Squarefree powers of edge ideals of forests

TL;DR

This paper investigates the algebraic properties of squarefree powers of edge ideals in forests, providing bounds, classifications, and formulas for their regularity and resolutions.

Contribution

It offers a sharp upper bound for regularity, classifies forests with linear resolutions, and derives a combinatorial formula for the regularity of the second squarefree power.

Findings

Sharp upper bound for regularity in terms of the k-admissible matching number

Complete classification of forests with linear resolution for all k

Explicit combinatorial formula for the regularity of I(G)^{[2]}

Abstract

Let $I(G)^{[k]}$ denote the $k$th squarefree power of the edge ideal of $G$. When $G$ is a forest, we provide a sharp upper bound for the regularity of $I(G)^{[k]}$ in terms of the $k$-admissable matching number of $G$. For any positive integer $k$, we classify all forests $G$ such that $I(G)^{[k]}$ has linear resolution. We also give a combinatorial formula for the regularity of $I(G)^{[2]}$ for any forest $G$.