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Unformatted text preview: n , let G n be the graph whose vertices are all binary strings of length n that have at most one block of 1s, where two vertices are adjacent if and only if the corresponding strings dier in exactly one position. (a) Determine the number of vertices p in G n . (b) Draw graphs G 3 and G 4 . (c) Determine the number of edges q in G n . 4. or n a positive integer, dene the prime graph P n as follows: V ( P n ) = { 1 , 2 ,... ,n } and E ( P n ) = {{ u,v } { 1 , 2 ,... ,n }  u + v is a prime } . (a) Prove that P n is bipartite for all n . (b) Prove that P n is connected for all n . (You may assume, without proof, that for every integer k > 1, there is a prime number r such that k < r < 2 k .) 5. Prove that every graph with at least 2 vertices has two vertices of the same degree. 2...
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This note was uploaded on 10/23/2009 for the course MATH 239 taught by Professor M.pei during the Spring '09 term at Waterloo.
 Spring '09
 M.PEI
 Math, Combinatorics

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