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 \(s =2,3\).

中文翻译：

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边理想的小幂具有线性自由分辨率的图的表征

令I ( G ) 为简单图G的理想边。我们证明\(I(G)^2\)具有线性自由分辨率当且仅当G是无间隙且\({{\,\textrm{reg}\,}}I(G) \le 3 \)。类似地，我们证明\(I(G)^3\)具有线性自由分辨率当且仅当G是无间隙且\({{\,\textrm{reg}\,}}I(G) \乐 4\)。我们通过建立无间隙图边理想幂正则性的通用公式来推导这些特征\({{\,\textrm{reg}\,}}(I(G)^s) = \max ({ {\,\textrm{reg}\,}}I(G) + s-1,2s)\)，对于\(s =2,3\)。