Embedding Quartic Eulerian Digraphs on the Plane

Minimal obstructions for embedding 4-regular Eulerian digraphs on the plane are considered in relation to the partial order defined by the cycle removal operation. Their basic properties are provided and all obstructions with parallel arcs are classified.Comment: 13 page

The genus distributions of 4-regular digraphs

An embedding of an Eulerian digraph in orientable surfaces was introduced by Bonnington et al. They gave some problems which need to be further studied. One of them is whether the embedding distribution of an embeddable digraph is always unimodal. In this paper, we first introduce the method of how to determine the faces and antifaces from a given rotation scheme of a digraph. The genus distributions of two new kinds of 4-regular digraphs in orientable surfaces are obtained. The genus distributions of one kind of digraph are strong unimodal, which gives a partial answer to the above problem