Coloring in essential annihilating-ideal graphs of commutative rings

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.