Combinatorial characterization of all-powers linear quotients

Identify natural combinatorial conditions on a gapfree graph G that ensure that I(G)^s has linear quotients for every s ≥ 2.

Background

The paper proves the all-powers linear-quotients property under the specific structural hypothesis that the gapfree graph has matching number three. Its reductions and graph operations suggest that other graph-theoretic conditions may also guarantee the same algebraic behavior.

The authors leave open the task of finding natural combinatorial conditions, beyond the cases treated in the paper, that characterize or systematically ensure linear quotients for every power beginning with the square.