NAME: 7 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 20th 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. Let G be a simple graph with §(G) ? 3. Prove that G contains a cycle of even length. (Hint: consider a longest path.) 2. (a) Which of the following graphs are bipartite G H K P d(v) P d(v) 3. Let G[X, Y] be bipartite. Show that Hence deduce that if G [X, Y] is bipartite and'k-regular, with k > 1, then |X| = |Y |. (a) For which values of n is Kn Hamiltonian, and for which values is it Eulerian? (b) For which values of m and n is the complete bipartite graph Km,n Hamiltonian, and for which values is it Eulerian? (c) For which values of n is the wheel Wn Hamiltonian, and for which values is it Eulerian? 4. Prove that a connected graph is semi-Eulerian if and only if it has exactly two vertices of odd degree. 5. An edge e of a graph G is said to be a bridge if G - e has more components than G. Prove that if a graph G has a bridge, then it has a vertex of odd degree. 6. Let G = (V, E) be a Hamiltonian graph with |V| > 2, and let v E V. Show that the graph G - v is connected. 7. Let G be a bipartite graph with an odd number of vertices. Prove that G is non-Hamiltonian. 8. (a) Prove that in any tree with at least one edge there are at least two vertices of degree one. (Hint: considera longest path.) (b) Let T be a tree and v be a leaf of T. Show that the graph obtained from T by deleting v is a tree. 9. Please fill in in the coversheet to let me know what