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

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 $t\ge 1$ when $\mathrm{\mid }Ass(J_G)\mathrm{\mid }=2$. 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.