Setting Port Numbers for Fast Graph Exploration

We consider the problem of periodic graph exploration by a finite automaton in which an automaton with a constant number of states has to explore all unknown anonymous graphs of arbitrary size and arbitrary maximum degree. In anonymous graphs, nodes are not labeled but edges are labeled in a local manner (called local orientation) so that the automaton is able to distinguish them. Precisely, the edges incident to a node v are given port numbers from 1 to dv, where dv is the degree of v. Periodic graph exploration means visiting every node infinitely often. We are interested in the length of the period, i.e., the maximum number of edge traversals between two consecutive visits of any node by the automaton in the same state and entering the node by the same port. This problem is unsolvable if local orientations are set arbitrarily. Given this impossibility result, we address the following problem: what is the mimimum function π(n) such that there exist an algorithm for setting the local orientation, and a finite automaton using it, such that the automaton explores all graphs of size n within the period π(n)? The best result so far is the upper bound π(n) ≤ 10n, by Dobrev et al. [SIROCCO 2005], using an automaton with no memory (i.e. only one state). In this paper we prove a better upper bound π(n) ≤ 4n. Our automaton uses three states but performs periodic exploration independently of its starting position and initial state.