交换环的本质零化理想图的染色
Coloring in essential annihilating-ideal graphs of commutative rings
摘要：
交换酉环$R$的本质零化理想图$\mathcal{EG}(R)$是一个单图，其顶点是$R$的非零理想，且零化子为非零，且两个不同顶点之间存在边$I,J$当且仅当$Ann(IJ)$与$R$的任何非零理想有非零交.本文证明了当$R$是Noether时，$\mathcal{EG}(R)$是弱完美的，并给出了$\mathcal{EG}(R)$团数的一个显式表达式.此外，还完全确定了所有本质零化理想图的色数为$2$的环的结构。在其他结果中，研究了$\mathcal{EG}(R)$的无孪晶团数和边色数。
英文摘要：
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.
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