Comparison of symbolic and ordinary powers of parity binomial edge ideals

In this paper, we investigate when symbolic and ordinary powers of the parity binomial edge ideal of a graph fail to be equal. It turns out that if ℐ_G is the parity binomial edge ideal of a graph G , then in each of the following cases the symbolic power ℐ_G^(t) and the ordinary power ℐ_G^t are not equal for some t : (i) the clique number of G is greater than 3; (ii) G has a net; or (iii) G has a PT as an induced subgraph.