I am studying graphs on surfaces. But I am confused about the definition of "Ribbon Graph".
Let $(\Gamma, I)$ be a graph. For $v\in V$ the star of $v$ $$E_v=\{e\in E \mid v= e_-\}$$ is the set of edges starting from $v$. A ribbon graph is the graph $(\Gamma, I)$, together a cyclic ordering on the star of every vertex.
The definitions are given here.
I am confused how to define cyclic ordering on the star of every vertex. I have tried the following diagram. Here, $v$ be a vertex, and we have $8$ vertex starting from $v$. Now I am confused how to define cyclic ordering on $v$?
Please help.
The cyclic ordering in a ribbon graph is meant to be a discrete representation of a geometric difference between two embeddings of a graph in the same surface.
For example, consider the two planar embeddings below:
The vertices and the adjacencies between them are the same in both pictures, so as abstract graphs they are the same. However, as embeddings they are different: in the first picture, vertices $3$ and $4$ are on the same side of the cycle $(0,1,2,0)$, and in the second picture, they are on different sides.
A ribbon graph also captures that difference, but in a different way. In this graph, the star of $0$ is the set of four edges $\{01, 02, 03, 04\}$. The cyclic ordering around vertex $0$ is $(01,02,03,04)$ in the first diagram and $(01,03,02,04)$ in the second. This is a cyclic ordering because the starting point of $1$ was arbitrary: $(01,02,03,04)$ and $(02,03,04,01)$ are the same cyclic ordering, both of them describing what happens in the left diagram. However, no cyclic shift will turn this into $(01,03,02,04)$, so the two cyclic orderings we get from different diagrams are different.