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arXiv · 1609.05792

A Variation on Chip-Firing: the diffusion game

C. Duffy·T. F. Lidbetter·M. E. Messinger·R. J. Nowakowski
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Abstract

We introduce a natural variant of the parallel chip-firing game, called the diffusion game. Chips are initially assigned to vertices of a graph. At every step, all vertices simultaneously send one chip to each neighbour with fewer chips. As the dynamics of the parallel chip-firing game occur on a finite set the process is inherently periodic. However the diffusion game is not obviously periodic: even if $2|E(G)|$ chips are assigned to vertices of graph G, there may exist time steps where some vertices have a negative number of chips. We investigate the process, prove periodicity for a number of graph classes, and pose some questions for future research.

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C. Duffy, T. F. Lidbetter, M. E. Messinger, R. J. Nowakowski. 2016-09-19. A Variation on Chip-Firing: the diffusion game. https://doi.org/10.23638/dmtcs-20-1-4

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