  A "folklore conjecture, probably due to Tutte" (as described in [P.D.
Seymour, Sums of circuits, Graph theory and related topics (Proc. Conf., Univ.
Waterloo, 1977), pp. 341-355, Academic Press, 1979]) asserts that every
bridgeless cubic graph can be embedded on a surface of its own genus in such a
way that the face boundaries are cycles of the graph. In this paper we consider
closed 2-cell embeddings of graphs and show that certain (cubic) graphs (of any
fixed genus) have closed 2-cell embedding only in surfaces whose genus is very
large (proportional to the order of these graphs), thus providing plethora of
strong counterexamples to the above conjecture. The main result yielding such
counterexamples may be of independent interest.
