No Polynomial Bound for the Period of the Parallel Chip Firing Game on Graphs

The following (solitaire) game is considered: Initially each node of a simple, connected , nite graph contains a nite number of chips. A move consists in ring all nodes with at least as many chips as their degree, where ring a node corresponds to sending one of the node's chips to each one of the node's neighbors. In Bi89] it was conjectured that every parallel chip ring game played on an N node connected and undirected graph nally evolves into a steady phase with a period that does not exceed N. It was later conjectured in Pr93] that over every strongly connected Eulerian multidigraph there is a parallel chip ring game that evolves into a steady state with period equal to the length of the longest dicycle of the underlying digraph. In this work we disprove both these conjectures by exhibiting a parallel chip ring game on an N node connected and undirected graph with period e ?p N log N .