Cohen-Macaulay squares of edge ideals

TL;DR

The paper addresses when the square of the edge ideal $I(G)^2$ is Cohen–Macaulay for a finite graph $G$ over a field $K$. It uses polarization to convert $I(G)^2$ into a squarefree ideal and applies Reisner's criterion to the Stanley–Reisner complex $\Gamma^2_G$, together with an explicit facet description of $\Gamma^2_G$ in terms of graph features. The main result shows that, within the classes of cycles, whisker graphs, trees, connected chordal graphs, and connected Cohen–Macaulay bipartite graphs, $I(G)^2$ is Cohen–Macaulay precisely when $G$ is the pentagon $C_5$ or has exactly one edge. The paper also provides a practical obstruction: if $G$ contains an induced path $P_3$ with two leaves, then $I(G)^2$ cannot be Cohen–Macaulay. This links combinatorial graph structure directly to homological properties of monomial ideals and yields a clear classification in key graph families.

Abstract

Let $G$ be a finite graph and $I(G)$ its edge ideal. The question in which we are interested is when the square $I(G)^2$ is Cohen--Macaulay. Via the polarization technique together with Reisner's criterion, it is shown 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.

Figures (2)

• Figure 1: \ref{['ex:stars']}
• Figure 2:

Theorems & Definitions (24)

• Example 1.1
• Definition 1.2: edge ideal
• Definition 1.3: Stanley--Reisner complex
• Definition 1.4: Cohen--Macaulay complexes
• Theorem 1.5: Reisner's criterion Rei
• Definition 1.6: Polarization
• Example 2.1: $\Gamma^2_{P_3}$ is not Cohen--Macaulay
• Theorem 2.2: The Stanley--Reisner complex of the polarization of $I(G)^2$
• proof
• Example 2.3: The case of a triangle