Candy-passing Games on General Graphs, I

We let G be an undirected graph and denote the vertex and edge sets of G by V (G) and E(G), respectively. The candy-passing game on G is defined by the following rules: • At the beginning of the game, c > 0 candies are distributed among |V (G)| students, each of whom is seated at some distinct vertex v ∈ V (G). • A whistle is sounded at a regular interval. • Each time the whistle is sounded, every student who is able to do so passes one candy to each of his neighbors. (If at the beginning of this step a student holds fewer candies than he has neighbors, he does nothing.) The candy-passing game was first introduced by Tanton [3], who defined the game for cyclic G. Tanton and Wagon proved that if G is an n-cycle then any candy-passing game on G with fewer than n candies terminates (see [2]). The first author [1] also studied the end behavior of candy-passing games on such G, showing that if the number of candies c is at least 3n− 2, then the configuration of candies eventually stabilizes. Here, we undertake the first study of the candy-passing game on arbitrary connected graphs G. We obtain a general stabilization result which encompasses the first author’s [1] results for c ≥ 3n.