(Question 2) Given two graphs G and H, the union G H is a graph with vertex set V (G) V (H) and edge set E(G) E(H).

(a) Suppose G and H are two bipartite graphs on the same set of vertices V . Describe how to construct a proper colouring f : V {1, 2} {1, 2} of G H.

(b) Let g, h be positive integers. Suppose G and H are two graphs on the same set of vertices V such that G is g-colourable and H is h-colourable. Prove that G H is gh-colourable. (Hint: Extend the proof in part (a) and construct a proper colouring f : V {1, 2, . . . , g} {1, 2, . . . , h} for G H.)