Regularity of Edge Ideals and Their Powers

TL;DR

This paper surveys recent research on the Castelnuovo-Mumford regularity of edge ideals of graphs and their powers, emphasizing bounds, exact values, and asymptotic behavior based on graph combinatorics.

Contribution

It provides a comprehensive overview of bounds and exact regularity values for edge ideals and their powers, highlighting recent advances in the field.

Findings

Bounds and exact values for $ ext{reg} I(G)$ and $ ext{reg} I(G)^q$ are summarized.

Asymptotic linear behavior of regularity for powers of edge ideals is discussed.

Connections between graph combinatorics and algebraic regularity are emphasized.

Abstract

We survey recent studies on the Castelnuovo-Mumford regularity of edge ideals of graphs and their powers. Our focus is on bounds and exact values of $\text{reg} I(G)$ and the asymptotic linear function $\text{reg} I(G)^q$, for $q \geq 1,$ in terms of combinatorial data of the given graph $G.$