Characterize when symbolic and ordinary powers coincide for binomial edge ideals

Determine a graph-theoretic characterization of simple graphs G for which the binomial edge ideal J_G ⊂ k[x_1, …, x_d, y_1, …, y_d] satisfies J_G^{(n)} = J_G^n for all integers n ≥ 1, analogous to the bipartite characterization known for monomial edge ideals, in order to identify precisely when symbolic and ordinary powers coincide in the binomial edge ideal setting.

Background

Symbolic powers and ordinary powers of ideals need not coincide in general. For monomial edge ideals, there is a complete characterization: equality holds if and only if the underlying graph is bipartite. The paper notes that such a characterization is not yet available for binomial edge ideals.

Although several families are known to have coinciding powers (e.g., closed graphs, complete multipartite graphs, and caterpillar graphs), a comprehensive characterization across all binomial edge ideals remains unknown. The authors paper related F-singularity properties as a path toward understanding this question.