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hw2 - CME 305 Discrete Mathematics and Algorithms...

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CME 305: Discrete Mathematics and Algorithms Instructor: Professor Amin Saberi ([email protected]) HW#2 – Due 02/09/10 1. Let T be a spanning tree of a graph G with an edge cost function c . We say that T has the cycle property if for any edge e 0 / T , c ( e 0 ) c ( e ) for all e in the cycle generated by adding e 0 to T . Also, T has the cut property if for any edge e T , c ( e ) c ( e 0 ) for all e 0 in the cut defined by e . Show that the following three statements are equivalent: (a) T has the cycle property. (b) T has the cut property. (c) T is a minimum cost spanning tree. Remark 1 : Note that removing e T creates two trees with vertex sets V 1 and V 2 . A cut defined by e T is the set of edges of G with one endpoint in V 1 and the other in V 2 (with the exception of e itself). Remark 2 : The cycle and cut properties we have defined in this problem are slightly different from what they usually mean. 2. Graph coloring and a magic trick: An edge coloring of a graph is an assignment of colors to each edge such that no two edges sharing a vertex have the same color. An edge coloring is optimal if a minimum number of colors is used, and that number of colors is called the chromatic index of G . (a) Recall, a graph is k -regular if all of its vertices have degree k . Show that the chromatic index of a k -regular bipartite graph is k . Hint: Use Hall’s theorem to show that every regular bipartite graph has a perfect matching. (b) Let G be a bipartite graph, and let k be the maximum degree among all vertices. Show that the chromatic index of G is k . Hint: Can you reduce the problem to the k -regular case? (c) Alice loves magic, and one day she invented a new magic trick: first, her assistant takes out the 13 cards of the same suit, and lets an audience pick out 6 cards.
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