On the Equality of Symbolic and Ordinary Powers of Binomial Edge Ideals

Abstract

In this paper, we investigate whether the symbolic and ordinary powers of a binomial edge ideal \(J_{G}\) are equal. We show that the equality \(J_{G}^{t}=J_{G}^{(t)}\) holds for every \(t \geq 1\) when \(|Ass(J_{G})|=2\). Moreover, if \(G\) is a caterpillar tree, then one has the same equality. Finally, we characterize the generalized caterpillar graphs which the equality of symbolic and ordinary powers of \(J_{G}\) occurs.