1. MAT 2720 Discrete Mathematics Section 8.2 Paths and Cycles http://myhome.spu.edu/lauw

2. Goals Paths and Cycles Definitions and Examples More Definitions

3. Definitions

6. Example 1 (a) Write down a path from b to e with length 4.

7. Example 1 (b) Write down a path from b to e with length 5.

8. Example 1 (c) Write down a path from b to e with length 6.

9. Definitions

10. Example 2 The graph is not connected because …

11. Definitions

14. Example 3 How many subgraphs are there with 3 edges?

15. Definitions

17. Connected Graph & Component What can we say about the components of a graph if it is connected?

18. Connected Graph & Component What can we say about the graph if it has exactly one component?

19. Theorem A graph is connected if and only if it has exactly one component

20. Definitions

23. The degree of a vertex v, denoted by  (v), is the number of edges incident on v

24. Definitions The degree of a vertex v, denoted by  (v), is the number of edges incident on v

25. The Königsberg bridge problem Euler (1736) Is it possible to cross all seven bridges just once and return to the starting point?

26. The Königsberg bridge problem Edges represent bridges and each vertex represents a region.

27. The Königsberg bridge problem Euler (1736) Is it possible to find a cycle that includes all the edges and vertices of the graph?

28. Definitions An Euler cycle is a cycle that includes all the edges and vertices of the graph

29. Theorems 8.2.17 & 8.2.18: G has an Euler cycle if and only if G is connected and every vertex has even degree.

30. Theorems 8.2.17 & 8.2.18: G has an Euler cycle if and only if G is connected and every vertex has even degree.

31. Example 4(a) Determine if the graph has an Euler cycle.

32. Example 4(b) Find an Euler cycle.

33. Observation The sum of the degrees of all the vertices is even.

34. Example 5 (a) What is the sum of the degrees of all the vertices?

35. Example 5 (b) What is the number of edges?

36. Example 5 (c) What is the relationship and why?

37. Theorem 8.2.21

38. Example 6 Is it possible to draw a graph with 6 vertices and degrees 1,1,2,2,2,3?

39. Corollary 8.2.22

40. Theorem 8.2.23

41. Theorem 8.2.24