Coloring of graphs associated to zero-divisors

Abstract

Let $G$ be a graph, $\chi(G)$ be the minimal number of colors which can be assigned to the vertices of $G$ in such a way that every two adjacent vertices have different colors and $\omega(G)$ to be the least upper bound of the size of the complete subgraphs contained in $G$. It is well-known that $\chi(G)\geq \omega(G)$. Beck in \cite{b} conjectured that $\chi(\Gamma_0(R))=\omega(\Gamma_0(R))$ if $\omega(\Gamma_0(R))<\infty$, where $\Gamma_0(R)$ is a graph associated to a commutative ring $R$. In this note, we provide some sufficient conditions for a ring $R$ to enjoy $\chi(\Gamma_0(R))=\omega(\Gamma_0(R))$. As a consequence, we verify Beck's conjecture for the homomorphic image of $\mathbb{Z}^n$.