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# hw1 - k> 0 and G = X,Y,E be a bipartite k-regular...

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AMS/MBA 546 (Fall, 2010) Estie Arkin Network Flows Homework Set # 1 Due Tuesday, September 14, 2010. Suggested reading Chapters 1 and 2 of Ahuja, Magnanti and Orlin. 1). Recall the definiton: A sequence S = d 1 ,d 2 ,...,d n is graphical if there exists an undirected (simple) graph with n nodes such that the degree of node i is equal to d i . A simple graph is a graph with no loops and no multiple edges. (a). Show that if a sequence is graphical, and n 2, then there must be two numbers d i , d j such that d i = d j . (In other words, graphical sequences cannot have all their elements be unique.) (b). Let S 1 = d 1 ,d 2 ,...,d n and S 2 = w 1 ,w 2 ,...,w n where w i = n 1 d i for i = 1 ,...,n . Show that S 1 is graphical if and only if S 2 is graphical. 2). Let G = ( V,E ) be a graph with n nodes and e edges, such that e > n 2 / 4. Prove that G is not bipartite. 3). Let k > 0, and
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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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