NAME: 6 MT263 problems Hand in your solutions, with this coversheet, for feedback in the Maths Office (McCrea 118) by 1pm on Monday 26th Feb I will keep a record of how many questions you made a decent attempt on. 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 YOUR FEEDBACK TO THE LECTURER (e.g., what was hard, interesting, fun, etc. this week): 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), ¿ < > (4, 6).
1 2 Solution: 6 5 3 4 (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). Solution: 1 2 5 3 4 (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? Solution: i. e and f are adjacent, f and g are adjacent, e and g are not. ii. 4 and 2 iii. e and f iv. 1 v. 3 (delete the edges incident to the vertex 5). 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. Solution: (a) (b) · (c) 3. Find the number of edges and vertices in Pn, Cn, Wn, and Kn. Solution: Pn has n + 1 vertices and n edges. Cn has n vertices and n edges. Wn has n vertices and 2n -