The intersection graph of ideals of ${\mathbb{Z}}_n$ is\\ weakly perfect

Abstract

A graph is called weakly perfect if its vertex chromatic number equals its clique number. Let $R$ be a ring and $I(R{)}^{\ast }$ be the set of all left proper non-trivial ideals of $R$. The intersection graph of ideals of $R$, denoted by $G(R)$, is a graph with the vertex set $I(R{)}^{\ast }$ and two distinct vertices $I$ and $J$ are adjacent if and only if $I\cap J\ne 0$. In this paper, it is shown that $G({\mathbb{Z}}_n)$, for every positive integer $n$, is a weakly perfect graph. Also, for some values of $n$, we give an explicit formula for the vertex chromatic number of $G({\mathbb{Z}}_n)$. Furthermore, it is proved that the edge chromatic number of $G({\mathbb{Z}}_n)$ is equal to the maximum degree of $G({\mathbb{Z}}_n)$ unless either $G({\mathbb{Z}}_n)$ is a null graph with two vertices or a complete graph of odd order.

8 pages

Cited by in corpus (1)

• On Perfectness of Intersection Graph of Ideals of ${\mathbb{Z}}_n$