Combinatorial characterizations of the saturation and the associated primes of the fourth power of edge ideals

TL;DR

This paper provides a combinatorial description of the saturation of the fourth power of edge ideals and classifies their associated primes based on graph properties, advancing understanding of local cohomology in algebraic combinatorics.

Contribution

It introduces a novel combinatorial characterization of the saturation of the fourth power of edge ideals and classifies associated primes in terms of graph structure.

Findings

Complete classification of associated primes of the fourth power of edge ideals.

Combinatorial description of the generators of the saturation of the fourth power.

Extension of previous results from second and third powers to the fourth power.

Abstract

To compute the local cohomology of powers of edge ideals one needs to know their saturations. The saturation of the second and third powers has been described in terms of the graph in [13] and [10]. In this article, we give a combinatorial description of the generators of the saturation of the fourth power. As a consequence, we are able to give a complete classification of the associated primes of the fourth power of edge ideals in terms of the graph.