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The $h$-vectors of toric ideals of odd cycle compositions revisited

Kieran Bhaskara, Adam Van Tuyl, Sasha Zotine

TL;DR

The paper addresses computing the $h$-polynomial of toric edge rings of graphs formed by $s$ odd cycles sharing a center vertex. It provides a new, purely algebraic proof by establishing that the toric ideals are geometrically vertex decomposable (GVD), which yields the same $h$-polynomial formula as in prior work and gives the Castelnuovo-Mumford regularity. The main contributions are the inductive GVD proof and the explicit $h$-polynomial and regularity formulas, offering a robust algebraic toolkit beyond initial ideals. The results strengthen the connection between graph structure and algebraic properties of toric edge rings and deliver exact, computable invariants for this family of graphs.

Abstract

Let $G$ be a graph consisting of $s$ odd cycles that all share a common vertex. Bhaskara, Higashitani, and Shibu Deepthi recently computed the $h$-polynomial for the quotient ring $R/I_G$, where $I_G$ is the toric ideal of $G$, in terms of the number and sizes of odd cycles in the graph. The purpose of this note is to prove the stronger result that these toric ideals are geometrically vertex decomposable, which allows us to deduce the result of Bhaskara, Higashitani, and Shibu Deepthi about the $h$-polyhomial as a corollary.

The $h$-vectors of toric ideals of odd cycle compositions revisited

TL;DR

The paper addresses computing the -polynomial of toric edge rings of graphs formed by odd cycles sharing a center vertex. It provides a new, purely algebraic proof by establishing that the toric ideals are geometrically vertex decomposable (GVD), which yields the same -polynomial formula as in prior work and gives the Castelnuovo-Mumford regularity. The main contributions are the inductive GVD proof and the explicit -polynomial and regularity formulas, offering a robust algebraic toolkit beyond initial ideals. The results strengthen the connection between graph structure and algebraic properties of toric edge rings and deliver exact, computable invariants for this family of graphs.

Abstract

Let be a graph consisting of odd cycles that all share a common vertex. Bhaskara, Higashitani, and Shibu Deepthi recently computed the -polynomial for the quotient ring , where is the toric ideal of , in terms of the number and sizes of odd cycles in the graph. The purpose of this note is to prove the stronger result that these toric ideals are geometrically vertex decomposable, which allows us to deduce the result of Bhaskara, Higashitani, and Shibu Deepthi about the -polyhomial as a corollary.

Paper Structure

This paper contains 3 sections, 8 theorems, 30 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Geometrically vertex decomposable ideals and Hilbert series
  3. Main results

Key Result

Theorem 1.1

The toric ideal $I_G$ of $G = G(t_1,\ldots,t_s)$ is geometrically vertex decomposable.

Figures (3)

  • Figure 1: The graph $G(2,1,2)$ consisting of two $2\cdot 2+1 = 5$-cycles and one $2\cdot 1 + 1 =3$-cycle meeting at a common vertex
  • Figure 2: The graph $G = G(t_1,t_2,\ldots,t_s)$ and the notation for its edges.
  • Figure 3: Diagram summarizing the argument for why $I_{G(t_1,\ldots,t_s)}$ is GVD.

Theorems & Definitions (15)

  • Theorem 1.1
  • Corollary 1.2: BHSD2023edgerings
  • Corollary 1.3
  • Definition 2.1: KR2021
  • Lemma 2.2
  • Lemma 2.3: CDSRVT2023
  • Lemma 2.4
  • Theorem 2.5: NRVT2024gvdideals
  • Remark 3.1
  • Lemma 3.2
  • ...and 5 more