Coloring in essential annihilating-ideal graphs of commutative rings

TL;DR

This paper investigates the coloring properties of essential annihilating-ideal graphs of commutative rings, establishing their weak perfection under certain conditions and characterizing rings with specific chromatic numbers.

Contribution

It proves that these graphs are weakly perfect for Noetherian rings, provides a formula for the clique number, and characterizes rings with chromatic number two.

Findings

Essential annihilating-ideal graphs are weakly perfect for Noetherian rings.

An explicit formula for the clique number of these graphs is provided.

Complete characterization of rings with chromatic number two.

Abstract

The essential annihilating-ideal graph $\mathcal{EG}(R)$ of a commutative unital ring $R$ is a simple graph whose vertices are non-zero ideals of $R$ with non-zero annihilator and there exists an edge between two distinct vertices $I,J$ if and only if $Ann(IJ)$ has a non-zero intersection with any non-zero ideal of $R$. In this paper, we show that $\mathcal{EG}(R)$ is weakly perfect, if $R$ is Noetherian and an explicit formula for the clique number of $\mathcal{EG}(R)$ is given. Moreover, the structures of all rings whose essential annihilating-ideal graphs have chromatic number $2$ are fully determined. Among other results, twin-free clique number and edge chromatic number of $\mathcal{EG}(R)$ are examined.