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hw5fa02 - S 4(10 pts A graph G = V E is called a bipartite...

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COT3100C-01, Fall 2002 Assigned: 11/05/2002 S. Lang Assignment #5 (40 pts.) Due: 11/14 in class by 8:45 am (Note changes to Question #4 added 11/05 after class) Instructions: Write your answer neatly and concisely. Show all your work and steps involved in manipulation of algebra expressions and in proofs. Illegible scribbles or lacking explanation will result in minimum credit . 1. (10 pts.) Compute GCD(1027, 373) using Euclid’s algorithm, and find two integers t and u such that 1027 t + 373 u = GCD(1027, 373). 2. (10 pts.) Suppose a , b , and c are positive integers. Prove that if GCD( a , b ) = GCD( a , c ) = 1, then GCD( a , bc ) = 1. ( Hint: Use proof by contradiction. Suppose GCD( a , bc ) = n > 1. Then n must have a prime factor p according to …) 3. (10 pts.) Suppose a and b are positive integers. Consider the set S = { am + bn | m and n are integers, and am + bn > 0}. (a) Prove a S and b S . (b) Prove that GCD( a , b ) S . (c) Prove if t S , then GCD( a , b ) | t . (Note: As a result, GCD( a , b ) is the smallest element of the set
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Unformatted text preview: S .) 4. (10 pts.) A graph G = ( V , E ) is called a bipartite graph if there exist subsets A and B of V such that V = A ∪ B , A ∩ B = ∅ , and every edge e connects a vertex of A to a vertex of B , that is, no edges connect two vertices belong to A or two vertices belong to B . An example of a bipartite graph is given below. b c d e V = A ∪ B , A = { a , b , c } and B = { d , e } (a) Prove that if a graph G is bipartite then every circuit of G (if exists) must have an even length (i.e., contains an even number of edges). (b) Prove if every circuit of a connected graph G = ( V , E ) contains an even length, where | V | > 1, then G is a bipartite graph. ( Hint: Pick any vertex, call it a . Consider the two subsets A = { t ∈ G | t is connected to vertex a by a simple path of an even length, and B = V – A . Note that a graph is connected if for every pair of vertices there exists a path connecting them. ) a...
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