Orientable embeddings and orientable cycle double covers of projective-planar graphs

In a closed 2 -cell embedding of a graph each face is homeomorphic to an open disk and is bounded by a cycle in the graph. The Orientable Strong Embedding Conjecture says that every 2 -connected graph has a closed 2 -cell embedding in some orientable surface. This implies both the Cycle Double Cover Conjecture and the Strong Embedding Conjecture. In this paper we prove that every 2 -connected projective-planar cubic graph has a closed 2 -cell embedding in some orientable surface. The three main ingredients of the proof are (1) a surgical method to convert nonorientable embeddings into orientable embeddings; (2) a reduction for 4 -cycles for orientable closed 2 -cell embeddings, or orientable cycle double covers, of cubic graphs; and (3) a structural result for projective-planar embeddings of cubic graphs. We deduce that every 2 -edge-connected projective-planar graph (not necessarily cubic) has an orientable cycle double cover.