Sortable simplicial complexes and $t$-independence ideals of proper interval graphs

TL;DR

This paper introduces the concepts of sortability and t-sortability for simplicial complexes, characterizes proper interval graphs via these properties, and studies the algebraic properties of associated t-independence ideals.

Contribution

It establishes that proper interval graphs are exactly those with sortable independence complexes and analyzes the algebraic properties of their t-independence ideals.

Findings

Proper interval graphs have sortable independence complexes.

The t-independence ideals of proper interval graphs have the strong persistence property.

All powers of these t-independence ideals have linear quotients.

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.