I am new to graph theory and I would like to have some insight into a method on how to solve questions like this.
Given a simple planar graph in a sphere satisfying the conditions that:
- Each vertex has degree five.
- Each face is either a triangle or a pentagon.
- Each pentagon shares edges with five triangles.
- Triangles are divided into Type I: sharing edges with one pentagon and two triangles, and Type II: sharing edges with three triangles.
- Each Type I triangle shares edges with one pentagon and two Type II triangles, and each Type II triangle shares edges with one Type I triangle and two Type II triangles.
Find the number of vertices, edges, and faces respectively for the graph.