Please help me out with this, work shown in great detail would be much appreciated

4. 20 Points] Planar Graphs (a) Prove that the complete graph on 5 vertices (notation Ks) is not planar. Hint: The proof is short and simple, all its ingredients are in lessons 7 and 9. Don't just state that you have not found a way to draw K5 (b) Show that every planar graph has a vertex of degree at most 5 (c) Use parts (a) and (b) to prove that each planar graph is 5-colorable. That is, it is possible to color each vertex with one of the colors 1,2,3, 4,5 so that no two adjacent vertices have the same color. Hint: use induction plus some ingenuity