On a regularity-conjecture of generalized binomial edge ideals

TL;DR

The paper addresses bounding the Castelnuovo-Mumford regularity of generalized binomial edge ideals $J_{K_m,G}$. It introduces $r$-compatible maps and leverages the clique-disjoint-edge invariant $\eta(G)$ to derive a tight combinatorial bound $reg(S/J_{K_m,G})\le\min\{(m-1)\eta(G), n-1\}$, improving previous conjectured bounds and proving sharpness via an infinite graph class. The approach unifies and extends prior results for binomial edge ideals and their generalizations, including a detailed analysis via induction on the number of internal vertices and a short exact sequence argument. The findings advance understanding of how graph structure governs algebraic invariants in generalized binomial edge ideals and have potential implications for algebraic statistics and combinatorial commutative algebra.

Abstract

In this paper, we prove the upper bound conjecture proposed by Saeedi Madani \& Kiani on the Castelnuovo-Mumford regularity of generalized binomial edge ideals. We give a combinatorial upper bound of regularity for generalized binomial edge ideals, which is better than the bound claimed in that conjecture. Also, we show that the bound is tight by providing an infinite class of graphs.

Figures (2)

• Figure 1: Graph $G$ such that $\mathrm{reg}(S/\mathcal{J}_{K_3,G})=(3-1)\eta(G)=6$.
• Figure 2: Graph $G\in\mathcal{B}\setminus \mathcal{B}_{3}$ and $\mathrm{reg}(S/\mathcal{J}_{K_3,G})<\mathrm{min}\{2\eta(G),5\}$.

Theorems & Definitions (14)

• Theorem 2.1: arvindgen20
• Definition 3.1
• Theorem 3.2
• proof
• Definition 3.3: rsmkconj21
• Theorem 3.4
• proof
• Remark 3.5
• Corollary 3.6
• proof