Associated primes of powers of edge ideals

TL;DR

This paper proves that for edge ideals of graphs, the associated primes of their powers form an ascending chain and their stable sets are equal, using combinatorial optimization and matching theory.

Contribution

It establishes the ascending chain property of associated primes for powers of edge ideals and shows their stable sets coincide, linking algebraic and combinatorial properties.

Findings

Associated primes of powers form an ascending chain

Stable sets of associated primes are equal for large powers

Uses combinatorial optimization and Berge's matching theory

Abstract

Let G be a graph and let I be its edge ideal. Our main result shows that the sets of associated primes of the powers of I form an ascending chain. It is known that the sets of associated primes of I(i) and intcl(I(i)) stabilize for large i, where "intcl" denotes integral closure and I(i) denotes the i-th power of I. We show that for edge ideals their corresponding stable sets are equal. To show our main result we use a classical result of Berge from matching theory and certain notions from combinatorial optimization.