Chip Firing on Directed $k$-ary Trees
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912096267010048 |
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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 a graph in which we place and disperse chips on vertices until a stable state is reached. We study a chip-firing variant played on an infinite rooted directed $k$-ary tree, where we place $k^\ell$ chips on the root for some positive integer $\ell$, and we say a vertex $v$ can fire if it has at least $k$ chips. A vertex fires by dispersing one chip to each out-neighbor. Once every vertex has less than $k$ chips, we reach a stable configuration since no vertex can fire. We determine the exact number and properties of the possible stable configurations of chips in the setting where chips are distinguishable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_23265 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Chip Firing on Directed $k$-ary Trees Inagaki, Ryota Khovanova, Tanya Luo, Austin Combinatorics 05C57, 05C63, 05A05, 05A15 Chip-firing is a combinatorial game played on a graph in which we place and disperse chips on vertices until a stable state is reached. We study a chip-firing variant played on an infinite rooted directed $k$-ary tree, where we place $k^\ell$ chips on the root for some positive integer $\ell$, and we say a vertex $v$ can fire if it has at least $k$ chips. A vertex fires by dispersing one chip to each out-neighbor. Once every vertex has less than $k$ chips, we reach a stable configuration since no vertex can fire. We determine the exact number and properties of the possible stable configurations of chips in the setting where chips are distinguishable. |
| title | Chip Firing on Directed $k$-ary Trees |
| topic | Combinatorics 05C57, 05C63, 05A05, 05A15 |
| url | https://arxiv.org/abs/2410.23265 |