This paper introduces the problem of n mobile agents that repeatedly visit all n nodes of a given network, subject to the constraint that no two agents can simultaneously occupy a node. This paper first presents a self-stabilizing phase-based protocol for a tree network on a synchronous model. The protocol realizes agent traversal with O(Δn) time where n is the number of nodes and Δ is the maximum degree of any vertex in the communication network. The phase-based protocol can also be applied to an asynchronous model and a ring network. This paper also presents a self-stabilizing link-alternator-based protocol with agent traversal time of O(Δn) for a tree network on an asynchronous model. The protocols are proved to be asymptotically optimal with respect to the agent traversal time.