General formulas for the instability minimum of Chip-firing games

TL;DR

This paper introduces three formulas to determine the minimal initial chips needed for an infinite chip-firing game on certain directed graphs, extending previous Eulerian case results and exploring algorithmic implications.

Contribution

The paper generalizes formulas for the instability minimum in chip-firing games beyond Eulerian graphs, applicable to strongly connected directed loop-free multigraphs.

Findings

Derived three formulas for the instability minimum.

Extended known results from Eulerian to more general graphs.

Investigated algorithmic consequences of the formulas.

Abstract

In this article, we provide three formulas allowing to compute the minimum amount of initial chips leading to an infinite Chip-firing game. These formulas hold for strongly connected directed loop-free multigraphs and generalize what was already known in the Eulerian case. In addition to the many theoretical aspects, some algorithmic consequences are also investigated.