sol1 - AMS 301 Spring, 2011 Homework Set # 1 Solution Notes...

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Unformatted text preview: AMS 301 Spring, 2011 Homework Set # 1 Solution Notes # 3, 1.1: (a) Nodes represent teams, and edges represent games between teams. The resulting graph is a circuit on 5 nodes. (b). Consider a graph as in part (a). It must have 5 nodes, each of degree 2. Start drawing such a graph, say with a node called A and 2 edges say to B,C (cant have 2 edges to the same node). Now node B must have another edge touching it, since its degree is 2. If this edge is (B,C) then the two other nodes D,E must each have degree 2, but can only have an edge to eachother, so this cannot happen. Thus Bs second edge is not to A nor C, say it is to D. Now look at node C, where does the second edge touching it go to? Cant be A nor B (who already have degree 2) nor D because then E has no edges touching it. Thus the edge must be (C,E). Now nodes E and D have only one edge touching each of them and A,B,C have 2 edges touching them, so the only way to have the degrees of D and E be 2 we must have an edge (D,E). We got a circuit onthem, so the only way to have the degrees of D and E be 2 we must have an edge (D,E)....
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