A characterization of graphs whose small powers of their edge ideals have a linear free resolution (Q6548020)

Let \(I(G)\) be the edge ideal of a simple graph \(G\), i.e., squarefree monomial ideal in a polynomial ring \(S = k[x_1, \ldots, x_n]\) over a field \(k\) generated by \(x_i x_ j\) where \(ij\) is an edge of \(G\). In this paper it is proved that \(I(G)^{2}\) has a linear free resolution if and only if \(G\) is gap-free and reg \(I(G) \leq 3\). Similarly, \(I(G)^{3}\) has a linear free resolution if and only if \(G\) is gap-free and reg \(I(G) \leq 4\). The authors deduce these characterizations by establishing a general formula for the regularity of powers of edge ideals of gap-free graphs as follows: reg\((I(G)^{s}) \)= max(reg \(I (G) + s - 1, 2s)\), for \(s = 2, 3\). This solves an open problem raised by \textit{N. Erey} et al. [J. Comb. Theory, Ser. A 188, Article ID 105585, 24 p. (2022; Zbl 1491.13029)].