3. For a graph G with n vertices and 4n edges, Alice claims to have a proper colouring with 3 colours Bob decides to try to find one by testing each of the possible ways to colour G with 3 colours. Bob can verify if a colouring is proper as soon as he looks at it. (a) Approximately how many equality operations will it take you to verify or disprove Alice's claim? (b) Approximately how many verifications will it take Bob to find a proper colouring with 3 colours, or prove one does not exist? 5. Prove that all acyclic graphs are planar. 7. A degree sequence of a graph is a list of the degrees of each vertex. Consider the following lists of six positive integers: B [5,5,5, 4, 3,2 C [5, 5,4, 4, 3,3 D=4.4, 2.2.2.2] E 3.3.3.3.3.3] Determine which sequence(s): (a) cannot be the degree sequence of a graph; (b) must be the degree sequence of an eulerian graph; (c) must be the degree sequence of a hamiltonian graph; (d) could be the degree sequence of a tree; and (e) could be the degree sequence of a graph, but cannot be a planar graplh You do not need to justify your answers for Problem 7