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On Toric Ideals Arising from the Chip-Firing Game

Rahul Karki

Abstract

We study certain groups and ideals arising from the chip-firing game on a generalisation of graphs called pargraphs. Several well-known families of toric ideals, including those defining rational normal curves and binomial edge ideals of complete graphs, arise as toppling ideals of pargraphs. We provide sufficient conditions under which the toppling ideal of a pargraph to be toric. In addition, we construct a Gröbner basis for the toppling ideal, a minimal cellular free resolution for a distinguished initial ideal known as the $G$-parking function ideal, and establish Cohen-Macaulay property for these ideals. We also study the Picard group of a pargraph and provide sufficient conditions ensuring its freeness.

On Toric Ideals Arising from the Chip-Firing Game

Abstract

We study certain groups and ideals arising from the chip-firing game on a generalisation of graphs called pargraphs. Several well-known families of toric ideals, including those defining rational normal curves and binomial edge ideals of complete graphs, arise as toppling ideals of pargraphs. We provide sufficient conditions under which the toppling ideal of a pargraph to be toric. In addition, we construct a Gröbner basis for the toppling ideal, a minimal cellular free resolution for a distinguished initial ideal known as the -parking function ideal, and establish Cohen-Macaulay property for these ideals. We also study the Picard group of a pargraph and provide sufficient conditions ensuring its freeness.
Paper Structure (7 sections, 14 theorems, 1 figure)

This paper contains 7 sections, 14 theorems, 1 figure.

Table of Contents

  1. Introduction
  2. Combinatorics of Pargraphs
  3. Chip-Firing Game on Pargraphs
  4. Algebraic Aspects of Pargraphs
  5. Picard Group of a Pargraph
  6. Gröbner Basis of the Toppling Ideal
  7. Cellular Free Resolutions of $G$-Parking Function Ideals

Key Result

Theorem 1.1

Let $(G,\Pi)$ be a simple pargraph where $\Pi=\{V_1,\dots,V_k\}$ is a connected partition of $V(G)$. If the graph obtained after removing the basic edges of $(G,\Pi)$ from $G$ is a forest, then the Picard group of $(G,\Pi)$ is free.

Figures (1)

  • Figure 1: The pargraph $(G,\Pi)$ with $\Pi=\{ \{2,5\},\{j\}\mid j \in \{1,3,4,6,7\} \}$.

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Definition 3.1
  • Theorem 3.2
  • ...and 21 more