CO 342 Harvard University Week 3 Basic Graph Theory Question

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Mathematics

CO 342

Harvard University

CO

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Proof problems about Menger's theorem and Hall's theorem. I have attached notes to help you out. I need 80%+ only bid if you can do this. Thank you.

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3. Recall that a matching is a set of edges such that no two edges share a vertex. A vertex that is used by an edge in the matching is saturated by the matching. A graph G is bipartite if the vertex set can be partitioned into two parts V (G) = (A, B), and all edges join a vertex in A and a vertex in B. The neighbour set of a subset of vertices S CV (G) is the set of all vertices that have at least one neighbour in S, i.e. NG(S) = {w € V(G) : ve S, vw E E(G)}. Hall's Theorem states the following: Let G be a bipartite graph with bipartition (A,B). Then there exists a matching that saturates all vertices in A if and only if |NG(S)| = |S| for all S S A. Use Menger's Theorem (any version) to prove Hall's Theorem. 9.1 Proof grading For proof problems, each problem is marked out of 10. Here is a rough guide to what the scores mean: 0: Nothing substantial turned in. 1-2: An attempt though wrong or misguided. 3-4: A partial solution or at least a justifiable idea at a solution, perhaps with a fatal flaw. 5-6: Generally the correct solution, though may be difficult to follow, partly incomplete, or have a num- ber of mistakes. 7-8: A correct solution, perhaps with a small number of missing and/or unnecessary details. 9: A correct solution with clean arguments that are well-presented. 10: An exceptional solution that has excellent presentation or has a very clever argument. Note that a good solution would receive 7 to 9 marks. A perfect mark is reserved for rare occasions when outstanding solutions are presented.
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