Characterize higher-degree graph-induced hypergraphs satisfying chordal formulas

Determine which classes of t-uniform hypergraphs induced from graphs satisfy simultaneously, for every integer t≥2: (i) reg(R/I(𝓗_t(G)))=(t−1)ν(𝓗_t(G)) when G is chordal, (ii) pd(R/I(𝓗_t(G)))=bight(I(𝓗_t(G))) when G is chordal, and (iii) I(𝓗_t(G)) has a linear resolution when the complement of G is chordal.

Background

The paper studies higher-degree edge ideals arising from t-uniform hypergraphs induced by graphs, with the t-connected ideals of chordal graphs as the principal example. For these ideals, the authors establish formulas for regularity and projective dimension and characterize linear resolutions under chordality hypotheses.

The concluding section asks for a broader classification of graph-induced hypergraph classes for which all three analogous properties remain valid simultaneously. This is posed as a generalization of the classical theory of edge ideals of chordal graphs.