The genus distributions of 4-regular digraphs

An embedding of an Eulerian digraph in orientable surfaces was intro- duced 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 in- troduce 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.