Edge ideals: algebraic and combinatorial properties

TL;DR

This survey explores the algebraic and combinatorial properties of edge ideals derived from clutters, providing new criteria and formulas for regularity, projective dimension, and associated primes, with implications for Cohen-Macaulayness.

Contribution

It introduces a new criterion for estimating regularity, offers formulas for the regularity of vertex cover ideals, and analyzes associated primes of powers of edge ideals.

Findings

New criterion for regularity estimation

Formulas for regularity of vertex cover ideals

Ascending chain of associated primes in graphs with leaves

Abstract

Let C be a clutter and let I(C) be its edge ideal. This is a survey paper on the algebraic and combinatorial properties of R/I(C) and C, respectively. We give a criterion to estimate the regularity of R/I(C) and apply this criterion to give new proofs of some formulas for the regularity. If C is a clutter and R/I(C) is sequentially Cohen-Macaulay, we present a formula for the regularity of the ideal of vertex covers of C and give a formula for the projective dimension of R/I(C). We also examine the associated primes of powers of edge ideals, and show that for a graph with a leaf, these sets form an ascending chain.