Associated primes of powers of edge ideals and ear decompositions of graphs

TL;DR

This paper characterizes the associated primes of all powers of edge ideals using generalized ear decompositions of graphs, revealing a novel link between algebraic and combinatorial graph properties.

Contribution

It provides a comprehensive description connecting associated primes of edge ideal powers with generalized ear decompositions, unifying previous results and offering new insights.

Findings

Complete description of associated primes via ear decompositions

Unification of previous results in the area

New connections between algebraic and combinatorial graph theory

Abstract

In this paper, we give a complete description of the associated primes of every power of the edge ideal in terms of generalized ear decompositions of the graph. This result establishes a surprising relationship between two seemingly unrelated notions of Commutative Algebra and Combinatorics. It covers all previous major results in this topic and has several interesting consequences.