The size of the Betti table of Binomial Edge Ideals

TL;DR

This work classifies the possible sizes of Betti tables for binomial edge ideals $J_G$ of graphs on $n$ non-isolated vertices by determining the set $\mathrm{pdreg}(n)$ of all pairs $(\mathrm{proj\,dim}(J_G),\mathrm{reg}(J_G))$. It develops universal bounds and then builds a comprehensive catalogue of special graph classes (joins, cones, decomposable graphs, and $D_5$-type graphs) to realize a wide range of pairs, using induction on $n$ to extend from small graphs. The main result provides an explicit description of $\mathrm{pdreg}(n)$ for all $n\ge 3$, with an exceptional set $A_n$ corresponding to reg=$n-1$ that remains partly unresolved, complemented by conjectures and computational evidence. Overall, the paper maps the full landscape of Betti-table sizes for binomial edge ideals, enabling precise predictions of homological invariants from graph structure and guiding further study of the reg=$n-1$ boundary.

Abstract

Let $G$ be a finite simple graph on $n$ non-isolated vertices, and let $J_G$ be its binomial edge ideal. We determine almost all pairs $(\text{projdim}(J_G),\text{reg}(J_G))$, where $G$ ranges over all finite simple graphs on $n$ non-isolated vertices, for any $n$.

Theorems & Definitions (35)

• Theorem 1.1
• proof
• Theorem 1.2
• proof
• Remark 1.3
• Proposition 1.4
• proof
• Theorem 2.1
• Corollary 2.2
• proof