A Chip-Firing Game on the Product of Two Graphs and the Tropical Picard Group

TL;DR

This paper proves a conjecture relating the tropical Picard groups of two graphs and their product, using concepts from tropical geometry, chip-firing games, and algebraic structures on tropical complexes.

Contribution

It establishes the injectivity and surjectivity of a map between the Picard groups of graphs and their product, confirming Cartwright's conjecture.

Findings

Proved the conjecture relating Picard groups of graphs and their product.

Analyzed properties of tropical complexes and their relation to chip-firing games.

Established conditions under which the map between Picard groups is surjective.

Abstract

In his preprint https://arxiv.org/abs/1308.3813, Cartwright introduced the notion of a weak tropical complex in order to generalize the concepts of divisors and the Picard group on graphs from Baker and Norine's paper Riemann-Roch and Abel-Jacobi Theory on a Finite Graph. A tropical complex $\Gamma$ is a $\Delta$-complex equipped with certain algebraic data. Divisors in a tropical complex are formal linear combinations of ridges, and piecewise-linear functions on a tropical complex give rise in a natural way to divisors. Divisors that arise from PL-functions are called principal, and divisors that are locally principal are called Cartier. Two divisors that differ by a principal divisor are said to be linearly equivalent. The linear equivalence classes of Cartier divisors on a tropical complex $\Gamma$ form a group called the Picard group of $\Gamma$, by analogy to the definition of the Picard group of a variety in algebraic geometry. Every graph has a unique tropical complex structure. If $G$ and $H$ are graphs, and $\Gamma$ is a triangulation of their product, then $\Gamma$ has a weak tropical complex structure that is compatible with the tropical complex structures on $G$ and $H$. Thus, divisors on $\Gamma$ can be thought of as states in a higher-dimensional chip-firing game on $\Gamma$. Cartwright conjectured that the Picard groups of $\Gamma$, $G$, and $H$ were closely related. Let $Pic(\Gamma)$ be the tropical Picard group of $\Gamma$, and $Pic(G)$ and $Pic(H)$ be the tropical Picard groups of $G$ and $H$. Then, it was conjectured that there is a map $\gamma: Pic(G) \times Pic(H) \to Pic(\Gamma)$ that is always injective and is surjective if at least one of $G$ or $H$ is a tree. In this paper, we prove the conjecture. In preparation, we discuss some basic properties of tropical complexes, along with some properties specific to the product-of-graphs case.