Chip Firing on Directed $k$-ary Trees

Ryota Inagaki; Tanya Khovanova; Austin Luo

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arXiv · 2410.23265

Chip Firing on Directed $k$-ary Trees

Ryota Inagaki·Tanya Khovanova·Austin Luo

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Abstract

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.

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BibTeX RIS

Ryota Inagaki, Tanya Khovanova, Austin Luo. 2024-10-30. Chip Firing on Directed $k$-ary Trees. https://arxiv.org/abs/2410.23265

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