3. (Graphs, planarity, and the Euler characteristic. (a) Suppose that G is a connected simple planar graph on at least three vertices. Let F denote the number of faces of G, and E the number of edges. Prove that 3F 2E. (b) Using (a), prove that if G is a connected simple planar graph on n 2 3 vertices, then G contains at most 3n -6 edges (c) Show that the complete graph on five vertices, Ks, is not a planar graph (d) Given a graph G, its complement is the graph G formed as follows: . The vertices of G are the same as the vertices of G. We connect two vertices in G with an edge if and only if they are not connected by an edge in G Prove that if G is a simple planar graph on at least eleven vertices, then C is nonplanar 16 marks