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Unformatted text preview: Examples of mathematical induction, PHP, invariance, extremal arguments ... ... and Thinking J.L.F. King Exercises Using Extema Here is an example argument. 1 : Obs. Consider two vertices, v,w , in a connected ( possibly infinite ) graph. Prove there exists a path be- tween them, with no repeated-vertex. Proof. Take a minimum-length path. Etc. 2 : Desegregation problem. A coloring of a graph as- signs to each vertex either black or white. It is desegregated , if each vertex has at least one neighbor of the opposite color from his. [ Two vertices are neigh- bors IFF they are connected by an edge. ] Prove that each finite connected graph G with N > 2 vertices, admits a desegregated coloring. Remark. Find two proofs of this. Can you generalize this problem? 3 : Bashful Boyfriends problem. [ In the future. ] Invariance The key underlying certain proofs, is that some quan- tity or some relation is preserved under the relevant operations. Eg: Invariant quantity. Have B be the 8 8 chess- board, but with the lower-right and upper-left cells removed; so | B | = 62. We start laying down domi- nos. Can we cover the board with 31 dominos? No! Why? Initially, the uncovered part of the board ( i.e, all of B ) has 32 black cells and 30 white cells. These numbers are not invariant under placing a domino. But the difference , # n Uncovered black cells o- # n Uncovered white cells o , : is unaltered by placing a domino it is invariant. Since the discrepancy is 2 initially, it will always be 2, no matter how many dominos we place. But a covered board would have a discrepancy of 0, not 2. Eg: Invariant relation. Our Lightning bolt alg. chose seeds for the s- and t- columns, so that r n = s n r + t n r 1 , : for n = 0 , 1. [ The n th : remainder , quotient , and B ezout multipliers are called r n ,q n ,s n ,t n . ] The LBolt update rule preserved relation ( ), in building row n from rows n- 2 and n- 1. When we found the index K where r K = Gcd ( r ,r 1 ) , this invariance handed us the GCD as a linear-combination of r and r 1 . 5.1 : usamo1994.2. Let R , B , Y denote the colors red , blue , yellow , respectively. The sides of a 99-gon are initially colored so that, traveling CW ( clockwise ) , consecutive sides are R , B , R , B , ..., R , B , R , B , Y . : Is it possible, still traveling CW, to obtain R , B , R , B , ..., R , B , R , Y , B : by a sequence of modifications? A modification changes the color of one side ( to one of R , B , Y ) under the constraint that at no time may two adjacent sides have the same color. Proof. No such modification-sequence exists. In a coloring, let-- B Y denote the number of B Y adjacent-pairs, when traveling CW. In ( ), then,-- B Y is 1, and-- Y B = 0. Still in ( ),-- R Y = 0 and-- Y R = 1 ( because CW from the end Y is the starting R )....
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- Fall '09