Rings are $\chi$-bounded

Abstract

We prove that there is a function $f:\mathbb{N}\to \mathbb{N}$ such that $\chi (\Gamma (R))\le f(\omega (\Gamma (R)))$ for the zero-divisor graph of any ring $R$, if the clique number is finite. On the one hand, this consolidates a disproved conjecture of Beck from 1988, claiming $\chi (\Gamma (R))=\omega (\Gamma (R))$ for unital commutative rings. While previous counterexamples satisfy $\chi (\Gamma (R))\le \omega (\Gamma (R))+2$, we obtain the lower bound $f\ge k^{\Omega (\log k)}$ among finite commutative rings. Finally, we show that neither zero-divisor graphs of finite commutative semirings nor those of finite commutative nonassociative rings are $\chi$-bounded.