Regularity and ℎ-polynomials of toric ideals of graphs

Favacchio, Giuseppe; Keiper, Graham; Van Tuyl, Adam

doi:10.1090/proc/15126

Regularity and $h$-polynomials of toric ideals of graphs
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by Giuseppe Favacchio, Graham Keiper and Adam Van Tuyl PDF

Proc. Amer. Math. Soc. 148 (2020), 4665-4677 Request permission

Abstract:

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, 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.

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