1. The Tutte Polynomial Graph Polynomials 238900 winter 05/06

2. The Rank Generation Polynomial Reminder Number of connected components in G(V,F)

3. The Rank Generation Polynomial Theorem 2 Is the graph obtained by omitting edge e Is the graph obtained by contracting edge e In addition, S(E n ;x,y) = 1 for G(V,E) where |V|=n and |E|=0

4. The Rank Generation Polynomial Theorem 2 – Proof Let us define : G’=G – e G’’=G / e r’ =r n’ =n r’’ =r n’’ =n

5. The Rank Generation Polynomial Theorem 2 – Proof Let us denote :

6. The Rank Generation Polynomial Observations

7. The Rank Generation Polynomial Observations

8. The Rank Generation Polynomial Observations

9. The Rank Generation Polynomial Observations 3

10. The Rank Generation Polynomial Observations 1 2

11. The Rank Generation Polynomial Reminder

12. The Rank Generation Polynomial Let us define

13. The Rank Generation Polynomial 1

15. 2 3

17. +

19. Obvious

20. The Rank Generation Polynomial Obvious

21. The Rank Generation Polynomial

22. Obvious Q.E.D

23. The Tutte Polynomial

24. Reminder 4 5

25. The Universal Tutte Polynomial

26. Theorem 6

27. The Universal Tutte Polynomial Theorem 6 - Proof 5

28. The Universal Tutte Polynomial If e is a bridge 4

29. The Universal Tutte Polynomial If e is a bridge

30. The Universal Tutte Polynomial If e is a loop 4

31. The Universal Tutte Polynomial If e is a bridge

32. The Universal Tutte Polynomial If e is neither a loop nor a bridge

33. The Universal Tutte Polynomial If e is neither a loop nor a bridge 4

34. The Universal Tutte Polynomial If e is neither a loop nor a bridge

35. The Universal Tutte Polynomial If e is neither a loop nor a bridge

36. The Universal Tutte Polynomial Theorem 6 - Proof Q.E.D

37. The Universal Tutte Polynomial Theorem 7 then

38. The Universal Tutte Polynomial Theorem 7 – proof V(G) is dependant entirely on V(G-e) and V(G/e). In addition – Q.E.D

39. Evaluations of the Tutte Polynomial Proposition 8 Let G be a connected graph. is the number of spanning trees of G Shown in the previous lecture

40. Evaluations of the Tutte Polynomial Proposition 8 Let G be a connected graph. is the number of connected spanning sub-graphs of G

41. Evaluations of the Tutte Polynomial Proposition 8 Let G be a connected graph. is the number of connected spanning sub-graphs of G sub-graphs of Gconnected

42. Evaluations of the Tutte Polynomial Proposition 8 Let G be a connected graph. is the number of (edge sets forming) spanning forests of G Spanning forests

43. Evaluations of the Tutte Polynomial Proposition 8 Let G be a connected graph. is the number of spanning sub-graphs of G

44. The Universal Tutte Polynomial Theorem 9 The chromatic polynomial and the Tutte polynomial are related by the equation :

45. The Universal Tutte Polynomial Theorem 9 – proof Claim :

46. The Universal Tutte Polynomial Theorem 9 – proof We must show that :

47. The Universal Tutte Polynomial Theorem 9 – proof and :

48. The Universal Tutte Polynomial Theorem 9 – proof Obvious. (empty graphs)

49. The Universal Tutte Polynomial Theorem 9 – proof Obvious Chromatic polynomial property

50. The Universal Tutte Polynomial Theorem 9 – proof Thus :

51. The Universal Tutte Polynomial Theorem 9 – proof And so, due to the uniqueness we shown in Theorem 7 :

52. The Universal Tutte Polynomial Theorem 9 – proof Remembering that :

53. The Universal Tutte Polynomial Theorem 9 – proof Remembering that : Q.E.D

54. Evaluations of the Tutte Polynomial Let G be a connected graph. is the number of acyclic orientations of G. We know that (Theorem 9) –

55. Evaluations of the Tutte Polynomial So – And thus –

56. Acyclic Orientations of Graphs Let G be a connected graph without loops or multiple edges. An orientation of a graph is received after assigning a direction to each edge. An orientation of a graph is acyclic if it does not contain any directed cycles.

57. Acyclic Orientations of Graphs is the number of pairs where is a map Proposition 1.1 and is an orientation of G, subject to the following : The orientation is acyclic

58. Acyclic Orientations of Graphs Proof The second condition forces the map to be a proper coloring. The second condition is immediately implied from the first one.

59. Acyclic Orientations of Graphs Proof Conversely, if the map is proper, than the second condition defines a unique acyclic orientation of G. Hence, the number of allowed mappings is simply the number of proper coloring with x colors, which is by definition

60. Acyclic Orientations of Graphs be the number of pairs where is a mapand is an orientation of G, subject to the following : The orientation is acyclic

61. Acyclic Orientations of Graphs Theorem 1.2

62. Acyclic Orientations of Graphs Proof The chromatic polynomial is uniquely determined by the following : G 0 is the one vertex graph Disjoint union

63. Acyclic Orientations of Graphs Proof We now have to show for the new polynomial : G 0 is the one vertex graph Disjoint union Obvious

64. Acyclic Orientations of Graphs Proof We need to show that : Let :

65. Acyclic Orientations of Graphs Proof Let : Let be an acyclic orientation of G-e compatible with Let :

66. Acyclic Orientations of Graphs Proof Let be an orientation of G after adding {u  v} to Let be an orientation of G after adding {v  u} to

67. Acyclic Orientations of Graphs Proof We will show that for each pair exactly one of the orientations is acyclic and compatible with,expect for of them, in which case both are acyclic orientations compatible with

68. Acyclic Orientations of Graphs Proof Once this is done, we will know that – due to the definition of

69. Acyclic Orientations of Graphs Proof For each pair where - and is an acyclic orientation compatible with one of these three scenarios must hold :

70. Acyclic Orientations of Graphs Proof Case 1 – Clearly is not compatible with while is compatible. Moreover, is acyclic : Impossible cicle

71. Acyclic Orientations of Graphs Proof Case 2 – Clearly is not compatible with while is compatible. Moreover, is acyclic : Impossible cicle

72. Acyclic Orientations of Graphs Proof Case 3 – Both are compatible with At least one is also acyclic. Suppose not, then: contains

73. Acyclic Orientations of Graphs Proof contains Impossible cicle

74. Acyclic Orientations of Graphs Proof We now have to show that both and are acyclic for exactly pairs of with Let z denote the vertex identifying {u,v} in

75. Acyclic Orientations of Graphs Proof Z some acyclic orientation compatible with uv impossible to add a circle by the new edge {u,v} two acyclic orientations, compatible with

76. Acyclic Orientations of Graphs Proof Z exactly one, necessarily acyclic, compatible with uv Some two acyclic orientations, compatible with All other vertices of G remains the same

77. Acyclic Orientations of Graphs Proof And so both and are acyclic for exactly pairs of with And so –

78. Acyclic Orientations of Graphs Proof It is obvious that for x = 1 every orientation is compatible with And so the expression count the number of acyclic orientations in G Q.E.D