Regularity and h-polynomials of toric ideals of graphs

For all integers \(4 \leq r \leq d\), we show that there exists a finite simple graph \(G= G_{r,d}\) with toric ideal \(I_G \subset R\) such that \(R/I_G\) has (Castelnuovo-Mumford) regularity \(r\) and \(h\)-polynomial of degree \(d\). To achieve this goal, we identify a family of graphs such that the graded Betti numbers of the associated toric ideal agree with its initial ideal, and furthermore, this initial ideal has linear quotients. As a corollary, we can recover a result of Hibi, Higashitani, Kimura, and O'Keefe that compares the depth and dimension of toric ideals of graphs.