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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 Spring '09
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