Regularity of powers of path ideals of line graphs

Abstract

Let $L_n$ be a line graph with $n$ vertices and let $I$ be its $t$-path ideal. It is shown 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 $s\ge 1$) if and only if $t\le n\le 2t$. In addition, we present an explicit formula for the regularity of $I^s$ for all $s\ge 1$. It turns out it is linear in $s$ from the very beginning.

arXiv comment10 pages to appear in Communications in Algebra