Motors and Impossible Firing Patterns in the Parallel Chip-Firing Game

TL;DR

This paper explores the behavior of the parallel chip-firing game on graphs, introducing motorized variants, characterizing possible firing patterns, and analyzing how motors influence the game's periodic firing sequences.

Contribution

It introduces motorized versions of the game, characterizes achievable firing patterns, and shows how motors can be simulated within ordinary game dynamics.

Findings

Motorized games can be transformed into ordinary games under certain conditions.

Characterization of exactly which firing patterns are possible in the standard game.

Analysis of the impact of motors connected to trees on firing sequences.

Abstract

The parallel chip-firing game is an automaton on graphs in which vertices "fire" chips to their neighbors when they have enough chips to do so. The game is always periodic, and we concern ourselves with the firing sequences of vertices. We introduce the concepts of motorized parallel chip-firing games and motor vertices, study the effects of motors connected to a tree and show that motorized games can be transformed into ordinary games if the motors' firing sequences occur in some ordinary game. We then characterize exactly which periodic firing patterns can occur in an ordinary game and state some implications of the finding.