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hw4 - AMS/MBA 546(Fall 2010 Estie Arkin Network Flows...

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AMS/MBA 546 (Fall 2010) Estie Arkin Network Flows Homework Set # 4 Due Thursday, October 7, 2010. 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. 2). (a). Prove that if a directed graph has an Euler cycle then the directed graph is strongly connected. (b). Is the converse true? (Does every strongly connected directed graph have an Euler cycle?) Prove or give a counterexample. 3). The k th power G k of a graph G is a graph with the same set of vertices as G , two vertices have an edge between them in G k if and only if they are joined by a path of at most k edges in G . We claimed in class that if G is biconnected then G 2 has a Hamilton cycle. (This is non trivial to show!) (a). Give an example of a graph G that is connected, but G 2 does NOT have a Hamilton cycle. (b). Let P be a path on three or more nodes. Show that
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