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Question

Question: 1. Prove that for a planar graph with k components, |V(G)|-|E(G)|+|F(G)| = 1+k.

2. Prove that the Petersen graph (shown in Figure) is not planar.

3. Show that if every face of a planar graph has four edges, then |E(G)| = 2|V(G)| -4.

4. For which n is Kn planar and for which n is Kn non planar? Make and prove a conjecture.