PS05-100406 - MATH 682 Problem Set #5 This problem set is...

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Unformatted text preview: MATH 682 Problem Set #5 This problem set is due at the beginning of class on April 6 . Below, graph means simple finite graph except where otherwise noted. 1. (10 points) Complete the proof that the Harary graphs are k-connected. You may use the case presented in class of even values of k , either by citation or imitation. (a) (5 points) Show that H n,k is k-connected for even n and odd k . This graph is symmetric, so you may specifically demonstrate connectedness between v 1 and an arbitrary v i in H n,k- S for v 1 ,v i 6 S and | S | < k . (b) (5 points) Show that H n,k is k-connected for odd n and odd k . This graph is not symmetric, so you must distinguish between the cases where v 1 S and v 1 / S . 2. (15 points) Answer the following questions about Tur an numbers and extremal graphs. (a) (5 points) Show that the extremal number ex( n,K 1 ,r ) is ( r- 1) n 2 if r is odd or n is even, and ( r- 1) n- 1 2 if r is even and n is even....
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This note was uploaded on 01/12/2012 for the course MATH 682 taught by Professor Wildstrom during the Spring '09 term at University of Louisville.

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