Squarefree powers of closed neighborhood ideals

TL;DR

This paper characterizes trees with certain linear properties of their neighborhood ideals, explores the regularity of their squarefree powers, and provides formulas for specific graph classes, advancing understanding in algebraic graph theory.

Contribution

It offers a complete characterization of trees with componentwise linear highest non-vanishing squarefree powers of neighborhood ideals and derives regularity formulas for caterpillar graphs.

Findings

Characterization of trees with componentwise linear highest non-vanishing squarefree powers

Regularity of squarefree powers can be arbitrarily large compared to the ideal's degree

Explicit formula for the regularity of caterpillar graphs' neighborhood ideals

Abstract

In this article, we characterize all trees whose highest non-vanishing squarefree power of the closed neighborhood ideal is componentwise linear. In addition, we investigate the Castelnuovo-Mumford regularity of the $\nu$-th squarefree power of the closed neighborhood ideal of trees and show that this number can be arbitrarily larger than the degree of the ideal. Finally, we give a formula for the regularity of $\nu$-th squarefree power of the closed neighborhood ideal of caterpillar graphs.