Abstract

We introduce the notion of sortability and $t$-sortability for a simplicial complex and study the graphs for which their independence complexes are either sortable or $t$-sortable. We show that the proper interval graphs are precisely the graphs whose independence complex is sortable. By using this characterization, we show that the ideal generated by all squarefree monomials corresponding to independent sets of vertices of $G$ of size $t$ (for a given positive integer $t$) has the strong persistence property, when $G$ is a proper interval graph. Moreover, all of its powers have linear quotients.

Highlights

  • The notion of strong persistence property for an ideal in a Noetherian ring R has been defined in [8]

  • It is known that any monomial ideal with the strong persistence property has the persistence property

  • We introduce a new class of monomial ideals associated to proper interval graphs with the strong persistence property

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Summary

Introduction

The notion of strong persistence property for an ideal in a Noetherian ring R has been defined in [8]. We introduce a new class of monomial ideals associated to proper interval graphs with the strong persistence property. To this aim, we introduce the notion of a sortable simplicial complex and show that the independence complex of a graph G is sortable if and only if G is a proper interval graph. We show that for any ideal in this class, all of its powers have linear quotients and linear resolutions

Sortable simplicial complexes
Algebraic properties of t-independence ideals of proper interval graphs

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