  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 $I\cap J\neq 0$. 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.
