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

Abstract

In a recent work, Kaiser, Stehlík and Škrekovski 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.

14 pp; minor revision; to appear in Acta Mathematica Vietnamica