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 embeddings of graphs and of embeddings of graphs such that the corresponding medial graph is Eulerian embedded. We prove that in the case of 4-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. © 2003 Published by Elsevier B.V. All rights reserved.
Pisanski, T., Tucker, T. W., & Žitnik, A. (2004). Straight-ahead walks in Eulerian graphs. Discrete Mathematics, 281(1–3), 237–246. https://doi.org/10.1016/j.disc.2003.09.011