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

Abstract

A closed 2-cell embedding of a graph embedded in some surface is an embedding such that each face is bounded by a circuit in the graph. The strong embedding conjecture says that every 2-connected graph has a closed 2-cell embedding in some surface. A graph is called k cross-cap embeddable if it can be embedded in the non-orientable surface of k cross-caps. In this paper, we prove that every 2-connected 4 cross-cap embeddable graph G has a closed 2-cell embedding in some surface. As a corollary, G has a cycle double cover, i.e., G has a set of circuits containing every edge exactly twice.