Chromatic uniqueness of zero-divisor graphs

Abstract

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