A characterization of the embeddability of graphs on the surface of given genus (Q1352472)

The following theorem is proved: A connected graph \(G\) can be 2-cell imbedded in the closed orientable 2-manifold of genus \(k\) if and only if there is a \(k\)-eulerian circuit (defined in the paper) with associated graph (obtained by adding certain edges to a spanning tree for the circuit) being planar. A similar result is stated for closed nonorientable 2-manifolds. A corollary to the latter result is given; this corollary applies to nonorientable imbeddings, although that is not explicitly stated.