3. For an integer k0 and graph G, let f(G, k) be the number of k-colourings of G, where the set of colours is {1, 2, , (a) What is f(Kn, k)? (b) Prove that f(G, k)-f(G - e,k) f(G/e, k) for every graph G and edge e of G. (c) Prove that f(T, k) k(k - 1)V(nl-1 for every tree T and inter0. 5 marks] 5 marks (5 marks (d) Prove that f(G, k) k(k-1)IV(G)-1 for every connected graph G and integer k > 0. 5 marks] (e) Let G be a connected graph. Prove that f(G, k) = k(k-IMG-1 for every integer k0 if and only if G is a tree. [5 marks] 3. For an integer k0 and graph G, let f(G, k) be the number of k-colourings of G, where the set of colours is {1, 2, , (a) What is f(Kn, k)? (b) Prove that f(G, k)-f(G - e,k) f(G/e, k) for every graph G and edge e of G. (c) Prove that f(T, k) k(k - 1)V(nl-1 for every tree T and inter0. 5 marks] 5 marks (5 marks (d) Prove that f(G, k) k(k-1)IV(G)-1 for every connected graph G and integer k > 0. 5 marks] (e) Let G be a connected graph. Prove that f(G, k) = k(k-IMG-1 for every integer k0 if and only if G is a tree. [5 marks]