The intersection graph of ideals of ℤn 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)^∗ 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)^∗ and two distinct vertices I and J are adjacent if and only if I ∩ J ≠ 0. In this paper, it is shown that G(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(ℤ_n). Furthermore, it is proved that the edge chromatic number of G(ℤ_n) is equal to the maximum degree of G(ℤ_n) unless either G(ℤ_n) is a null graph with two vertices or a complete graph of odd order.