# Applications of parity formula on connected planar graph

I've been given the following problem as homework:

A graph is drawn in the plane and has 78 faces, all of them triangles.
Prove the outer face is not a 19-gon.


We're also given a hint to use the formula: |V| - |E| + |F| = 2 (for planar, connected graph, where |F| = the number of faces).

What I've tried so far: I realize the number of faces is 78, so I can plug "78" in for |F|. I realize there's probably a way to calculate |E| given that the outer face is a 19-gon, but I can't figure this out. Further, I imagine if I calculate |E|, I'll find that it doesn't satisfy this equation which proves the outer face can't be a 19-gon.

This is homework, so I'm not looking for an answer or proof. Could you give some useful hints to point me in the right direction?

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Hmm ... The problem says "all of them [faces are] triangles". Presumably, "The outer face is not a 19-gon because you just said that it's a triangle!" isn't the answer they want, in which case the problem seems poorly-posed. At any rate, you need to consider whether the outer face is supposed to be counted in the "$78$ faces" before substituting-in for $|F|$. (Once you do that, note that each edge of a planar graph belongs to two faces, so it's like each of those two faces owns "half" of the edge. Thus, knowing the valence of each face allows you to compute the number of edges.) – Blue May 16 '12 at 18:18
"Proof: 3 ≠ 19." – JeffE May 17 '12 at 16:43

## 1 Answer

(I'll assume "all the faces" means "all the bounded faces".)

Suppose the outer face actually is a 19-gon. How many new edges would you need to add to the outer face to cut it up into triangles? How many faces would that planar graph have?

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