Unique and faithful embeddings of projective‐planar graphs

A graph G is uniquely embeddable in a surface F 2 if for any two embeddings f 1 , f 2 : G → F 2 , there exists an isomorphism σ: G → G and a homeomorphism h : F 2 → F 2 for which h → f 1 = f 2 σ. A graph G is faithfully embeddable in a surface F 2 if G admits an embedding f: G → F 2 such that for any isomorphism σ: G → G , there is a homeomorphism h : F 2 → F 2 with h → f = f → σ. It will be shown that if a projective‐planar graph G is 5‐connected and contains a subdivision of the complete graph K 6 as its subgraph, then G is uniquely embeddable in a projective plane, and that moreover if G is not isomorphic to K 6 , then G is faithfully embeddable in a projective plane.