Reply to comment

An alternate proof

Using the fact that every vertex 'u' is connected to d(u) (degree of 'u') faces of the polyhedron, we try and see what happens to |V|,|F| and |E| when a vertex is removed and a new polyhedron is formed with |V'| , |F'| and |E'| (# of vertices, faces and edges).

|V'|= |V|-1 (trivial)

|F'|=|F|-d(u)+1
all the d(u) faces vertex 'u' is connected to will merge into a single face

|E'|=|E|-d(u) (trivial)

observe that |V|+|F|-|E|=|V'|+|F'|-|E'|

repeat this process on the given polyhedron until only a cycle is left.
And we can see that |V|+|F|-|E|=2
(the face formed by merging the faces to be removed and the face already present is the same, i.e our cycle has two faces)

Reply

  • Web page addresses and e-mail addresses turn into links automatically.
  • Allowed HTML tags: <a> <em> <strong> <cite> <code> <ul> <ol> <li> <dl> <dt> <dd>
  • Lines and paragraphs break automatically.

More information about formatting options

To prevent automated spam submissions leave this field empty.
By submitting this form, you accept the Mollom privacy policy.