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Labeled Chip-Firing on Undirected $k$-ary Trees

Ryota Inagaki, Aaron Lin

TL;DR

This work generalizes labeled chip-firing from the binary to undirected $k$-ary trees under the root-initial regime with $N_{k,\ell}=\frac{k^\ell-1}{k-1}$ chips. It extends endgame analysis to $k$-ary trees, introduces the zigzag bound to tightly bound the number of stable configurations, and provides a nontrivial lower bound through constructive symmetric firings. Together, these results advance understanding of how chip-labels distribute across large, hierarchical graphs and establish groundwork for further questions on ballot properties and permutation structures of stabilized configurations. The findings have potential implications for combinatorial dynamics on trees and related permutation-structure questions in labeled chip-firing.

Abstract

We explore labeled chip-firing on undirected $k$-ary trees, trees where every vertex has degree $k+1$. First, we extend known results for binary trees from Musiker and Nguyen, including the endgame and the locations of the smallest and largest chips, as well as relations between chips at different vertices. Then, inspired by recent work on the binary tree by the first author, Khovanova, and Luo, we use these properties to construct an upper bound, which we call the zigzag bound, on the number of stable configurations in labeled chip-firing on $k$-ary trees with $\frac{k^{\ell}-1}{k-1}$ labeled chips starting at the root. We further provide a novel lower bound on the number of stable configurations of $k$-ary trees, complementing our upper bounds.

Labeled Chip-Firing on Undirected $k$-ary Trees

TL;DR

This work generalizes labeled chip-firing from the binary to undirected -ary trees under the root-initial regime with chips. It extends endgame analysis to -ary trees, introduces the zigzag bound to tightly bound the number of stable configurations, and provides a nontrivial lower bound through constructive symmetric firings. Together, these results advance understanding of how chip-labels distribute across large, hierarchical graphs and establish groundwork for further questions on ballot properties and permutation structures of stabilized configurations. The findings have potential implications for combinatorial dynamics on trees and related permutation-structure questions in labeled chip-firing.

Abstract

We explore labeled chip-firing on undirected -ary trees, trees where every vertex has degree . First, we extend known results for binary trees from Musiker and Nguyen, including the endgame and the locations of the smallest and largest chips, as well as relations between chips at different vertices. Then, inspired by recent work on the binary tree by the first author, Khovanova, and Luo, we use these properties to construct an upper bound, which we call the zigzag bound, on the number of stable configurations in labeled chip-firing on -ary trees with labeled chips starting at the root. We further provide a novel lower bound on the number of stable configurations of -ary trees, complementing our upper bounds.

Paper Structure

This paper contains 15 sections, 18 theorems, 30 equations, 14 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Unlabeled Chip-Firing
  3. Labeled-chip firing
  4. Motivations and Roadmap
  5. Preliminaries and Prior Results
  6. Definitions and Notations
  7. Prior Results
  8. An Upper Bound on the Number of Stable Configurations
  9. Generalizing the Endgame of Musiker and Nguyen
  10. The Zigzag Bound
  11. Lower Bounds on the Number of Stable Configurations
  12. Further Directions
  13. Ballot Property
  14. The Stable Configurations as a Permutation of $1, 2, \dots, N_{k, \ell}$
  15. Acknowledgements

Key Result

Theorem 1.2

Consider configuration $\mathcal{C}$ and stable configuration $\mathcal{C}'$. If $\mathcal{C}'$ can be reached from $\mathcal{C}$ after finitely many legal firings, then $\mathcal{C}'$ is the unique stable configuration that is obtained from $\mathcal{C}$.

Figures (14)

  • Figure 1: An example unlabeled chip-firing process on a binary tree starting with 7 chips at the root.
  • Figure 2: A labeled chip-firing process on a binary tree with 7 chips at the root. At each step, we bold the chips that get fired from the vertex.
  • Figure 3: Two possible labeled chip-firing processes starting from the same stable configuration and ending with different stable configurations. This demonstrates that confluence, as in Theorem \ref{['thm:Confluence']}, does not always hold in the labeled chip-firing game on undirected $k$-ary trees.
  • Figure 4: Vertex labelings for the first three layers of a 4-ary tree.
  • Figure 5: A stable configuration on a 4-ary tree.
  • ...and 9 more figures

Theorems & Definitions (51)

  • Example 1.1
  • Theorem 1.2: Theorem 2.2.2, klivans2018mathematics
  • Example 1.3
  • Example 1.4
  • Theorem 2.1: Agrawal et al. agrawal2025chipfiringinfinitekarytrees
  • Definition 2.2
  • Example 2.3
  • Example 2.4
  • Proposition 2.5: Proposition 4.4 in MR4827886
  • Proposition 2.6: Proposition 4.5 in MR4827886
  • ...and 41 more