Coloring of graphs associated to zero-divisors
Abstract
Description
Let $G$ be a graph, $χ(G)$ 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 $ω(G)$ to be the least upper bound of the size of the complete subgraphs contained in $G$. It is well-known that $χ(G)\geq ω(G)$. Beck in \cite{b} conjectured that $χ(Γ_0(R))=ω(Γ_0(R))$ if $ω(Γ_0(R))<\infty$, where $Γ_0(R)$ is a graph associated to a commutative ring $R$. In this note, we provide some sufficient conditions for a ring $R$ to enjoy $χ(Γ_0(R))=ω(Γ_0(R))$. As a consequence, we verify Beck's conjecture for the homomorphic image of $\mathbb{Z}^n$.
13 pages
13 pages