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

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

2009-11-13

arxiv.org/abs/0911.2713v1

Mathematics

Combinatorics

16 pages, 3 figures

Scientific paper

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.

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