Regularity of powers of edge ideals: from local properties to global bounds

Let $I = I(G)$ be the edge ideal of a graph $G$. We give various general upper bounds for the regularity function $\text{reg} I^s$, for $s \ge 1$, addressing a conjecture made by the authors and Alilooee. When $G$ is a gap-free graph and locally of regularity 2, we show that $\text{reg} I^s = 2s$ for all $s \ge 2$. This is a slightly weaker version of a conjecture of Nevo and Peeva. Our method is to investigate the regularity function $\text{reg}I^s$, for $s \ge 1$, via local information of $I$.