Betti numbers of toric ideals of graphs: A case study

TL;DR

This paper calculates the Betti numbers, Hilbert series, and h-vector for a specific family of graph toric ideals, revealing their algebraic structure through initial ideals with linear quotients.

Contribution

It introduces a method to compute Betti numbers and related invariants for toric ideals of graphs formed by adjoining cycles to bipartite graphs, using initial ideals with linear quotients.

Findings

Betti numbers are explicitly computed for the family of graphs.

Hilbert series and h-vector formulas are derived for these toric ideals.

The family admits initial ideals with linear quotients, facilitating calculations.

Abstract

We compute the graded Betti numbers for the toric ideal of a family of graphs constructed by adjoining a cycle to a complete bipartite graph. The key observation is that this family admits an initial ideal which has linear quotients. As a corollary, we compute the Hilbert series and $h$-vector for all the toric ideals of graphs in this family.