Classify graph classes satisfying both t-connected ideal formulas

Find all classes of graphs G for which, for every integer t≥2, the t-connected ideal I_t(G) satisfies both reg(R/I_t(G))=(t−1)ν_t(G) and pd(R/I_t(G))=bight(I_t(G)).

Background

The paper proves both equalities for t-connected ideals of chordal graphs. It also observes that analogous equalities for edge ideals hold in additional graph classes, motivating a search for broader graph-theoretic conditions.

The concluding question asks for a classification beyond chordal graphs, requiring both the regularity and projective-dimension formulas to hold uniformly for all t≥2.

References

Find those classes of graph $G$ for which $reg(R/I_{t}(G))=(t-1)\nu_{t}(G)$ and $pd(R/I_{t}(G))=bight(I_t(G))$ for all $t\geq 2$, where $I_t(G)$ denotes the $t$-connected ideal of $G$.

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