Regularity and h-polynomials of toric ideals of graphs

For all integers 4 ≤ r ≤ d, we show that there exists a finite simple graph G = Gr,d with toric ideal IG ⊂ R such that R/IG 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, that 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.