On Chip-Firing on Undirected Binary Trees

Ryota Inagaki; Tanya Khovanova; Austin Luo

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arXiv · 2410.00039

On Chip-Firing on Undirected Binary Trees

Ryota Inagaki·Tanya Khovanova·Austin Luo

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Abstract

Chip-firing is a combinatorial game played on an undirected graph in which we place chips on vertices. 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 3 chips, and it fires by dispersing $1$ 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 $2^\ell - 1$ 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 $2^\ell - 1$. We also calculate the total number of fires in this case.

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Ryota Inagaki, Tanya Khovanova, Austin Luo. 2024-09-26. On Chip-Firing on Undirected Binary Trees. https://arxiv.org/abs/2410.00039

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