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Unformatted text preview: AMS/MBA 546 (Fall 2010) Estie Arkin Network Flows: Solution sketch to homework set # 4 1). Let G = ( V,E ) be a connected multigraph in which every vertex has degree 4. Show that G can be decomposed into two graphs, each containing all nodes, such that the degree of every vertex is 2 in each graph. In other words, show that there exist E 1 and E 2 , where E i ⊂ E , E 1 ∪ E 2 = E , E 1 ∩ E 2 = ∅ and G 1 = ( V,E 1 ), G 2 = ( V,E 2 ) have all nodes of degree 2. Since G is connected and every node has an even degree (4) we know that G has an Euler tour. Further, since G has n nodes and m edges, we have that 2 m = ∑ vinV deg ( v ) = 4 n we have that m is even. Now imagine walking around the Euler tour colouring edges red and blue alternatingly. Define E 1 to be the red edges and E 2 to be the blue edges. Since the Euler tour enters each node twice and leaves it twice, each node is adjacent to 2 red edges and 2 blue edges, so we get the desired decomposition. 2). (a). Prove that if a directed graph has an Euler cycle then the directed graph is strongly connected. To show that there is a path from a node i to a node j, we first show that there is a walk from i to j, by simply following part of the Euler cycle from node i until it reaches j. Since all nodes are visited by the Euler cycle, such a walk exists. Since there is a walk from i to j, there is also a path from i to j, for every pair of nodes, and therefore the directed graph is strongly connected....
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This note was uploaded on 01/31/2011 for the course AMS 546 taught by Professor Arkin,e during the Fall '08 term at SUNY Stony Brook.
- Fall '08