I have a rotation system and try to draw the planar graph. However, after I do a facial walk of the rotation system, and the number of vertices, number of edges, and number of faces don't meet the Euler's formula. Where are my mistakes?
Here are some slides talking about the facial walk on rotation system(page 10 to 11)
The following is the rotation system (clockwise)
$$ \begin{matrix} a: & b & d & g \\ b: & d & c & a \\ c: & b & e & d \\ d: & a & c & b \\ e: & h & c & f \\ f: & h & g & e \\ g: & a & h & f \\ h: & f & e & g \\ \end{matrix} $$
The following are the facial walk (4 faces)
$F_0 = (a, b)(b, d)(d, a)(a, g)(g, h)(h, f)(f, g)(g, a)(a, b)$
$F_1 = (a, d)(d, c)(c, d)(b, a)(a, d)$
$F_2 = (b, c)(c, e)(e, f)(f, h)(h, e)(e, c)(c, d)(d, b)(b, c)$
$F_3 = (e, h)(h, g)(g, f)(f, e)(e, h)$
The following the planar graph I produced based on the rotation system
However, there are 8 vertices, 12 edges, 4 faces, which don't meet the Euler's formula ($n - m + f = 2$).
Anything suggestions would be appreciated.
For the graph that you have drawn, I count $\color{blue}{ 8 \text{ vertices}}$ (numbered in blue), $\color{green}{ 12 \text{ edges}}$ (numbered in green), and $\color{red}{ 6 \text{ faces}}$ (numbered in red; note that one of the faces is the exterior face, i.e. the region outside of the graph). Euler's formula then gives us $$ V - E + F = 8 - 12 + 6 = 2, $$ which is the expected result.
Note that this graph does not correspond to the rotation system that you have written down. This graph corresponds to the rotation system $$\begin{matrix} a: & b & d & g \\ b: & c & d & a & * \\ c: & e & d & b \\ d: & c & a & b & * \\ e: & h & f & c & * \\ f: & e & h & g \\ g: & f & h & a & *\\ h: & g & f & e \end{matrix}$$ The marked entries are those that differ from what you have written. I believe that the rotation system you have written down does not correspond to a planar graph. Indeed, the fact that it has Euler characteristic not equal to 2 indicates this fact. I would be willing to bet that the rotation system you have written down could be embedded in a surface of genus 1, i.e. a torus. However, I do not have a donut or bagel handy, nor would I want to mark up such a thing with a marker, so I cannot confirm that this would work.