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

TL;DR

This paper proves that all 2-connected projective-planar cubic graphs can be embedded in orientable surfaces with cycle double covers, advancing the understanding of graph embeddings and cycle covers in topological graph theory.

Contribution

It introduces a surgical method to convert nonorientable embeddings into orientable ones and establishes that all 2-connected projective-planar cubic graphs have orientable embeddings with cycle double covers.

Findings

Every 2-connected projective-planar cubic graph has an orientable 2-cell embedding.

Every 2-edge-connected projective-planar graph admits an orientable cycle double cover.

Abstract

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.