On the minimum number of chips allowing an infinite Chip-firing game

In this article, we provide three formulas allowing to compute the minimum amount of initial chips leading to an infinite Chip-firing game, answering a question originally posed by Björner and Lovász in 1992. These formulas hold for strongly connected directed loop-free multigraphs and generalize what was already known in the Eulerian case. The various proofs heavily rely on a notion of dynamical bound, which allows to encode some specific sequences of chip configurations.