, Professor of Mathematics, Illinois College
"Perfect Zero-Divisor Graphs of Z_n,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 17
, Article 6.
Available at: https://scholar.rose-hulman.edu/rhumj/vol17/iss2/6