Straight-ahead walks in Eulerian graphs

AbstractA 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

Links and Graphs

In this thesis we derive some basic properties of graphs G embedded in a surface determining a link diagram D(G), having a specified number μ(D(G)) of components. ( The relationship between the graph and the link diagram comes from the tangle which replaces each edge of the graph). Firstly, we prove that μ (D(G)) ≤ f (G) + 2g, where f (G) is the number of faces in the embedding of G and g is the genus of the surface. Then we focus on the extremal case, where μ (D(G)) = f (G) + 2g. We note that μ (D(G)) does not change when undergoing graph Reidemeister moves or embedded ∆ ↔ Y exchanges. It is also useful that μ(D(G)) changes only very slightly when an edge is added to the graph. We finish with some observations on other possible values of μ(D(G)). We comment on two cases: when μ = 1, and the Petersen and Heawood families of graphs. These two families are obtained from K6 and K7 respectively by using ∆ ↔ Y exchanges.The Iraqi Ministry of Higher Educations