Characterization of graphs whose a small power of their edge ideals has a linear free resolution

TL;DR

This paper characterizes when small powers of edge ideals of graphs have linear resolutions, linking it to the graph being gap-free and bounds on the regularity of the edge ideal.

Contribution

It provides a precise characterization of graphs whose squared and cubed edge ideals have linear resolutions, based on gap-freeness and regularity bounds.

Findings

$I(G)^2$ has a linear resolution iff $G$ is gap-free and reg$I(G) \\le 3$.

$I(G)^3$ has a linear resolution iff $G$ is gap-free and reg$I(G) \\le 4$.

Derived a formula for the regularity of powers of edge ideals of gap-free graphs.

Abstract

Let $I(G)$ be the edge ideal of a simple graph $G$. We prove that $I(G)^2$ has a linear free resolution if and only if $G$ is gap-free and reg$I(G) \le 3$. Similarly, we show that $I(G)^3$ has a linear free resolution if and only if $G$ is gap-free and reg$I(G) \le 4$. We deduce these characterizations from a general formula for the regularity of powers of edge ideals of gap-free graphs $${\rm reg}(I(G)^s) = \max({\rm reg} I(G) + s-1,2s),$$ for $s =2,3$.