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On the Period Lengths of the Parallel Chip-Firing Game

The parallel chip-firing game is a periodic automaton on graphs in which vertices "fire" chips to their neighbors. In 1989, Bitar conjectured that the period of a parallel chip-firing game with n vertices is at most n. Though this conjecture was disproven in 1994 by Kiwi et. al., it has been proven for particular classes of graphs, specifically trees (Bitar and Goles, 1992) and the complete graph K_n (Levine, 2008). We prove Bitar's conjecture for complete bipartite graphs and characterize completely all possible periods for positions of the parallel chip-firing game on such graphs. Furthermore, we extend our construction of all possible periods for games on the bipartite graph to games on complete c-partite graphs, c>2, and prove some pertinent lemmas about games on general simple connected graphs.