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Use the havel hakimi theorem to show that the

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Use the Havel-Hakimi Theorem to show that the sequence s: 5,5,4,4,3,3 is the degree sequence of some graph. Then show there is a plane graph with this sequence as its degree sequence. [You may need to do a second drawing.] _________________________________________________________________ 4. (10 pts.) (a) Explain why is Hamiltonian for n 5. C n (b) Give an example of a simple connected graph G of order n with (G) < n/2 that is Hamiltonian. What does this tell us about Ore’s Theorem?
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MAD3305/Final Exam Page 3 of 8 _________________________________________________________________ 5. (15 pts.) Each of the following propositions may be proved by induction. Prove exactly one of them after clearly indicating which you are proving. (a) Every nontrivial connected graph G has a spanning tree. [Hint: If the order of the graph G is at least 3, Theorem 1.10 implies that G has a vertex v with G - v connected.] (b) If G is a connected plane graph of order n, size m, and having r regions, then n - m + r = 2. [Hint: Do induction on the size of G.] _________________________________________________________________ 6. (10 pts.) (a) What can you say about the order of any nontrivial digraph in which no two vertices have the same outdegree, but every two vertices have the same indegree? Why?? (b) What degree condition characterizes Eulerian digraphs? Is there an Eulerian digraph whose underlying graph is not Eulerian?
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