Closed 2-cell embeddings of 4 cross-cap embeddable graphs

The strong embedding conjecture states that every 2-connected graph has a closed 2-cell embedding in some surface , i . e . an embedding that each face is bounded by a circuit in the graph . A graph is called k -crosscap embeddable if it can be embedded in the surface of non-orientable genus k . We confirm the strong embedding conjecture for 5-crosscap embeddable graphs . As a corollary , every such graph has a cycle double cover , i . e . a set of circuits containing every edge exactly twice . We classify simple closed curves in the surface of 3-crosscap graphs and study some topological properties of simple closed curves in the torus and the punctured torus .