# If yes show the simplest example you can think of

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[If "yes", show the simplest example you can think of.]

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MAD3305/Final Exam Page 4 of 8 _________________________________________________________________ 7. (10 pts.) Apply Kruskal’s algorithm to find a minimum spanning tree in the weighted graph below. When you do this, list the edges in the order that you select them from left to right. What is the weight w(T) of your minimum spanning tree T? _________________________________________________________________ 8. (15 pts.) (a) If G is a nontrivial graph, how is κ (G), the vertex connectivity of G, defined? (b) If G is a nontrivial graph and v is a vertex of G, κ (G - v) κ (G) - 1. Provide the simple proof for this. (c) If G is a Hamiltonian graph, what can you say about κ (G)? Why??
MAD3305/Final Exam Page 5 of 8 _________________________________________________________________ 9. (15 pts.) Find a minimum spanning tree for the weighted graph below by using only Prim’s algorithm and starting with the vertex a. When you do this, list the edges in the order that you select them from left to right. What is the weight w(T) of your minimum spanning tree T? ________________________________________________________________ 10. (10 pts.) (a) What is the chromatic number of any nontrivial tree? Why? (b) Give an example of a nonplanar graph G with χ (G) = 4.

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MAD3305/Final Exam Page 6 of 8 _________________________________________________________________ 11. (10 pts.) (a) Suppose that G is a bipartite graph with partite sets U and W with U W . What does it mean to say that U is neighborly? (b) K 3,3 is given below with partite sets U = {a, b, c} and W = {A, B, C}. Give a connected bipartite subgraph of K 3,3 with the same partite sets but with U failing to be neighborly. Explain why your example satisfies the requirements. _________________________________________________________________ 12. (15 pts.) (a) State Kuratowski’s Theorem that characterizes planar graphs. (b) Without using Kuratowski’s Theorem, explain briefly how one can see that K 5 is not planar.
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