Nevo–Peeva eventual linear-resolution conjecture for gapfree graphs

Prove that for every gapfree graph G, all sufficiently large powers of its edge ideal I(G) have linear resolutions.

Background

The paper explains that gapfreeness is necessary for powers of an edge ideal to have linear resolutions, but that gapfreeness alone does not guarantee a linear resolution for the square. It then places the authors’ matching-number-three theorem in the context of a broader conjecture attributed to Nevo and Peeva.

The conjecture remains unresolved in general; the paper establishes it for gapfree graphs with matching number three and, in fact, proves linear quotients for every power beginning with the square in that case.

References

They conjectured that it does force $I(G)q$ to have a linear resolution for all sufficiently large $q$ Open Problem~1.11(1).

— Edge ideals of gapfree graphs with matching number $3$  (2610.21866 - Erey et al., 15 Oct 2026) in Introduction

Their stronger regularity conjecture Open Problem~1.11(2) would imply linear resolutions for every $q\geqreg I(G)-1$.

— Edge ideals of gapfree graphs with matching number $3$  (2610.21866 - Erey et al., 15 Oct 2026) in Introduction

If $G$ is gapfree with $\nu(G)=4$, does $I(G)s$ have a linear resolution for every $s\geq2$? Can one prove linear quotients for all these powers?

— Edge ideals of gapfree graphs with matching number $3$  (2610.21866 - Erey et al., 15 Oct 2026) in Question 1, Section Further directions

For every gapfree graph $G$ with at least one edge, do the following implications hold? \begin{enumerate}[(a)] \item If $\nu(G)\leq5$, then $I(G)s$ has a linear resolution for every $s\geq2$. \item If $\nu(G)\leq6$, then $I(G)s$ has a linear resolution for every $s\geq3$. \end{enumerate}

— Edge ideals of gapfree graphs with matching number $3$  (2610.21866 - Erey et al., 15 Oct 2026) in Question 2, Section Further directions