A combinatorial characterization of S2 binomial edge ideals

Abstract

Several algebraic properties of a binomial edge ideal $J_G$ can be interpreted in terms of combinatorial properties of its associated graph $G$. In particular, the so-called cut sets of a graph $G$, special sets of vertices that disconnect $G$, play an important role since they are in bijection with the minimal prime ideals of $J_G$. In this paper we establish the first graph-theoretical characterization of binomial edge ideals $J_G$ satisfying Serre’s condition $\left(S_2\right)$ by proving that this is equivalent to having $G$ accessible, which means that $J_G$ is unmixed and the cut-point sets of $G$ form an accessible set system. The proof relies on the combinatorial structure of the Stanley–Reisner simplicial complex of a multigraded generic initial ideal of $J_G$, whose facets can be described in terms of cut-point sets. Another key step in the proof consists in proving the equivalence between accessibility and strong accessibility for the collection of cut sets of $G$ with $J_G$ unmixed. This result, interesting on its own, provides the first relevant class of set systems for which the previous two notions are equivalent.