Straight-ahead walks in Eulerian graphs

A straight ahead walk in an embedded Eulerian graph G always passes from an edge to the opposite edge in the rotation at the same vertex A straight ahead walk is called Eulerian if all the edges of the embedded graph G are traversed in this way starting from an arbitrary edge An embedding that contains an Eulerian straight ahead walk is called an Eulerian embedding In this article we characterize some properties of Eulerian embed dings of graphs and of embeddings of graphs such that the correspond ing medial graph is Eulerian embedded We prove that in the case of valent planar graphs the number of straight ahead walks does not depend on the actual embedding in the plane Finally we show that the minimal genus over Eulerian embeddings of a graph can be quite close to the minimal genus over all embeddings