Orientation by assigning a direction to each edge so

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orientation by assigning a direction to each edge so that the resulting digraph is strong. (b) The graph to the left has many strong orientations. Does the graph have an Eulerian orientation? Explain briefly. (c) If you were asked to give me an example of a connected graph which has no orientations that are strong, what feature(s) would you include in your example to ensure that the example satisfies the requirement? Why?? _________________________________________________________________ 4. (10 pts.) Theorem 5.17, a corollary of sorts to Menger’s Theorem allows you to deal with the vertex connectivity of the graph below easily. Explain briefly. [Hint: Look north-south as well as east-west after considering δ (G).]
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TEST3/MAD3305 Page 3 of 4 _________________________________________________________________ 5. (15 pts.) (a) What is a clique? (b) Sketch the shadow graph S(C 5 ) of a generic 5-cycle below. What is χ (S(C 5 ))?? (c) How is the Grötzsch graph, which we will denote by G here, obtained from the shadow graph of Part (b) above?? It turns out that ω (G) = 2 and χ (G) = 4. What is the significance of this?? _________________________________________________________________ 6. (10 pts.) Prove, by induction on the size of the graph, that if G is a connected plane graph of order n, size m, and having r regions, then n - m + r = 2.
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TEST3/MAD3305 Page 4 of 4 _________________________________________________________________ 7. (15 pts.) (a) What is a legal (or feasible) flow in a network N ? [ Hint: Definition. ] (b) Obtain a maximum flow f in the network below, and verify the flow is a maximum by producing a set of vertices S that produces a minimum cut. Check that the total capacity of that cut is the same as the value of your max flow. _________________________________________________________________ 8. (10 pts.) Which complete bipartite graphs K r,s are Hamiltonian and which are not? Explain briefly. [Hint: When can you use Dirac? What is the well-known necessary condition?]
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