Generating Functions of Eulerian and Separtating Eulerian Subgraphs

Throughout this paper, all graphs are assumed to be embedded into an orientable surface. A graph is Eulerian if the degree of every vertex is even. An Eulerian graph is separating if the regions into which the surface is divided by the graph are 2-colorable. Let G be a graph and G^[*] its dual. We show an identity which relates the generating function of Eulerian subgraphs of G and the generating function of separating Eulerian subgraphs of G^[*].