Combinatorics2008aug - Qualifying Exam in Combinatorics 19...

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Qualifying Exam in Combinatorics 19 August, 2008 1. Let Γ be a 3-connected planar graph with planar dual Γ * . Prove or disprove: If Γ is a Cayley graph and Γ * is is vertex-transitive, then Γ * is a Cayley graph. If you believe the statement to be true, give a proof, or at least discuss what must be done to give a correct proof. If you believe the statement to be false, then give a counter-example, proving that exactly one of Γ and Γ * is a Cayley graph. 2. (a) Starting with the definition of derangement , derive an explicit formula for the n th derangement number D n . (Assume D 0 = 1 and D 1 = 0.) (b) Obtain an exponential generating function for the sequence { D n : n 0 } . (c) Give a combinatorial proof (not an algebaic proof) of the recurrence D n = ( n - 1)( D n - 1 + D n - 2 ) ( n 2) . 3. Describe how to “complete” an affine plane of order n to obtain a projective plane of order n . Then explain what this has to do with
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This note was uploaded on 06/19/2011 for the course MATH 680 taught by Professor Na during the Spring '11 term at University of North Carolina School of the Arts.

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