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

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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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