On Chip-Firing on Undirected Binary Trees

Abstract

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.