  Let $R$ be a commutative ring with non-zero identity. We describe all $C_3$-
and $C_4$-free intersection graph of non-trivial ideals of $R$ as well as
$C_n$-free intersection graph when $R$ is a reduced ring. Also, we shall
describe all complete, regular and $n$-claw-free intersection graphs. Finally,
we shall prove that almost all Artin rings $R$ have Hamiltonian intersection
graphs. We show that such graphs are indeed pancyclic.
