Abstract
Kominers and Kominers showed that any parallel chip-firing game on G(V, E) with at least \(4|E|-|V|\) chips stabilizes with an eventual period of length 1. We make this bound exact: we prove that any parallel chip-firing game with more than \(3|E|-|V|\) or less than |E| chips must stabilize and that if the number of chips is outside this range, then there exists some parallel chip-firing game with that many chips that does not stabilize. In addition, as do Kominers and Kominers, we provide an upper bound on the number of rounds before the game stabilizes.
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Acknowledgements
We thank Scott Kominers and Kevin Cong for useful discussions. The second author gratefully acknowledges the support from the Harvard College Research Fund.
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Bu, A., Choi, Y. & Xu, M. An exact bound on the number of chips of parallel chip-firing games that stabilize. Arch. Math. 119, 471–478 (2022). https://doi.org/10.1007/s00013-022-01777-3
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DOI: https://doi.org/10.1007/s00013-022-01777-3