A Variation on Chip-Firing: the diffusion game
Discrete mathematics & theoretical computer science, Tome 20 (2018) no. 1
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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.
@article{DMTCS_2018_20_1_a4,
author = {Duffy, C. and Lidbetter, T. F. and Messinger, M. E. and Nowakowski, R. J.},
title = {A {Variation} on {Chip-Firing:} the diffusion game},
journal = {Discrete mathematics & theoretical computer science},
publisher = {mathdoc},
volume = {20},
number = {1},
year = {2018},
doi = {10.23638/DMTCS-20-1-4},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.23638/DMTCS-20-1-4/}
}
TY - JOUR AU - Duffy, C. AU - Lidbetter, T. F. AU - Messinger, M. E. AU - Nowakowski, R. J. TI - A Variation on Chip-Firing: the diffusion game JO - Discrete mathematics & theoretical computer science PY - 2018 VL - 20 IS - 1 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.23638/DMTCS-20-1-4/ DO - 10.23638/DMTCS-20-1-4 LA - en ID - DMTCS_2018_20_1_a4 ER -
%0 Journal Article %A Duffy, C. %A Lidbetter, T. F. %A Messinger, M. E. %A Nowakowski, R. J. %T A Variation on Chip-Firing: the diffusion game %J Discrete mathematics & theoretical computer science %D 2018 %V 20 %N 1 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.23638/DMTCS-20-1-4/ %R 10.23638/DMTCS-20-1-4 %G en %F DMTCS_2018_20_1_a4
Duffy, C.; Lidbetter, T. F.; Messinger, M. E.; Nowakowski, R. J. A Variation on Chip-Firing: the diffusion game. Discrete mathematics & theoretical computer science, Tome 20 (2018) no. 1. doi: 10.23638/DMTCS-20-1-4
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