Comparision between regularity of small symbolic powers and ordinary powers of an edge ideal

TL;DR

This paper investigates the regularity of small symbolic and ordinary powers of edge ideals of graphs, establishing equality for s=2,3 and providing bounds for their regularities.

Contribution

It proves the equality of regularities for small symbolic and ordinary powers of edge ideals and establishes new bounds for these regularities.

Findings

${ m reg}(I^{(s)}) = { m reg}(I^s)$ for s=2,3

Bounds on regularity: ${ m reg} I^{s} extless{}= { m reg} I + 2s - 2$ for s=2,3

Bounds on symbolic powers: ${ m reg} I^{(s)} extless{}= { m reg} I + 2s - 2$ for s=2,3,4

Abstract

Let $G$ be a simple graph and $I$ its edge ideal. We prove that $${\rm reg}(I^{(s)}) = {\rm reg}(I^s)$$ for $s = 2,3$, where $I^{(s)}$ is the $s$-th symbolic power of $I$. As a consequence, we prove the following bounds \begin{align*} {\rm reg} I^{s} & \le {\rm reg} I + 2s - 2, \text{ for } s = 2,3, {\rm reg} I^{(s)} & \le {\rm reg} I + 2s - 2,\text{ for } s = 2,3,4. \end{align*}