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.

Source : oai:arXiv.org:1609.05792

DOI : 10.23638/DMTCS-20-1-4

Volume: Vol. 20 no. 1

Section: Graph Theory

Published on: January 17, 2018

Submitted on: April 19, 2017

Keywords: Computer Science - Discrete Mathematics,Mathematics - Combinatorics,91A50

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