English

General formulas for the instability minimum of Chip-firing games

Combinatorics 2026-04-16 v1

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.

Keywords

Cite

@article{arxiv.2604.13949,
  title  = {General formulas for the instability minimum of Chip-firing games},
  author = {Merlin Michalski and Christophe Dubussy},
  journal= {arXiv preprint arXiv:2604.13949},
  year   = {2026}
}

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31 pages