I have this question on an exam. I just need explanation on what's meant by rotation system?
Below is the rotation system of a plane graph.
0: -> 6 -> 2
1: -> 4 -> 3
2: -> 0 -> 4 -> 5
3: -> 4 -> 1
4: -> 1 -> 2 -> 3
5: -> 2 -> 6
6: -> 0 -> 5
How many faces does the graph have? Draw the graph so that the largest (incident on the most edges) face is the unbounded outer face.
A rotation system is simply an ordering of neighbours around each vertex that defines a particular embedding of the graph in the plane. So for your rotation system you have (say) "2: -> 0 -> 4 -> 5" which specifies the three neighbours of vertex 2 are 0 then 4 then 5 in clockwise order.
I say 'clockwise' because your diagram is drawn that way, but you can get a mirror image drawing by following the ordering the other direction. It really doesn't matter so long as you stick to one way for the whole embedding.