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

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 Inagaki, 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.