1. Chap. 11 Graph Theory and Applications 1

2. Directed Graph 2

3. (Undirected) Graph 3

4. Vertex and Edge Sets 4

5. Walk 5

6. Closed (Open) Walk 6

7. Trail, Path, Circuit, and Cycle 7

8. Comparison of Walk, Trail, Path, Circuit, and Cycle 8

9. Theorem 11.1 Observation: 9

10. Theorem 11.1 1.It suffices to show from a to b, the shortest trail is the shortest path. 2. Let be the shortest trail from a to b. 3. 4.4. 10

11. Connected Graph connected graphdisconnected graph 11

12. Multigraph 12

13. Subgraph 13

14. Spanning Subgraph 14

15. Induced Subgraph 15

16. Which of the following is an induced subgraph of G? O Induced Subgraph O X 16

17. Components of a Graph 12 connected sugraph 17

18. G-vG-v 18

19. G-eG-e 19

20. Complete Graph 20

21. Complement of a Graph 21

22. Isomorphic Graphs 22

23. Isomorphic Graphs Which of the following function define a graph isomorphism for the graphs shown below? O X 23

24. Isomorphic Graphs 24

25. Isomorphic Graphs Are the following two graphs isomorphic? In (a), a and d each adjacent to two other vertices. In (b), u, x, and z each adjacent to two other vertices. X 25

26. Vertex Degree 26

27. Theorem 11.2 27

28. Corollary 11.1 28

29. 29

30. a b c d 30

31. Euler Circuit and Euler Trail 31

32. Theorem 11.3 ( ⇒) 1. 2. 3. 4. 5. 6. 7. 32

33. Theorem 11.3 8. 9. 33

34. Theorem 11.3 ( ⇐) 1. 2. 3. 34

35. Theorem 11.3 4. 5. 6. 7. 8. 35

36. Theorem 11.3 9. 10. 11. 12. 13. 14. 36

37. Corollary 11.2 ( ⇐) 1. 2. 3. 4. ( ⇒) The proof of only if part is similar to that of Theorem 11.3 and omitted. 37

38. Incoming and Outgoing Degrees 2 38

39. Theorem 11.4 The proof is similar to that of Theorem 11.3 and omitted. 39

40. Planar Graph Which of the following is a planar graph? OO 40

41. Euler’s Theorem v = e = r = v – e + r = 2 7 8 3 41

42. Euler’s Theorem Proof. 1. Use induction on v (number of vertices). 2. Basis (v = 1): –G is a “bouquet” of loops, each a closed curve in the embedding. –If e = 0, then r = 1, and the formula holds. –Each added loop passes through a region and cuts it into 2 regions. This augments the edge count and the region count each by 1. Thus the formula holds when v = 1 for any number of edges. 42

43. Euler’s Theorem 3. Induction step (v>1): –There exists an edge e that is not a loop because G is connected. –Obtain a graph G’ with v’ vertices, e’ edges, and r’ regions by contracting e. –Clearly, v’=v–1, e’=e–1, and r’=r. –v’– e’+ r’ = 2. –Therefore, v-e+r=2. e (induction hypothesis) 43

44. Corollary 11.3 1. It suffices to consider connected graphs; otherwise, we could add edges. 2. If v  3, every region contains at least three edges (  L(R i )  3r). 3. 2e=  L(R i ), implying 2e  3r. 4. By Euler’s Theorem, v–e+r=2, implying e≤ 3v– 6. If also G is triangle-free, then e ≤ 2v–4. (  L(R i )  4r) (2e  4r) (e≤ 2v–4) If G is a simple planar graph with at least three vertices, then e≤3v–6. (A simple graph is not a multigraph and does not contain any loop.) 44

45. Bipartite Graph 45

46. Nonplanarity of K 5 and K 3,3 K 5 (e = 10, n = 5) K 3,3 (e = 9, n = 6) These graphs have too many edges to be planar. –For K 5, we have e = 10>9 = 3n-6. –Since K 3,3 is triangle-free, we have e = 9>8 = 2n-4. 46

47. Subdivision of a Graph 47

48. Subdivision of a Graph 48

49. 49

50. 50

51. Hamilton Cycle 51

52. Hamilton Cycle Does the following graph contain a hamiltion cycle? X 52

53. Theorem 11.8 1. 2. 3. 4. 53

54. Theorem 11.8 5. 6. 7. 8. 9. 10. 11. 12. 13. 54

55. Theorem 11.8 14. 15. 16. 17. 18. 19. 55

56. Theorem 11.8 17. 18. 19. 20. 56

57. Theorem 11.9 1. 2. 3. 4. 5. 57

58. Theorem 11.9 6. 7. 8. 9. 10. 11. 58

59. Proper Coloring and Chromatic Number 59

60. Counting Proper Colors 1. 2. 3. 4. 60

61. 61

62. Theorem 11.10 1. 2. 3. 4. 5. 6. 62

63. Example 11.36 63

64. Example 11.37 64