Linear-resolution property for t-path ideals under co-chordality

Determine whether t-path ideals satisfy the linear-resolution condition I(๐“Ÿ_t(G)) having a linear resolution whenever the complement of G is chordal.

Background

The paper notes that t-path ideals do not satisfy the regularity and projective-dimension formulas under consideration, even though they are natural higher-degree generalizations of edge ideals. The remaining issue is whether they satisfy the third condition associated with linear resolutions.

The authors explicitly identify this property as unresolved for t-path ideals, making it a concrete open problem within the broader classification question.

References

However, to the best of our knowledge, it is still not known whether $t$-path ideals satisfy the condition (iii) or not.

— Connected ideals of chordal graphs  (2501.01112 - Das et al., 2 Jan 2025) in Section 5, Concluding Remarks