Maximum genus orientable embeddings from circuit decompositions of dense eulerian graphs and digraphs

Abstract

Suppose we have an eulerian (di)graph with a (directed) circuit decomposition. We show that if the (di)graph is sufficiently dense, then it has an orientable embedding in which the given circuits are facial walks and there are exactly one or two other faces. This embedding has maximum genus subject to the given circuits being facial walks. When there is only one other face, it is necessarily bounded by an euler circuit. Thus, if the numbers of vertices and edges have the same parity, a sufficiently dense (di)graph $D$ with a given (directed) euler circuit $C$ has an orientable embedding with exactly two faces, each bounded by an euler circuit, one of which is $C$. The main theorem encompasses several special cases in the literature, for example, when the digraph is a tournament.