Edge ideals and their asymptotic syzygies

Let $G$ be a finite simple graph, and let $I(G)$ denote its edge ideal. In this paper, we investigate the asymptotic behavior of the syzygies of powers of edge ideals through the lens of homological shift ideals ${\text{HS}}_i(I(G{)}^k)$. We introduce the notion of the $i$th homological strong persistence property for monomial ideals $I$, providing an algebraic characterization that ensures the chain of inclusions $\text{Ass}{\text{HS}}_i(I)\subseteq \text{Ass}{\text{HS}}_i(I^2)\subseteq \text{Ass}{\text{HS}}_i(I^3)\subseteq \cdots$. We prove that edge ideals possess both the $0$th and $1$st homological strong persistence properties. To this end, we explicitly describe the first homological shift algebra of $I(G)$ and show that ${\text{HS}}_1(I(G{)}^{k+1})=I(G)\cdot {\text{HS}}_1(I(G{)}^k)$ for all $k\ge 1$. Finally, we conjecture that if $I(G)$ has a linear resolution, then ${\text{HS}}_i(I(G{)}^k)$ also has a linear resolution for all $k\gg 0$, and we present partial results supporting this conjecture.