Cohen-Macaulay squares of edge ideals

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.