The size of the Betti table of Binomial Edge Ideals

TL;DR

This paper investigates the algebraic properties of binomial edge ideals associated with finite simple graphs, specifically focusing on the size of their Betti tables by analyzing projective dimension and regularity across all such graphs.

Contribution

It provides a comprehensive determination of almost all pairs of projective dimension and regularity for binomial edge ideals of graphs with a fixed number of vertices.

Findings

Almost all pairs of projective dimension and regularity are characterized.

Results apply to all finite simple graphs with non-isolated vertices.

The study advances understanding of the algebraic invariants of binomial edge ideals.

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 $n$.