On the extremal Betti numbers of the binomial edge ideal of closed graphs

TL;DR

This paper investigates the extremal Betti numbers of binomial edge ideals of closed graphs, establishing conditions for their uniqueness and equality between the ideal and its initial ideal.

Contribution

It proves the existence and uniqueness of extremal Betti numbers for certain binomial edge ideals and their initial ideals in closed graphs.

Findings

Unique extremal Betti number for initial ideal in some cases

Equality of extremal Betti numbers between ideal and initial ideal

Conditions under which extremal Betti numbers are equal

Abstract

We study the equality of the extremal Betti numbers of the binomial edge ideal $J_G$ and those of its initial ideal ${\rm in}(J_G)$ of a closed graph $G$. We prove that in some cases there is an unique extremal Betti number for ${\rm in}(J_G)$ and as a consequence there is an unique extremal Betti number for $J_G$ and these extremal Betti numbers are equal