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<p>Chip-firing is a combinatorial game played on an undirected graph in which we place chips on vertices and disperse them. 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 three chips, and it fires by dispersing one 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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(2^\ell - 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mn>2</mn> <mi>ℓ</mi> </msup> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> labeled chips at the root. When starting with <i>N</i> chips at the root where <i>N</i> is a positive integer, we determine the number of times each vertex fires when <i>N</i> is not necessarily of the form <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(2^\ell - 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mn>2</mn> <mi>ℓ</mi> </msup> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. We also calculate the total number of fires in this case.</p>

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On Chip-Firing on Undirected Binary Trees

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年8月18日

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  • Ryota Inagaki,
  • Tanya Khovanova,
  • Austin Luo

摘要

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