Gapfree graphs and powers of edge ideals with linear quotients

Abstract

Let $I\left(G\right)$ be the edge ideal of a gapfree graph G. An open conjecture of Nevo and Peeva states that $I{\left(G\right)}^q$ has a linear resolution for $q\gg 0$. We present a promising approach to this challenging conjecture by investigating the stronger property of linear quotients. Specifically, we make the conjecture that if $I{\left(G\right)}^q$ has linear quotients for some integer $q\ge 1$, then $I{\left(G\right)}^s$ has linear quotients for all $s\ge q$. We give a partial solution to this conjecture by considering a special order of the generators of $I{\left(G\right)}^q$. It is known that if G does not contain a cricket, a diamond, or a cycle $C_4$ of length 4, then $I{\left(G\right)}^q$ has a linear resolution for $q\ge 2$. We construct a family of gapfree graphs G containing a cricket, a diamond, a $C_4$ together with a cycle $C_5$ of length 5 as induced subgraphs of G for which $I{\left(G\right)}^q$ has linear quotients for $q\ge 2$.