  A connected digraph in which the in-degree of any vertex equals its
out-degree is Eulerian; this baseline result is used as the basis of existence
proofs for universal cycles (also known as deBruijn cycles or $U$-cycles) of
several combinatorial objects. We present new results on the existence of
universal cycles of certain classes of functions. These include onto functions,
and 1-inequitable sequences on a binary alphabet. In each case the
connectedness of the underlying graph is the non-trivial aspect to be
established.
