NAME: 8 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 27th 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 saturated hydrocarbon is a molecule Cm Hn in which every carbon atom C has four bonds, every hydrogen atom H has one bond, and no sequence of bonds forms a cycle. Show that for every m E N, the molecule Cm Hn can exist only if n = 2m + 2. 2. Find a minimum weight spanning tree for each of the following graphs. 2 x x a 4 a a 7 d 1 4 6 e 8 6 7 1 2 5 b f 2 5 1 6 4 1 6 b x 2 8 1 5 4 e 5 6 8 f c 2 b d 8 c c 3. (a) By adapting the definition of isomorphism for graphs, formulate a definition of what it means for digraphs D = (V, A) and Dº = (Vº, Aº) to be isomorphic. (b) Which pair of the following digraphs are isomorphic? (c) Explain why no two of the following digraphs are isomorphic. 4. Prove that a (weakly) connected digraph D is Eulerian if and only if d+(v) = d" (v) for every vertex v of D. (Hint: adapt one of the proofs you have seen for the similar result for graphs.) 5. A digraph D is balanced if |d+(v) - dª(v)| ? 1 for each vertex v of D. (a) Show that every tree has a balanced orientation. (Hint: use induction on the number of edges.) (b) Show that every graph has a balanced orientation. (Hint: use induction on the number of cycles.) (Note that we say that a digraph D is an orientation of a graph G if D can be obtained from G by directing each of its edges.