On Chip-Firing on Undirected Binary Trees
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866917791796297728 |
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| author | Inagaki, Ryota Khovanova, Tanya Luo, Austin |
| author_facet | Inagaki, Ryota Khovanova, Tanya Luo, Austin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_00039 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Chip-Firing on Undirected Binary Trees Inagaki, Ryota Khovanova, Tanya Luo, Austin Combinatorics 05C57, Secondary 05C63, 05A15 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. |
| title | On Chip-Firing on Undirected Binary Trees |
| topic | Combinatorics 05C57, Secondary 05C63, 05A15 |
| url | https://arxiv.org/abs/2410.00039 |