Exercise 3 Let G = (V,E) be a graph. We define its total graph T (G) by taking as vertices V ∪ E, and defining the adjacency relation (i.e., the edge set) via (i) v ∈ V and u ∈ V are adjacent in T (G) if they are neighbors in G. (ii) v ∈ V and e ∈ E are adjacent in T (G) if v ∈ e (that is, v is an endpoint of e in G). (iii) e ∈ E and f ∈ E are adjacent in T (G) if they are coincident in G (that is, they share an endpoint). Prove that if G is connected and has at least two vertices, then T (T (G)) is Hamiltonian. Hint: First prove that T (G) contains a spanning subgraph which is Eulerian.