## Duffy, C. and Lidbetter, T. F. and Messinger, M. E. and Nowakowski, R. J. - A Variation on Chip-Firing: the diffusion game

dmtcs:2039 - Discrete Mathematics & Theoretical Computer Science, January 17, 2018, Vol. 20 no. 1 - https://doi.org/10.23638/DMTCS-20-1-4
A Variation on Chip-Firing: the diffusion game

Authors: Duffy, C. and Lidbetter, T. F. and Messinger, M. E. and Nowakowski, R. J.

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.

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