Graphs and Euler cycles

1. Graphs and Euler cycles Let Maths take you Further…

2. edges vertices What is a graph? • A graph is a set of points called vertices (or nodes) connected by lines called edges (or arcs) • In a graph a line from point A to point B is considered to be the same thing as a line from point B to point A

3. Graphs Traversable graphs Which of these graphs can be drawn without taking your pen off the paper or repeating any edges?

4. Graphs • What is significant about the results? • Can you explain why? No Yes - start and finish in different places Yes - start and finish in the same place Semi-Eulerian Eulerian

5. Graphs Some important ideas • The degree of a vertex is the number of edges that meet at that vertex • Can you draw a graph with 3 odd vertices? • What about 5 odd vertices? • Why are there are always an even number of odd vertices?

6. What is the sum of the degrees of the vertices for each graph? • What do you notice if you compare these totals to the number of edges in the graph? 12 20 20

7. Graphs The sum of the degrees of all the vertices in a graph is twice the number of edges. This is more formally called the Handshaking Theorem and is written Σ deg v = 2e

8. Konigsberg bridges Is it possible to find a route that • Starts and finishes at the same place? • Crosses each bridge exactly once?

9. Konigsberg bridges The Königsberg bridges is a famous mathematics problem inspired by an actual place and situation. The city of Königsberg on the River Pregel in Prussia (now Kaliningrad, Russia) includes two large islands which were connected to each other and the mainland by seven bridges. The citizens of Königsberg allegedly walked about on Sundays trying to find a route that crosses each bridge exactly once, and return to the starting point.

10. Konigsberg bridges Simplify the problem

11. Konigsberg bridges Model it as a graph where the edges represent the bridges and the vertices represent the islands.

12. Konigsberg bridges In 1736 Leonard Euler proved that it was not possible because all the vertices of the graph are odd.

13. 8 Depot 5 4 5 2 6 6 4 3 7 9 5 A problem A postman starts his rounds at the depot. He needs to deliver letters along the all the streets and return to the depot. What is the shortest route he can take?

14. Identify the odd nodes E 8 D 5 4 5 F 2 6 6 G C 4 3 7 9 A 5 B

15. Investigate possible pairings E 8 BC, DE, FG has length 9+8+2=19 D 5 4 5 F 2 6 6 G C 4 3 7 9 A 5 B

16. Investigate possible pairings E 8 D BC, EF, DG has length 9+4+5=18 5 4 5 F 2 6 6 G C 4 3 7 9 A 5 B

17. Investigate possible pairings E BF, CE, DG has length 3+5+5=13 8 D 5 4 5 F 2 6 6 G C 4 3 7 9 A 5 B

18. Investigate possible pairings BC, DE, FG has length 9+8+2=19 E 8 D 5 4 5 BC, EF, DG has length 9+4+5=18 F 2 6 6 G C 4 3 7 BF, CE, DG has length 3+5+5=13 9 A 5 B No other pairings are sensible (Why?). BF, CE, DG is the shortest pairing found.

19. Solution E 8 Total length of all the edges plus the lengths of BF, CE and DG = 64 + 13 = 77 D 5 4 5 F 2 6 6 G C 4 3 7 9 A 5 B A possible route is: DGFAGDABFBCECFED

20. Algorithms • An algorithm is a set of instructions for solving a type of problem • Finding cycles that go along every edge at least once is called a Route Inspection problem • The algorithm for finding the shortest such cycle was published in 1962 by the Chinese mathematician Mei Ko Kwan (so it is sometimes called the Chinese postman problem) • Can you think of some more situations where this algorithm would be useful?