NAME: 6 MT263 problems Hand in your solutions, with this coversheet, for feedback to the MT263 homework drop box in the foyer of the Bedford Building (beside the door of room 1-33) by 11am on Wednesday 6th November. I can provide you with additional feedback if you want it. Many students don't collect their feedback so if you don't want additional feedback, please tick the relevant box below. Otherwise, if you make a decent attempt of N questions, you can request additional feedback on [+] questions (note the use of the floor function here). Please indicate which questions you'd like feedback on in the relevant box below (if you don't indicate what feedback you want, I'll assume you don't want any). • Put a tick in this box if you don't want additional written feedback on your solutions. · I attempted questions. Please provide me with feedback on questions WHAT FEEDBACK WOULD YOU LIKE? 1. Is there anything in particular you'd like feedback on? (E.g., a proof that you're not sure about, somewhere you're not sure that you've expressed yourself clearly, etc .. ) 2. Were there any questions that you couldn't figure out? If so, what did you try? FEEDBACK ON YOUR SOLUTIONS FROM THE MARKER:
1. (a) Draw the graph G = (V, E), where V = {1, 2, 3, 4, 5, 6}, E={a, b, c, e, f, g, h, i} a >> (1, 2), b <> (1, 3), c <> (1, 4), d <> (2, 5), e <> (2, 6), f <> (3,5), g <> (3, 6), h +> (4, 5), i <> (4, 6). (b) Draw the graph G = (V, E), where V ={1,2, 3, 4, 5}, E={a, b, c, d, e, f, g} a >> (1, 2), b <> (1, 4), c <> (2, 3), d <> (2, 4), e <> (2, 5), f <> (3, 4), g <> (3, 5). (c) Consider the following graph. 6 2 1 g f 3 5 e 4 i. Which of the edges e, f, and g are adjacent? ii. Which vertices is edge f is incident to? iii. Which of the edges e, f, and g is vertex 4 incident to? iv. How many components does the graph have? v. What is the smallest number of edges that you need to delete from the graph to obtain a graph with exactly 3 components? 2. (a) List all of the non-isomorphic simple graphs with 4 vertices and 3 edges. (b) List all simple graphs with 3 vertices up to isomorphism. (c) List all simple graphs with 4 vertices and 2 or 4 edges up to isomorphism. 3. Find the number of edges and vertices in Pn, Cn, Wn, and Kn. 4. (a) Show that the number of vertices in a k-regular graph is even if k is odd. (b) Hence show that it is not possible to have a group of seven people such that each person in the group knows exactly three other people in the