Regularity of powers of path ideals of line graphs

Let I be the t-path ideal of Ln, the line graph with n-vertices. It is shown that Is has a linear resolution for some s≥1 (or equivalently for all s≥1) if and only if Is has linear quotients for some s≥1 (or equivalently for all s≥1) if and only if t≤n≤2t. In addition, we present an explicit formula for the regularity of Is for all s≥1. It turns out it is linear in s from the very beginning.