1. MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 5,Wednesday, September 10

2. A.2 Mathematical Induction Let p n denote a statement involving n objects. Induction proof of p n, for all n ¸ 0: Initial step (Induction Basis): Verify that p 0 is true. Induction step: Show that if p 0, p 1,..., p n-1 are true, then p n must be true. Note: You have to prove p 0. You also have to prove p n, but in the proof you may “pretend” that p n-1 or any other p k, 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,...

3. Example 1 Let s n = 1 + 2 +... + n. (A) Prove s n = n(n+1)/2. (B) Proof by induction. Initial step: s 1 = 1. (A). s 1 = 1(1+1)/2 = 1 (B). Induction step: s n = [1 + 2 +... + (n-1)] + n = s n-1 + n. Now assume s n-1 = (n-1)n/2 s n = s n-1 + n = (n-1)n/2 + n = n(n+1)/2.

4. 2.1 Euler Cylces Homework (MATH 310#2W): 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).

5. Multigraph In a multigraph we may have: Parallel edges Loop edges (= loops). A B C D

6. Königsberg Bridges 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. A B C D

7. Trails and Cycles Path P = x 1 - x 2 -... – x n (all vertices distinct). Circuit C = x 1 - x 2 -... – x n – x 1 [a path with an extra edge (x n,, x 1 )]. Trail T = x 1 - x 2 -... – x n (vertices may repeat but all edges are distinct). Cycle E = x 1 - x 2 -... – x n – x 1 [a trail with an extra edge (x n,, x 1 )].

8. Euler Cycles and Trails A cycle that uses every edge of a graph is called an Euler cycle (and visits every vertex). A trail that uses every edge of a graph is called an Euler trail (and visits every vertex at least once).

9. Theorem 1 (Euler, 1736) An (undirected) multigraph has an Euler cycle if and only if: it is connected and has all vertices of even degree.

10. Example 3: Routing Street Sweepers Solid red edges represent a collection of blocks to be swept.

11. Corollary A multigraph has an Euler trail, but not an Euler cycle, if and only if it is connected and has exactly two vertices of odd degree.