Saturation and associated primes of powers of edge ideals

TL;DR

This paper characterizes the embedded associated primes of powers of edge ideals of graphs using weighted graph representations, providing a complete classification for the second and third powers.

Contribution

It introduces a novel description of monomials in the saturation of powers of edge ideals via weighted graphs, enabling precise prime classification.

Findings

Complete classification of associated primes for I^2 and I^3

Characterization of embedded primes using subgraph covers

Description of monomials in saturation through weighted graphs

Abstract

For the edge ideal I of an arbitrary simple graph G we describe the monomials of the saturation of a power of I in terms of (vertex) weighted graphs associated with the monomials. This description allows us to characterize the embedded associated primes of a power of I as covers of G which contain certain types of subgraphs of G. As an application, we completely classify the associated primes of the second and the third power of I in terms of G.