Application of Betti Splittings to the Regularity of Binomial Edge Ideals

TL;DR

This paper addresses determining the Castelnuovo–Mumford regularity of binomial edge ideals $S/J_G$ for trees. It combines Betti splittings with the Ene–Herzog–Hibi process to obtain sharper upper and lower bounds depending on combinatorial graph invariants and to identify classes of trees with exact regularity. The main contributions include explicit upper and lower bounds: $reg(S/J_G)\le iv(G)+1+D_G-2s-\sum_{i=1}^{p}\left\lfloor \frac{s_i}{3}\right\rfloor-e_G$ and $reg(S/J_G)\ge iv(G)+1+D_G-2s-\mu_G-\sum_{i=1}^{p}(s_i-1)$, as well as corollaries giving exact values in jewel-free or sparsely jewel-influenced trees and a refined notion of jewels (generalized jewels). The results illuminate how tree structure (jewels, spine length, degree distribution) governs BEI regularity and extend prior bounds in the literature.

Abstract

In this paper, we use Betti splittings of binomial edge ideals to establish improved upper and lower bounds for their regularity in the case of trees. As a consequence, we determine the exact regularity for certain classes of trees.

Figures (3)

• Figure 1: A tree $G$ containing a generalized jewel $\mathcal{J}_v$
• Figure 2:
• Figure 3:

Theorems & Definitions (27)

• Definition 2.1
• Definition 2.2
• Remark 2.3
• Remark 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof