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In a recent work, Kaiser, Stehl\'ik and \v{S}krekovski provide a family ofcritically 3-chromatic graphs whose expansions do not result in critically4-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 alsogives 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 ofgraphs indeed fail to have the persistence and non-increasing depth properties.

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