Squarefree powers of closed 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.

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