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

TL;DR

This paper extends core results of parallel chip-firing games from undirected to directed graphs, analyzing periods, firing sequences, and establishing bounds using algebraic methods.

Contribution

It proves analogs of key properties for directed graphs, introduces Gauss-Jordan elimination on Laplacian-like matrices, and characterizes atomic firing sequences in directed settings.

Findings

Possible periods on directed simple cycles identified.

Lower bounds on maximum periods for complete and bipartite graphs established.

Characterization of atomic firing sequences in strongly connected directed graphs.

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 $s=0$.