Powers of edge ideals

TL;DR

This paper computes Betti numbers for powers of certain edge ideals, proves linear quotients for anti-d-path edge ideals, and characterizes normally torsion-free ideals, including some non-squarefree cases from specific graphs.

Contribution

It provides explicit Betti number calculations, proves linear quotient properties, and characterizes normally torsion-free ideals for a class of edge ideals, including non-squarefree examples.

Findings

Betti numbers computed for all powers of initial and final lexsegment edge ideals.

Edge ideals of anti-d-paths have linear quotients.

Identified classes of normally torsion-free non-squarefree ideals.

Abstract

We compute the Betti numbers for all the powers of initial and final lexsegment edge ideals. For the powers of the edge ideal of an anti-$d-$path, we prove that they have linear quotients and we characterize the normally torsion-free ideals. We determine a class of non-squarefree ideals, arising from some particular graphs, which are normally torsion-free.