On the equality of symbolic and ordinary powers of binomial edge ideals (Q2106567)

In this paper, the authors study the equality of symbolic and ordinary powers of binomial edge ideals. More precisely let \(G\) be a simple graph on the vertex set \(\{1,\ldots ,n\}\), and \(S = K[x_1,\ldots, x_n, y_1,\ldots, y_n]\) the polynomial ring over the field \(K\). The binomial edge ideal \(J_G\subset S\) is defined to be \(J_G = \langle x_i y_j - x_j y_i \mid \{i, j\} \in E(G)\rangle\) where \(E(G)\) is the edge set of \(G\). This kind of ideals was introduced by \textit{J. Herzog} et al. [Adv. Appl. Math. 45, No. 3, 317--333 (2010; Zbl 1196.13018)]. If \(I\subset S\) is an ideal and \(t\) is a positive integer then the \(t\)-th symbolic power of \(I\) is \[ I^{(t)}=\bigcap_{P\in \mathrm{MinAss}(I)} (I^tS_P\cap S) \] where \(\mathrm{MinAss}(I)\) is the set of minimal associated primes of \(I\). The aim of this paper is to check the equality \(I^t=I^{(t)}\) for each \(t\) for binomial edge ideals. It has been shown that if the number of associated primes of \(J_G\) is two then this property holds. Furthermore, if \(G\) is a caterpillar tree, then the same property is true. Finally, they characterize the generalized caterpillar graphs such that the equality of symbolic and ordinary powers holds true.