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Combinatorics2007jan - with at most 4 objects of any one...

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Qualifying Exam in Combinatorics 12 January, 2007 1. Suppose that G is a graph with no vertex of valence < 5, exactly 13 vertices of valence 5, no vertex of valence 7, and possibly an assortment of vertices of other valences. (a) What is the least number e 0 of edges that G may have? (b) If G has exactly e 0 edges, determine whether it is possible for G to be planar. 2. Suppose that ( P, P ) is a cut in a network N with capacity function w . where P contains the source and P contains the sink. Suppose that a flow ϕ in N has the property that whenever x P and y P , then ϕ ( x, y ) = w ( x, y ). Can you conclude that ϕ must be a maximum flow, or that ( P, P ) is a minimum cut? If your answer is yes, prove it; if your answer is no, verify it by constructing a counterexample. 3. Count the number of selections of 15 objects of five different kinds
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Unformatted text preview: with at most 4 objects of any one kind in each of the following two ways: [Obtain a numerical answer each time.] (a) by the Principle of Inclusion-Exclusion; (b) by constructing an appropriate generating function and evalu-ating an appropriate coefficient. 4. Construct two non-isomorphic, self-dual, rank 3 matroids on 6 ele-ments; also construct a rank three matroid on 6 elements that is iso-morphic to its dual but not self- dual. 5. (a) Define a projective plane, define Latin square and define or-thogonal Latin squares. (b) State a theorem that relates projective planes and orthogonal Latin squares and sketch its proof. 1...
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