Let (G, φ) be a connected 4-regular plane simple graph in which every vertex lies on
two (opposite) faces of length 5 and on two (opposite) faces of length 3. Use Euler’s formula
to find the number of edges and the number of faces of (G, φ)
So euler's formula says that e – v + f = 2. And with the question it seems to give 4 faces (2 opposing pairs). How do I figure out the number of vertices to be able to find the number of edges?
Best Answer
Denote the number of faces of size $k$ by $f_k$. Then we find the following equations to hold $$\begin{align*} v-e+f &=2&(\text{Euler})\\ f_3+f_5&=f&(\text{Def})\\ 4v&=2e\\ 3f_3+5f_5&=2e\\ 3f_3&=2v\\ 5f_5&=2v \end{align*}$$ Solving the corresponding matrix equation we obtain $v=30, e=60, f=32$.