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Squarefree Monomial Ideals that Fail the Persistence Property and Non-increasing Depth

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Abstract

In a recent work (Kaiser et al., J. Comb. Theory Ser. A 123, 239–251, 2014), Kaiser et al. 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 et al. (Discrete Math. 310, 2176–2182, 2010). 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.

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Acknowledgments

Hà is partially supported by the Simons Foundation (grant #279786).

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Authors and Affiliations

  1. Department of Mathematics, Tulane University, 6823 St. Charles Ave., New Orleans, LA, 70118, USA

    Huy Tài Hà & Mengyao Sun

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  1. Huy Tài Hà
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  2. Mengyao Sun
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Correspondence to Huy Tài Hà.

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Dedicated to Professor Ngô Viêt Trung on the occasion of his sixtieth birthday

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Hà, H.T., Sun, M. Squarefree Monomial Ideals that Fail the Persistence Property and Non-increasing Depth. Acta Math Vietnam 40, 125–137 (2015). https://doi.org/10.1007/s40306-014-0104-x

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  • DOI: https://doi.org/10.1007/s40306-014-0104-x

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