Question: 1. Might the graph in Figure be planar?
2. The goal of this problem is to use Euler's formula to list all possible regular polyhedra. (In this case, regular means that every face has the same number of edges and every vertex has the same degree.)
(a) Convert the idea of a regular polyhedron into a graph in some way.
(b) Find an upper bound on the possible degrees of the vertices.
(c) Find a lower bound on the possible degrees of the vertices.
(d) Find a relationship between the number of vertices and the number of edges.
(e) Find a relationship between the number of faces and the number of edges.
(f) Now use Euler's formula and solve some equations.
(g) Find graphs, and polyhedra, that correspond to your solutions.