Enumeration and Quasipolynomiality of Chip-Firing Configurations

TL;DR

This paper investigates the enumeration of reachable configurations in chip-firing games on graphs, proving quasipolynomiality of the count and providing explicit formulas and bounds for specific cases like cycle graphs.

Contribution

It introduces debt-reachability, proves quasipolynomiality of configuration counts, and derives explicit formulas and bounds for cycle graphs.

Findings

Number of debt-reachable configurations is quasipolynomial in c.

Explicit near formulas for cycle graphs C_n.

Polynomial asymptotic bounds for configurations.

Abstract

In this paper we explore enumeration problems related to the number of reachable configurations in a chip-firing game on a finite connected graph G. We define an auxiliary notion of debt-reachability and prove that the number of debt-reachable configurations from an initial configuration with c chips on one vertex is a quasipolynomial in c. For the cycle graph C_n, we apply these results to compute a near explicit formula for the number of debt-reachable configurations. We then derive polynomial asymptotic bounds for the number of debt-reachable and reachable configurations, and finally provide evidence for a quasipolynomiality conjecture regarding the number of reachable configurations.