Bounds on the regularity and projective dimension of ideals associated to graphs

TL;DR

This paper establishes new logarithmic upper bounds on the regularity and bounds on the projective dimension of certain graph-associated ideals, connecting these results to classical commutative algebra topics.

Contribution

It introduces novel logarithmic bounds on regularity for k-step linear edge ideals and extends bounds on projective dimension, linking to Alexander duality and classical algebra.

Findings

Logarithmic upper bounds on regularity for k-step linear edge ideals

Bounds on projective dimension for these ideals

Connections to classical commutative algebra topics

Abstract

In this paper we give new upper bounds on the regularity of edge ideals whose resolutions are k-steps linear; surprisingly, the bounds are logarithmic in the number of variables. We also give various bounds for the projective dimension of such ideals, generalizing other recent results. By Alexander duality, our results also apply to unmixed square-free monomial ideals of codimension two. We also discuss and connect these results to more classical topics in commutative algebra.