Let R be a ring and I⋆(R) be the set of all nontrivial left ideals of R. The Co-intersection graph of ideals of R, denoted by Ω(R), is an undirected simple graph with the vertex set I⋆(R), and two distinct vertices I and J are adjacent if and only if I + J ̸= R. This paper derives a sufcient and necessary condition for Ω(R) to be a connected graph. We characterize the values of n for which the graph Ω(Zn) is Eulerian and Hamiltonian. Furthermore, the bad (and nice) decision number of Ω(Zn) are studied in the paper.
