On the Depth of Generalized Binomial Edge Ideals

TL;DR

The paper investigates the depth of generalized binomial edge ideals $\mathcal{J}_{K_m,G}$ by extending the $d$-compatible map framework to pairs $(K_m,G)$, yielding a combinatorial lower bound $\mathrm{depth}(S/\mathcal{J}_{K_m,G}) \ge (m-2)t+f(G)+d(G)$ and a vertex-connectivity–based upper bound $\mathrm{depth}(S/\mathcal{J}_{K_m,G}) \le m+n-\kappa(G)$. The authors establish the lower bound via a $d$-compatible map and an induction on the number of non-free vertices, using a key short exact sequence and Depth Lemma. The upper bound is derived via cohomological dimension arguments and characteristic-p reductions, extended to characteristic zero. They further compute exact depths for several graph classes, including cycles and strongly unmixed graphs, and discuss cases where the gap between bounds is arbitrarily large, as well as implications for Cohen–Macaulay and accessible/strongly unmixed structures. Overall, the work provides a combinatorial framework for depth of generalized binomial edge ideals and connects graph-theoretic invariants with homological depth properties.

Abstract

This research focuses on analyzing the depth of generalized binomial edge ideals. We extend the notion of $d$-compatible map for the pairs of a complete graph and an arbitrary graph, and using it, we give a combinatorial lower bound for the depth of generalized binomial edge ideals. Subsequently, we determine an upper bound for the depth of generalized binomial edge ideals in terms of the vertex-connectivity of graphs. We demonstrate that the difference between the upper and lower bounds can be arbitrarily large, even in cases when one of the bounds is sharp. In addition, we calculate the depth of generalized binomial edge ideals of certain classes of graphs, including cyclic graphs and graphs with Cohen-Macaulay binomial edge ideals.

Figures (5)

• Figure 1: A graph $G\in \mathcal{H}_1$ with $K_s=K_3=G- \{2,5\}$.
• Figure 2: The graph $G\in\mathcal{H}_{2}$ such that $G=(((C_6)_{1})_{2})_{4}$.
• Figure 3: The graph $G_3\in\mathcal{H}_{3}$ with $6$ maximal cliques.
• Figure 4: The graph $G_1$ considered in the proof of Theorem \ref{['thmd=lb=ub-k']}.
• Figure 5: A graph $G$ with $(m-2)t+f(G)+d(G)<\mathrm{depth}(S/\mathcal{J}_{K_m,G})<m+n-\kappa(G)$.

Theorems & Definitions (31)

• Theorem 1.1: Theorem \ref{['thmd-comp']}, \ref{['thmlb']}
• Theorem 1.2: Theorem \ref{['thmupperbound']}
• Definition 2.1
• Definition 2.2
• Theorem 2.3: arvindgen20
• Lemma 2.4: arvindgen20
• Theorem 2.5: ci20
• Theorem 2.6: Auslander-Buchbaum Formula
• Definition 3.1
• Theorem 3.2