Integral closure of small powers of edge ideals and their regularity

TL;DR

This paper investigates the regularity of the integral closure of small powers of edge ideals of graphs, establishing equality with ordinary powers for s ≤ 4 and exploring characteristic-dependent behavior for larger s.

Contribution

It proves the equality of regularity between integral closures and ordinary powers for small powers and provides examples illustrating characteristic-dependent regularity growth.

Findings

Regularity of integral closures equals that of ordinary powers for s ≤ 4.

Characteristic of the field affects the regularity growth for larger powers.

Provides explicit examples demonstrating the regularity behavior in different characteristics.

Abstract

Let $I(G)$ be the edge ideal of a simple graph $G$ over a field k. We prove that $${\rm reg}(\overline {I(G)^s}) = {\rm reg}(I(G)^s),$$ for all $s \le 4$. Furthermore, we provide an example of a graph $G$ such that $${\rm reg} I(G)^s = {\rm reg} \overline{I(G)^s} = {\rm reg} I(G)^{(s)} = \begin{cases} 5 + 2s & \text{ if char k} = 2 \\ 4 + 2s & \text{ if char k} \neq 2, \end{cases}$$ for all $s \ge 1.$