MATH 310 FALL 2003 Combinatorial Problem Solving Lecture
MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 5, Wednesday, September 10
A. 2 Mathematical Induction n n Let pn denote a statement involving n objects. Induction proof of pn, for all n ¸ 0: • Initial step (Induction Basis): Verify that p 0 is true. • Induction step: Show that if p 0, p 1, . . . , pn-1 are true, then pn must be true. n n Note: You have to prove p 0. You also have to prove pn, but in the proof you may “pretend” that pn-1 or any other pk, k < n is true. Note: Induction comes in various forms. For instance, sometimes the initial step involves some other small number, say, p 1, or p 3, . . .
Example 1 n n Let sn = 1 + 2 +. . . + n. (A) Prove sn = n(n+1)/2. (B) Proof by induction. Initial step: n n n s 1 = 1. (A). s 1 = 1(1+1)/2 = 1 (B). Induction step: n n n sn = [1 + 2 +. . . + (n-1)] + n = sn-1 + n. Now assume sn-1 = (n-1)n/2 sn = sn-1 + n = (n-1)n/2 + n = n(n+1)/2.
2. 1 Euler Cylces n Homework (MATH 310#2 W): • Read 2. 2. Read Supplement I. (pp 46 -48) Write down a list of all newly introduced terms (printed in boldface or italic) • Do Exercises A. 2: 4, 12, 17, 24 • Do Exercises 2. 1: 2, 10, 12, 17 • Volunteers: • ____________ • Problem: 2. 1: 17. Challenge (up to 5 + 5 points): Do Exerecise 2. 1: # 20 (requires computer programming).
Multigraph n A n n B C D In a multigraph we may have: Parallel edges Loop edges (= loops).
Königsberg Bridges n A B C D Great Swiss mathematician Leonhard Euler solved the problem of Seven Bridges of Königsberg by showing that it is impossible to walk across each bridge just once.
Trails and Cycles n n Path P = x 1 - x 2 -. . . – xn (all vertices distinct). Circuit C = x 1 - x 2 -. . . – xn – x 1 [a path with an extra edge (xn , , x 1 )]. Trail T = x 1 - x 2 -. . . – xn (vertices may repeat but all edges are distinct). Cycle E = x 1 - x 2 -. . . – xn – x 1 [a trail with an extra edge (xn , , x 1 )].
Euler Cycles and Trails A cycle that uses every edge of a graph is called an Euler cycle (and visits every vertex). n A trail that uses every edge of a graph is called an Euler trail (and visits every vertex at least once). n
Theorem 1 (Euler, 1736) n An (undirected) multigraph has an Euler cycle if and only if: • it is connected and • has all vertices of even degree.
Example 3: Routing Street Sweepers n Solid red edges represent a collection of blocks to be swept.
Corollary n A multigraph has an Euler trail, but not an Euler cycle, if and only if it is connected and n has exactly two vertices of odd degree. n
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