On cycles in intersection graphs of rings

Let R be a commutative ring with non-zero identity. ‎We describe all C3‎- ‎and C4-free intersection graph of non-trivial ideals of R as well as Cn-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.