Let x = {1, 2, 3, ... , n}, where n ≥ 2. Construct the loop- free undirected graph G = (V, E) as follows:
• (V): Each two-element subset of X determines a vertex of G.
• (E): If v1, v2 ∈ V correspond to subsets {a, b} and {c, d}, respectively, of X, draw the edge {v1, v2] in G when {a, b} ∩ {c, d} = 0.
(a) Show that G is an isolated vertex when n = 2 and that G is disconnected for ft = 3,4.
(b) Show that for ft > 5, G is connected. (In fact, for all v1, v2 ∈ V, either {v1, v2] ∈ E or there is a path of length 2 connecting v1 and v2.)
(c) Prove that G is nonplanar for n ≥5.