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For disk topology the euler formula is V - E + F = 1, for sphere it is V - E + F = 2. Is there a simple and elegant way to prove these?

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There are a lot of ways to prove it. I bet if you type $$\rm Euler\ formula\ for\ maps$$ into your favorite search engine, many proofs will come up. –  Gerry Myerson Oct 4 '12 at 13:28
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By induction:

  • Start with a triangle (or square or however your mesh is build up) and verify that $V+F-E=1$ (e.g. triangle: $3-1-3=1$)
  • Add another point $V\to V+1$ to get another triangle, means 2 more edges $E\to E+2$ and one more face $F\to F+1$. In total $V+F-E\to (V+1)+ (F+1) - (E-2)=V+F-E $
  • Or add an edge connecting to vertices. Then $E\to E+1$ and $F\to F+1$ and in total $V+F-E \to V+(F+1)-(E-1)=V+F-E$
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