Reducing space usage of st-connectivity with multiple passes?
It is a long standing open problem to find an algorithm for st-connectivity that runs in simultaneously sub-linear space and polynomial time, an easier task that what you are aiming at.  Such algorithms are known for the un-directed version, but even these require a large polynomial time (rather than O(km) time which would be implied by a k-pass algorithm).  See especially the reference to Tompa's paper on why the directed case seems hard.