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

Authors:
M. N. Ellingham;Xiaoya Zha
Affiliations:
Department of Mathematics, 1326 Stevenson Center, Vanderbilt University, Nashville, TN 37240, USA;Department of Mathematical Sciences, Middle Tennessee State University, Murfreesboro, TN 37132, USA
Venue:
European Journal of Combinatorics
Year:
2011

Citing 6
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Abstract

In a closed2-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.