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We give various conditions on pinched-torus polyhedral maps which are necessary for their graphs to be embeddable in the projective plane. Our other main result is that even if the graph of a polyhedral map in the pinched torus is embeddable in a projective plane, the map induced by the embedding cannot be polyhedral, but must have all faces bounded by cycles. Finally, we give a class of examples of graphs which have polyhedral embeddings on the pinched torus and also on orientable surfaces of arbitrary high genus.
Projective plane embeddings of polyhedral pinched maps. (1994). Discrete Mathematics, 126(1), 281-291. http://dx.doi.org/10.1016/0012-365X(94)90272-0