Linear strands of powers of certain binomial edge ideals

TL;DR

This paper derives a closed formula for the graded Betti numbers in the linear strands of all powers of binomial edge ideals from certain closed graphs, confirming a conjecture in a specific case.

Contribution

It provides a closed formula for Betti numbers of powers of binomial edge ideals for a class of graphs and verifies a conjecture relating these to initial ideals.

Findings

Betti numbers match those of the initial ideal for the specified graphs

Confirmed a conjecture of Ene--Rinaldo--Terai in a special case

Established a formula applicable to all powers of the ideals in question

Abstract

We provide a closed formula for the graded Betti numbers in the linear strands of all powers of binomial edge ideals $J_G$ arising from closed graphs $G$ that do not have the complete graph $K_4$ as an induced subgraph. We show that these agree with the corresponding Betti numbers for the powers of the lexicographic initial ideal of $J_G$, thereby confirming a conjecture of Ene--Rinaldo--Terai in a special case.