  We consider the problem of periodic graph exploration in which a mobile
entity with constant memory, an agent, has to visit all n nodes of an arbitrary
undirected graph G in a periodic manner. Graphs are supposed to be anonymous,
that is, nodes are unlabeled. However, while visiting a node, the robot has to
distinguish between edges incident to it. For each node v the endpoints of the
edges incident to v are uniquely identified by different integer labels called
port numbers. We are interested in minimisation of the length of the
exploration period.
  This problem is unsolvable if the local port numbers are set arbitrarily.
However, surprisingly small periods can be achieved when assigning carefully
the local port numbers. Dobrev et al. described an algorithm for assigning port
numbers, and an oblivious agent (i.e. agent with no memory) using it, such that
the agent explores all graphs of size n within period 10n. Providing the agent
with a constant number of memory bits, the optimal length of the period was
previously proved to be no more than 3.75n (using a different assignment of the
port numbers). In this paper, we improve both these bounds. More precisely, we
show a period of length at most 4 1/3 n for oblivious agents, and a period of
length at most 3.5n for agents with constant memory. Moreover, we give the
first non-trivial lower bound, 2.8n, on the period length for the oblivious
case.
