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# 10 - Fall 2009 CS131 Combinatorial Structures Homework 10...

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Fall 2009 CS131 – Combinatorial Structures Homework 10 Homework 10, due Dec 8 You must prove your answer to every question. Do not rely only on the homework for exercise: there are several self-check ex- ercises of the easier kind in the book, try to solve them, too! Problem 1. (a) (5pts) Which expression grows faster: ( n 4 ) or ( n + 3 n ) ? Solution. We have n 4 = n ( n - 1)( n - 2)( n - 3) 4! , n + 3 n = n + 3 3 = ( n + 1)( n + 2)( n + 3) 3! . The first expression is a polynomial of degree 4, the second one a polynomial of degree 3, so the first expression grows faster. (b) (10pts) Show that the expression ( 3 n n ) grows exponentially. Solution. We have 3 n n = 3 n (3 n - 1) ··· (2 n + 1) n · ( n - 1) ··· 1 . Every factor of the top is at least twice larger than any factor at the bottom. So the whole expression is 2 n , since we can write it as 3 n n 3 n - 1 n - 1 ··· 2 n + 1 1 , each of the n factors of which is greater than 2. For an upper bound, we know ( 3 n n ) 2 3 n = 8 n . Problem 2. (15 pts) Decide which of the following graphs has a Hamilton cycle. The answer only counts if you prove your statements.

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Fall 2009 CS131 – Combinatorial Structures Homework 10 a 1 a 2 a 3 a 4 a 5 b 1 b 2 b 3 b 4 b 5 c 1 c 2 c 3 c 4 c 5 d 1 d 2 d 3 d 4 d 5 A 1 A 2 A 3 A 4 A 5 B 1 B 2 B 3 B 4 B 5 Solution. For the first graph, a Hamilton cycle is given by a 1 , b 1 , d 3 , c 3 , c 2 , c 1 , c 5 , c 4 , d 4 , b 2 , d 5 , b 3 , d 1 , b 4 , d 2 , b 5 , a 5 , a 4 , a 3 , a 2 .
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10 - Fall 2009 CS131 Combinatorial Structures Homework 10...

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