Unformatted text preview: k > 0, and G = ( X,Y,E ) be a bipartite kregular graph (i.e., each node of G has degree exactly k ). Prove that  X  =  Y  . Give an example for k = 0 for which  X  n =  Y  . 4). For a graph G = ( V,E ), de±ne c ( G ) = the number of connected components of G . Prove that c ( G )+  E  ≥  V  for every graph G . (Hint: Use induction  E  ) 5). A directed graph is strongly connected if there is a directed path from each vertex to every other vertex. Prove that directed graph G = ( V,A ) is strongly connected if and only if the following is true: For eve ry proper subset W of vertices, there exists a directed edge from a vertex w in W to a vertex y in V − W ....
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 Fall '08
 Arkin,E
 Graph Theory, #, Estie Arkin, Orlin

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