Unformatted text preview: k > 0, and G = ( X,Y,E ) be a bipartite k-regular 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
- Graph Theory, #, Estie Arkin, Orlin