Cohen-Macaulay squares of edge ideals

TL;DR

This paper characterizes when the square of an edge ideal of a graph is Cohen--Macaulay by analyzing the Stanley--Reisner complex of its polarization, providing a combinatorial criterion based on graph classes.

Contribution

It offers a complete description of the Stanley--Reisner complex of the polarization of $I(G)^2$ and applies Reisner's criterion to determine Cohen--Macaulayness for various graph classes.

Findings

$I(G)^2$ is Cohen--Macaulay only for specific graphs like the pentagon or a single edge.

The paper introduces tools from Stanley--Reisner theory to study powers of edge ideals.

It provides a combinatorial characterization for Cohen--Macaulay squares of edge ideals.

Abstract

Let $G$ be a finite graph and $I(G)$ its edge ideal. We give a full description of the Stanley--Reisner complex of the polarization of $I(G)^2$, naturally introducing the tools of Stanley--Reisner theory in the study of the algebraic behaviour of powers of edge ideals. As an application, we demonstrate how Reisner's criterion can be applied directly to check if $I(G)^2$ is Cohen--Macaulay. We can show that if $G$ belongs to the class of finite graphs which consists of cycles, whisker graphs, trees, connected chordal graphs and connected Cohen--Macaulay bipartite graphs, then the square $I(G)^2$ is Cohen--Macaulay if and only if either $G$ is the pentagon, the cycle of length $5$, or $G$ consists of exactly one edge.