1. 9.5 Euler and Hamilton graphs

2. 9.5: Euler and Hamilton paths Konigsberg problem

3. Graph of the Konigsberg problem Also see handout

4. Terminology - Euler Euler circuit – a simple circuit containing every edge of G Euler path – a simple path containing every edge of G

5. Are there Euler paths or circuits for these graphs? (see handout) A A B AB CA C C BD C D DE FB E AB CD

6. … A A BCAB BD C D ECD

7. Q—When is there an Euler circuit or path? A connected multigraph has an Euler circuit iff each of its vertices has _______. A connected multigraph has an Euler path but not an Euler circuit iff it has exactly _____.

8. Thm. 1 Theorem 1: A connected multigraph has an Euler circuit iff each of its vertices has _______. Partial proof:  Assume the circuit begins with a vertex a. Show that the degree of a is ___... Why?: The circuit contributes __ when it begins, __ when it ends, and ______ Next, consider a vertex other than a. Show that the degree is_______

9. …proof outline of Thm. 1  If all vertices are even, must an Euler circuit exist? Show it does by constructing one. Form a circuit beginning at a=x 0. If it is an Euler circuit, we are done. If not,…

10. Theorem 2: Thm 2 : A connected multigraph has an Euler path but not an Euler circuit iff it has exactly _____. Proof:  If a graph has a Euler path, but not circuit, from a to b, deg (a) is ____ because_____ deg (b) is ____ because _______ all other degrees are ____ because_____  Suppose a graph has exactly 2 vertices of odd degree, say a and b. Consider __________/ Now all vertices have ______ degree. By Thm. 1, ___________ The removal of the new edge produces an __________

11. Euler- misc. Applications of Euler: – Highway inspector – … Directed graphs: seen in hw

12. Hamilton paths and circuits Questions: Can we find simple paths or circuits that contain every vertex of the graph exactly once? Def: A path x 0, x 1, …x n in a graph G=(V,E) is called a Hamilton path if V={ x 0, x 1, …x n } and x i ≠x j for 0≤i ≤ j ≤ n. A circuit x 0, x 1, …x n, x 0 in a graph G=(V,E) is called a Hamilton circuit if, x 1, …x n is a Hamilton path. Uses for Hamilton circuits: …

13. Do these graphs have Hamilton paths or circuits? (see handout) A AB AB CA C C B D CD DE FB E AB CD

14. …examples A A BCAB B D C D EC D

15. Hamilton paths and circuits A BA B ABC C DC D DEF G E

16. 12-sided example (see handout too)

17. Any necessary and sufficient conditions for Hamilton? No… but there are sufficient conditions for a Hamilton circuit, such as – If G is a simple graph with n vertices where n>=3 where the degree of every vertex in G is at least n/2.

18. KnKn Ex: Show that K n has a Hamilton circuit whenever n≥3.

19. exercises See. P. 644 in book: #7