Chromatic uniqueness of zero-divisor graphs

The zero-divisor graph Π(R) of a commutative ring R is the whose vertices are elements such that u and v adjacent if only uv = 0. If graphs G H have same chromatic polynomial, then we say they chromatically equivalent (or χ−equivalent), written as ∼ H. Suppose uniquely determined by its polynomial. Then it said to be unique χ-unique). In this paper, discuss question: For which numbers n Π(Zn) χ-unique? While Zn one simplest rings, proved for any A0, some n, contains an induced subgraph isomorphic A0. first result in subject states ≥ 10 even, not χ-unique (Gehet, Khalaf). By definition, square-free prime or product different numbers. Our main following. neither nor square χ-unique. Here our preceding work, use common method. odd non-prime problem open, though on structure know much case.