Enumeration of digraph embeddings

A cellular embedding of an Eulerian digraph D into a closed surface is said to be directed if the boundary of each face is a directed closed walk in D. The directed genus polynomial of an Eulerian digraph D is the polynomial ΓD(x) = ∑ h≥0 gh(D)x h where gh(D) is the number of directed embeddings into the orientable surface Sh, of genus h, for h = 0, 1, . . . . The sequence {gh(D)|h ≥ 0}, which is called the directed genus distribution of the digraph D, is known for very few classes of graphs, compared to the genus distribution of a graph. This paper introduces a variety of methods for calculating the directed genus distributions of Eulerian digraphs. We use them to derive an explicit formula for the directed genus distribution of any 4-regular outerplanar digraph. We show that the directed genus distribution of such a digraph is determined by the red-blue star decompositions of the characteristic tree for an outerplanar embedding. The directed genus distribution of a 4-regular outerplanar digraph is proved to be log-concave, which is consistent with an affirmative answer to a question of Bonnington, et al. [2]. Indeed, the corresponding genus polynomial is realrooted. We introduce Eulerian splitting at a vertex of a digraph, and we prove a splitting theorem for digraph embedding distributions that is analogous to the splitting theorem for (undirected) graph embedding distributions. This new splitting theorem allows conversion of the enumeration of embeddings of a digraph with vertex degrees larger than 4 into a problem of enumerating the embeddings of some 4-regular digraphs.