Powers of squarefree monomial ideals and combinatorics

TL;DR

This paper surveys the connection between algebraic properties of powers of squarefree monomial ideals and combinatorial structures, highlighting new algebraic characterizations of perfect graphs and links to linear programming conjectures.

Contribution

It introduces algebraic characterizations of perfect graphs without relying on the Strong Perfect Graph Theorem and explores the equivalence between a linear programming conjecture and ideal power properties.

Findings

Algebraic characterization of perfect graphs independent of the Strong Perfect Graph Theorem

Detection of graph properties via ideal membership and associated primes

Equivalence between the Conforti-Cornuejols conjecture and symbolic power equality

Abstract

We survey research relating algebraic properties of powers of squarefree monomial ideals to combinatorial structures. In particular, we describe how to detect important properties of (hyper)graphs by solving ideal membership problems and computing associated primes. This work leads to algebraic characterizations of perfect graphs independent of the Strong Perfect Graph Theorem. In addition, we discuss the equivalence between the Conforti-Cornuejols conjecture from linear programming and the question of when symbolic and ordinary powers of squarefree monomial ideals coincide.