On Chip-Firing on Undirected Binary Trees
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26_2025_Article_779.pdf
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623.6 KB
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Author(s)
Inagaki, Ryota
Khovanova, Tanya
Luo, Austin
Date Issued
August 18, 2025
Journal
Annals of Combinatorics
Publisher
Springer International Publishing
Citation
Inagaki, R., Khovanova, T. & Luo, A. On Chip-Firing on Undirected Binary Trees. Ann. Comb. (2025).
Version
Final published version
Abstract
Chip-firing is a combinatorial game played on an undirected graph in which we place chips on vertices and disperse them. We study chip-firing on an infinite binary tree in which we add a self-loop to the root to ensure each vertex has degree 3. A vertex can fire if the number of chips placed on it is at least its degree. In our case, a vertex can fire if it has at least three chips, and it fires by dispersing one chip to each neighbor. Motivated by a 2023 paper by Musiker and Nguyen on this setting of chip-firing, we give an upper bound for the number of stable configurations when we place 2 ℓ - 1 labeled chips at the root. When starting with N chips at the root where N is a positive integer, we determine the number of times each vertex fires when N is not necessarily of the form 2 ℓ - 1 . We also calculate the total number of fires in this case.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution
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DOI of Published Version
https://doi.org/10.1007/s00026-025-00779-6
