Bi-eulerian embeddings of graphs and digraphs

In 1965 Edmonds showed that every eulerian graph has a bi-eulerian embedding, i.e., an embedding with exactly two faces, each bounded by an euler circuit. We refine this result by giving conditions for a graph to have a bi-eulerian embedding that is specifically orientable or nonorientable. We give connections to the maximum genus problem for directed embeddings of digraphs, in which every face is bounded by a directed circuit. Given an eulerian digraph $D$ with all vertices of degree 2 mod 4 and a directed euler circuit $T$ of $D$, we show that $D$ has an orientable bi-eulerian directed embedding with one of the faces bounded by $T$; this is a maximum genus directed embedding. This result also holds when $D$ has exactly two vertices of degree $0$ mod $4$, provided they are interlaced by $T$. More generally, if $D$ has $\ell$ vertices of degree 0 mod 4, we can find an orientable directed embedding with a face bounded by $T$ and with at most $\ell+1$ other faces. We show that given an eulerian graph $G$ and a circuit decomposition $C$ of $G$, there is an nonorientable embedding of $G$ with the elements of $C$ bounding faces and with one additional face bounded by an euler circuit, unless every block of $G$ is a cycle and $C$ is the collection of cycles of $G$. In particular, every eulerian graph that is not edgeless or a cycle has a nonorientable bi-eulerian embedding with a given euler circuit $T$ bounding one of the faces. Polynomial-time algorithms giving the specified embeddings are implicit in our proofs.