hw1 - CME 305: Discrete Mathematics and Algorithms...

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CME 305: Discrete Mathematics and Algorithms Instructor: Professor Amin Saberi ([email protected]) HW#1 – Due 01/26/10 1. Prove that at least one of G and G is connected. Here G is the complement of G i.e. a graph on the same vertex set such that two vertices in G are adjacent if and only if they are not adjacent in G . 2. Recall the deﬁnition of a bipartite graph. Let G ( V,E ) be a graph and ( A,B ) be a partition of V . We say that G is bipartite if all edges in E have one end-point in A and the other in B . More precisely, for all ( u,v ) E either u A,v B or u B,v A . (a) Prove that a graph is bipartite if and only if it doesn’t have an odd cycle. (b) A graph is called k -regular if all vertices have degree k . Prove that if a bipartite G is also k -regular with k 1 then | A | = | B | . 3. The stable roommate problem is similar to the stable marriage problem, except pairings are made within a single pool rather than between two genders. As in stable marriage, a rogue pair is a pair ( a,b ) such that a prefers b to his current partner and b prefers a to his current partner. A stable pairing is one that has no rogue pairs. Give an example of the stable roommate problem in which no stable pairing exists.

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hw1 - CME 305: Discrete Mathematics and Algorithms...

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