Rees algebras of complementary edge ideals

In this paper we investigate the Rees algebras of squarefree monomial ideals $I\subset S=K[x_1,\dots ,x_n]$ generated in degree $n-2$, where $K$ is a field. Every such ideal arises as the complementary edge ideal $I_c(G)$ of a finite simple graph $G$. We describe the defining equations of the Rees algebra $\mathcal{R}(I_c(G))$ in terms of the combinatorics of $G$. If $G$ is a tree or a unicyclic graph whose unique induced cycle has length $3$ or $4$, we prove that $\mathcal{R}(I_c(G))$ is Koszul. We also determine the asymptotic depth of the powers of $I_c(G)$, proving that ${\lim }_{k\to \mathrm{\infty }}\text{depth}S\mathrm{/}{I}_c(G{)}^k=b(G)$, where $b(G)$ is the number of bipartite connected components of $G$. Finally, we show that the index of depth stability of $I_c(G)$ is at most $n-2$, and equality holds when $G$ is a path graph.