Regularity of powers of path ideals of line graphs (Q6636345)

Let \(L_n\) be the line graph with \(n\) vertices \(x_i\), and \(I\) the \(t\)-path ideal of \(L_n\) with \(2\leq t\leq n\), where \(I=\langle \,\prod_{j=1}^t x_{i_j}\mid \{x_{i_1},x_{i_2},\dots,x_{i_t}\} \text{ is a }t \text{ step path in } L_n\,\rangle\) and it is an ideal of the polynomial ring \(k[x_1,\dots,x_n]\) over a field \(k\).\N\NIn this paper, the authors show that \(I^s\) has a linear resolution for some \(s \ge 1\) (or equivalently, for all \(s \ge 1\)) if and only if \(I^s\) has linear quotients for some \(s \ge 1\) (or equivalently for all \(s \ge 1\)), and the latter holds true if and only if \(\frac{n}{2}\leq t \leq n \). In addition, they present an explicit formula for the regularity of \(I^s\) for all \(s \ge 1\), and it turns out that it is linear in the variable \(s\) from the very beginning.