A combinatorial characterization of $S_2$ binomial edge ideals

TL;DR

The paper provides the first graph-theoretical characterization of binomial edge ideals satisfying Serre's condition $(S_2)$ by proving that $(S_2)$ is equivalent to graph accessibility. The authors translate the problem into the combinatorics of cut sets via the Stanley-Reisner complex Delta_G, which arises from the Z^n-graded gin(J_G), and prove that accessibility is equivalent to strong accessibility for unmixed graphs. A key methodological advance is showing that for accessible graphs, all nonzero-dimensional links in Delta_G are connected, yielding $(S_2)$ for gin(J_G) and thus for J_G. These results connect unmixedness, cut-set structure, and Serre conditions, advancing the program to characterize Cohen–Macaulay binomial edge ideals through combinatorial graph properties.

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 $G$ in a minimal way, 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 $(S_2)$ by proving that this is equivalent to having $G$ accessible, which means that $J_G$ is unmixed and the cut 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 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.

Figures (8)

• Figure 1: A graph $G$
• Figure 2: A graph $G$ with $\mathcal{C}(G)$ accessible and not strongly accessible
• Figure 3: A graph $G$ to illustrate Lemma \ref{['L.CutsetContained']}
• Figure 4: An example of Lemma \ref{['L.CutsetContained']}
• Figure 5: An example to illustrate Lemma \ref{['L.firstLemmaMainTheorem']}

Theorems & Definitions (33)

• Conjecture 1.1: BMS22
• Conjecture 1.2: LMRR23
• Lemma 2.1
• proof
• Example 2.2
• Proposition 2.3
• proof
• Definition 3.1
• Remark 3.2
• Example 3.3