2009/11/13 by M. N. Ellingham, Ellingham, M. N., Xiaoya Zha +1 · 1 citation
Engineering · Materials Science · #05C10 (Primary) 05C70 (Secondary) #Advanced Antenna and Metasurface Technologies #Advanced Cellulose Research Studies #Combinatorics (math.CO) #FOS: Mathematics #Structural Analysis and Optimization
paper · pdf · doi:10.48550/arxiv.0911.2713
openalex publication_date 2009/11/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.