The regularity–$h$-degree band conjecture for connected graphs

Let $G$ be a connected graph on $n$ vertices. Write $R$ for the polynomial ring associated with $G$, $J(G)$ for its cover ideal, $\operatorname{reg}(R/J(G))$ for the Castelnuovo–Mumford regularity, and $h_{R/J(G)}(t)$ for the numerator of the Hilbert series of $R/J(G)$. Regularity–$h$-degree band conjecture.

$$\bigl|\operatorname{reg}(R/J(G))-\deg h_{R/J(G)}(t)\bigr|\ \le\ \left\lceil\frac{n}{2}\right\rceil-2.$$

Computations for connected graphs with at most $12$ vertices suggest that the realizable pairs lie in a band around the diagonal. The conjecture gives a uniform bound on the discrepancy between regularity and the degree of the $h$-polynomial; its resolution remains open based on the supplied source.