No polynomial bound for the chip firing game on directed graphs

Tardos has proved a polynomial bound on the length of a convergent chip firing game on an undirected graph. This paper demonstrates a game with exponential growth on a directed graph. INTRODUCTION The chips game for general graphs was formulated by Bjorner, Lovasz, and Shor [1]. For undirected graphs, G. Tardos [2] gave an upper limit, 0(n4), on the number of moves in a game for which the number of nodes is bounded by n. In this note we prove that for directed graphs no polynomial bound on the number of moves exists. In fact, we give an example of a sequence of games with exponential growth. Definitions. The chips game is played in the following way. Place a pile of chips on each node of a connected finite graph. A move now consists of selecting a node with at least as many chips in its pile as it has outgoing edges, and firing that node by moving one chip to each of its neighbors (respecting the direction of edges). Consequently, a position where each node has too few chips to be fired is a final position. This game is known to be strongly convergent [1], that is, either every legal game from the initial position is infinite or every legal game terminates with the same final position in the same number of moves, whatever choices have been made during play. The exponential game. We define a game for every even positive number n. Take a bidirected circuit of n 1 nodes and add a center node with edges to all other nodes, all but one bidirected, the last one directed from the center node. The initial position is 3n 5 chips on the center node and no chips on the circuit nodes. This game is convergent, as will be seen later. Only one distribution of 3n 5 chips will do as a final position: n 2 on the center node, 1 on the node with the directed edge (call this node the top node), and 2 on all the others. Received by the editors July 29, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 05C35.