Gapfree graphs and powers of edge ideals with linear quotients

TL;DR

This paper explores the properties of powers of edge ideals of gapfree graphs, focusing on linear quotients and their implications for the conjecture that high powers have linear resolutions, providing partial solutions and specific graph examples.

Contribution

It proposes a conjecture linking linear quotients of powers of edge ideals to their higher powers, and offers partial results and conditions for verifying this in gapfree graphs.

Findings

Identified conditions where powers of edge ideals have linear quotients

Constructed gapfree graphs with specific subgraphs where powers have linear quotients

Provided partial solutions to the conjecture relating linear quotients and linear resolutions

Abstract

Let $I(G)$ be the edge ideal of a gapfree graph $G$. An open conjecture of Nevo and Peeva states that $I(G)^q$ has 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 $I(G)^q$ has linear quotients for some integer $q\geq 1$, then $I(G)^{s}$ has linear quotients for all $s\geq q$. We give a partial solution to this conjecture, and identify conditions under which only finitely many powers need to be checked. It is known that if $G$ does not contain a cricket, a diamond, or a $C_4$, then $I(G)^q$ has linear resolution for $q \geq 2$. We construct a family of gapfree graphs $G$ containing cricket, diamond, $C_4$ together with $C_5$ as induced subgraphs of $G$ for which $I(G)^q$ has linear quotients for $q \ge 2$.