General formula for the Castelnuovo–Mumford regularity of S/I(X) for arbitrary graphs when q > 3

Establish a general formula for the Castelnuovo–Mumford regularity reg S/I(X) of the quotient S/I(X) associated with the projective toric subset X parameterized by an arbitrary simple graph G over a finite field F, in the case q > 3.

Background

The regularity of S/I(X) governs when the Hilbert function (and hence the dimension of the codes) stabilizes, and many works have computed regularity for specific graph families. Known formulas include the complete intersection (torus) case, certain bipartite graphs with nested ear decompositions, the case q = 3 via a combinatorial characterization, and some special families like complete graphs and complete bipartite graphs.

Despite these advances, the authors state that no formula is currently known that applies to arbitrary graphs when q > 3, marking this as an explicit open problem.