Coalescing Walks on Rotor-Router Systems

We consider the rotor-router mechanism for distributing particles in an undirected graph. If the last particle passing through a vertex v took an edge v,u, then the next time a particle is at v, it will leave v along the next edge v,w according to a fixed cyclic order of edges adjacent to v. The system works in synchronized steps and when two or more particles meet at the same vertex, they coalesce into one particle. A k-particle configuration of such a system is stable, if it does not lead to any coalescing. For 2≤k≤n, we give the full characterization of stable k-particle configurations for cycles. We also show sufficient conditions for regular graphs with n vertices to admit n-particle stable configurations.