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Geometric vertex decomposition and liaison for toric ideals of graphs

Mike Cummings, Sergio Da Silva, Jenna Rajchgot, Adam Van Tuyl

Abstract

The geometric vertex decomposability property for polynomial ideals is an ideal-theoretic generalization of the vertex decomposability property for simplicial complexes. Indeed, a homogeneous geometrically vertex decomposable ideal is radical and Cohen-Macaulay, and is in the Gorenstein liaison class of a complete intersection (glicci). In this paper, we initiate an investigation into when the toric ideal $I_G$ of a finite simple graph $G$ is geometrically vertex decomposable. We first show how geometric vertex decomposability behaves under tensor products, which allows us to restrict to connected graphs. We then describe a graph operation that preserves geometric vertex decomposability, thus allowing us to build many graphs whose corresponding toric ideals are geometrically vertex decomposable. Using work of Constantinescu and Gorla, we prove that toric ideals of bipartite graphs are geometrically vertex decomposable. We also propose a conjecture that all toric ideals of graphs with a square-free degeneration with respect to a lexicographic order are geometrically vertex decomposable. As evidence, we prove the conjecture in the case that the universal Gröbner basis of $I_G$ is a set of quadratic binomials. We also prove that some other families of graphs have the property that $I_G$ is glicci.

Geometric vertex decomposition and liaison for toric ideals of graphs

Abstract

The geometric vertex decomposability property for polynomial ideals is an ideal-theoretic generalization of the vertex decomposability property for simplicial complexes. Indeed, a homogeneous geometrically vertex decomposable ideal is radical and Cohen-Macaulay, and is in the Gorenstein liaison class of a complete intersection (glicci). In this paper, we initiate an investigation into when the toric ideal of a finite simple graph is geometrically vertex decomposable. We first show how geometric vertex decomposability behaves under tensor products, which allows us to restrict to connected graphs. We then describe a graph operation that preserves geometric vertex decomposability, thus allowing us to build many graphs whose corresponding toric ideals are geometrically vertex decomposable. Using work of Constantinescu and Gorla, we prove that toric ideals of bipartite graphs are geometrically vertex decomposable. We also propose a conjecture that all toric ideals of graphs with a square-free degeneration with respect to a lexicographic order are geometrically vertex decomposable. As evidence, we prove the conjecture in the case that the universal Gröbner basis of is a set of quadratic binomials. We also prove that some other families of graphs have the property that is glicci.
Paper Structure (15 sections, 48 theorems, 61 equations, 5 figures)

This paper contains 15 sections, 48 theorems, 61 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Geometrically vertex decomposable ideals
  3. Toric ideals of graphs
  4. Toric ideals of graphs
  5. Structure results about $N_{y,I}$ and $C_{y,I}$
  6. Geometric vertex decomposability under graph operations
  7. Toric ideals of graphs and the glicci property
  8. Gorenstein liaison preliminaries
  9. Some toric ideals of graphs which are glicci
  10. Toric ideals of bipartite graphs
  11. Toric ideals of bipartite graphs are geometrically vertex decomposable
  12. Alternate proofs in special cases
  13. Toric ideals with a square-free degeneration
  14. Framework for the conjecture
  15. Proof of the conjecture in the quadratic case

Key Result

Theorem 1.1

Let $I \subsetneq R =\mathbb{K}[x_1,\ldots,x_n]$ and $J \subsetneq S=\mathbb{K}[y_1,\ldots,y_m]$ be proper ideals. Then $I$ and $J$ are geometrically vertex decomposable if and only if $I+J$ is geometrically vertex decomposable in $R \otimes S =\mathbb{K}[x_1,\ldots,x_n,y_1,\ldots,y_m]$.

Figures (5)

  • Figure 1: Gluing an even cycle $C$ to a graph $G$ along an edge.
  • Figure 2: A graph whose toric ideal is geometrically vertex decomposable
  • Figure 3: The graph $T_\lambda$ for $\lambda = (5,3,2,1)$
  • Figure 4: The graph $G_{3,5}$
  • Figure 5: The relation between the ideals $I^G_{E,F}$

Theorems & Definitions (99)

  • Theorem 1.1: Theorem \ref{['tensorproduct']}
  • Theorem 1.2: Theorem \ref{['gluetheorem']}
  • Theorem 1.3: Theorem \ref{['thm: gvdBipartite']}
  • Conjecture 1.4: Conjecture \ref{['mainconjecture']}
  • Theorem 1.5: Theorem \ref{['quadratic_GVD']}
  • Theorem 1.6: Corollary \ref{['glue=glicci']}
  • Theorem 1.7: Corollary \ref{['cor:square-freeDegenGlicciGraph']}
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3: KMY
  • ...and 89 more