Closed 2-cell Embeddings of 5-crosscap 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 calledk-crosscap embeddable if it can be embedded in the surface of non-orientable genusk.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.