Let k ≥ 1, and let G be a k-regular bipartite graph with bipartition X, Y . Prove that |X| is the minimum size of a vertex cover in G.

3. [10 marks] Let Go = (V,E) and G₁ = (V,E₁) be two graphs on the same set of vertices. Let (V, EU E1), so that (u, v) is an edge of H if and only if (u, v) is an edge of Go or of G1 (or of both). H = (a) Show that if Go and G₁ are both Eulerian and En E₁ = Ø (i.e., Go and G₁ have no edges in common), then H is also Eulerian. (b) Give an example where Go and G₁ are both Eulerian, but H is not Eulerian.

26. (a) Provide an example where X, X but E(X,) does not converge to E(X).