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

Alexander Lazar
2017 unpublished
Cartwright (2015) introduced the notion of a weak tropical complex in order to generalize the theory of divisors on graphs from Baker and Norine (2007). A weak tropical complex Γ is a ∆-complex equipped with algebraic data that allows it to be viewed as the dual complex to a certain kind of degeneration over a discrete valuation ring. Every graph has a unique tropical complex structure (which is the same structure studied by Baker and Norine) in which divisors correspond to states in the
more » ... tates in the chip-firing game on that graph. Let G and H be graphs, and let Γ be a triangulation of G × H obtained by adding in one diagonal of each resulting square. There is a particular weak tropical complex structure on Γ that Cartwright conjectured was closely related to the weak tropical complex structures on G and H. The main result of this paper is a proof of Cartwright's conjecture. In preparation, we discuss some basic properties of tropical complexes, along with some properties specific to the product-of-graphs case.