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Unformatted text preview: AMS 301.1 Fi rst Test, Fall 2009, SOLUT IONS 1. Graphs are isomorphic.See Figure 1.19 in text. 2. Test A: LB = 18 (subtracting first from rows). Don't use (1,3), LB=22; use (1.3) (deleting row 1 & col 3 and setting (entry (3,1) = infinity), LB = 19. Test B: LB = 14 (subtracting first from rows). Don't use (1,3), LB = 19; use (1,3) (deleting row 1 & col 3 and setting entru (3,1) = infinity), LB= 16 3. Test A: connected; test B: not connected 4. Test A and 5. Test B: vertices equal actuaries/statisticians, edges = personaly conflicts, colors = hotels; does 6 colors suffice? 5. Test A and 4. Test B: a) not possible, violates e <=3v6; b) possible, v = (Test A, 12), (Test B, 10) remember that all vertices musst have deg 3; c) many possibilities, e.g., t riangle with trailing path (like a kite), 6. As explained in class, by symmetry and need to use at least one vertical edge, you can assume you start with edge (c,h), at c by symmetry choose bc & delete cd, forcing at d, ed i. Two cases at b or h or e or i. We consider cases at h Case 1. Use hf, delete hj => at j use ejg, at e delete ea, => at a use baf, subcricuit abc hfa. Case 2. Use hj, delete hf => at f, use afi, at i delete ig => at g use bgj, subcircuit bchj gb. 7. a) Many possibilities. b) If a cutset K did not intersect a tree T, then GK would still be connected since GK would contain a spanning tree. c) If G1 and G2 are the two components of GK, and if circuit C starts in G1, then any time C uses an edge of K, it moves over to G2. I t must eventually return to G1, using a second edge of K. Thus crossing to G2 and returning to G1 cna occur many times, but each time the circuit uses 2 edges of K....
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 Spring '08
 ARKIN

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