A Characterization of Graphs Whose Small Powers of Their Edge Ideals Have a Linear Free Resolution

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