{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

This preview shows page 1. Sign up to view the full content.

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
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern