Squarefree monomial ideals that fail the persistence property and non-increasing depth

TL;DR

This paper demonstrates that certain squarefree monomial ideals, specifically cover ideals of a family of critically 3-chromatic graphs, fail the persistence property and have non-increasing depth, providing counterexamples to previous conjectures.

Contribution

It extends the known counterexamples to the persistence and non-increasing depth properties to an entire family of graphs, not just a single instance.

Findings

Cover ideals of all graphs in the family fail the persistence property.

Cover ideals of these graphs also fail the non-increasing depth property.

Counterexamples challenge existing conjectures in combinatorial commutative algebra.

Abstract

In a recent work, Kaiser, Stehl\'ik and \v{S}krekovski provide a family of critically 3-chromatic graphs whose expansions do not result in critically 4-chromatic graphs, and thus give counterexamples to a conjecture of Francisco, Ha and Van Tuyl. The cover ideal of the smallest member of this family also gives a counterexample to the persistence and non-increasing depth properties. In this paper, we show that the cover ideals of all members of their family of graphs indeed fail to have the persistence and non-increasing depth properties.