assgt5 - B n , n 1. (b) Determine the number of edges in B...

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MATH 239 ASSIGNMENT 5 Due Friday, October 31, at NOON 1. Let A n , n 1, be the graph whose vertices are given by the n -element subsets of { 1 ,..., 2 n + 1 } , where two vertices are adjacent if and only if they are disjoint subsets. (For example, when n = 3, vertex { 2 , 3 , 6 } is adjacent to { 1 , 5 , 7 } and { 4 , 5 , 7 } , but not to { 1 , 3 , 5 } .) (a) Determine the number of vertices in A n , n 1. (b) Determine the number of edges in A n , n 1. (c) Determine the values of n 1 for which A n contains a triangle. 2. Let B n , n 1, be the graph whose vertices are given by the { 0 , 1 } -strings of length n , and two vertices are adjacent if and only if they di±er in two consecutive positions. (For example, when n = 5, vertex 10110 is adjacent to 11010 and 10000, but not to 00110.) (a) Determine the number of vertices in
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Unformatted text preview: B n , n 1. (b) Determine the number of edges in B n , n 1. (c) Determine the values of n 1 for which B n is connected. 3. Suppose that G is a graph in which the minimum vertex degree is equal to d , for some d 2. (a) Prove that G has a path of length at least d . (b) Prove that G has a cycle of length at least d + 1. 4. If H has p vertices, p 1, and every vertex in H has degree greater than or equal to 1 2 p , prove that H is connected. 5. A quartic tree is a tree whose vertices have degrees 1 or 4 only. Prove that every quartic tree with m vertices of degree 4 has a total of 3 m + 2 vertices, for m 0....
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