Periods and atomic firing sequences of parallel chip-firing games on directed graphs

TL;DR

This work extends the theory of parallel chip-firing games from undirected to directed graphs, establishing directed-graph analogs of core results. It develops a Gauss-Jordan elimination framework on an out-degree Laplacian to bound periods for orientations of complete and bipartite graphs, and provides precise period classifications for directed cycles and acyclic structures. The study also characterizes atomic firing sequences, showing that any binary string with at least one '1' can occur (and clarifying the case of all-zero sequences), via explicit graph-construction methods. Together, these results reveal rich, graph-structure–dependent periodic dynamics in directed chip-firing games and suggest tight conjectures for the maximal periods in major graph families.

Abstract

In 1992, Bitar and Goles introduced the parallel chip-firing game on undirected graphs. Two years later, Prisner extended the game to directed graphs. While the properties of parallel chip-firing games on undirected graphs have been extensively studied, their analogs for parallel chip-firing games on directed graphs have been sporadic. In this paper, we prove the outstanding analogs of the core results of parallel chip-firing games on undirected graphs for those on directed graphs. We find the possible periods of a parallel chip-firing game on a directed simple cycle and introduce the method of Gauss-Jordan elimination on a Laplacian-like matrix to establish a lower bound on the maximum period of a parallel chip-firing game on an orientation of an undirected complete graph and an undirected complete bipartite graph. Finally, we expand the method of motors by Jiang, Scully, and Zhang to directed graphs to show that a binary string $s$ can be the atomic firing sequence of a vertex in a parallel chip-firing game on a strongly connected directed graph if and only if $s$ contains $1$ or $s=0$.

Figures (8)

• Figure 1: An undirected graph $G$ (left) and an orientation of $G$.
• Figure 2: A directed graph (left) with strongly connected components boxed and its condensation (right).
• Figure 3: A useful orientation of $K_5$.
• Figure 4: An example of a parallel chip-firing game on $K_4$ with $T=T_4=4$.
• Figure 5: A useful orientation of $K_{4,\,4}$.

Theorems & Definitions (42)

• Proposition 2.1: prisner
• Lemma 2.2
• proof
• Proposition 3.1: prisner
• Corollary 3.2
• Corollary 3.3
• Lemma 3.4
• proof
• Theorem 3.5
• proof