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The Tutte polynomial and a bijection between subgraphs and orientations Olivier Bernardi - CRM Combinatorics Seminar at CRM, October 2006, Barcelona Content of the talk The Tutte Polynomial and its specializations A new characterization of the

  1. The tour of a spanning tree Start from the root. Iterate the move : External half-edge = ⇒ Turn . Internal half-edge = ⇒ Cross + turn . Example : CRM, October 2006 ▽ Olivier Bernardi - CRM – p.15/31

  2. The tour of a spanning tree Start from the root. Iterate the move : External half-edge = ⇒ Turn . Internal half-edge = ⇒ Cross + turn . Example : CRM, October 2006 ▽ Olivier Bernardi - CRM – p.15/31

  3. The tour of a spanning tree Start from the root. Iterate the move : External half-edge = ⇒ Turn . Internal half-edge = ⇒ Cross + turn . Example : CRM, October 2006 ▽ Olivier Bernardi - CRM – p.15/31

  4. The tour of a spanning tree Start from the root. Iterate the move : External half-edge = ⇒ Turn . Internal half-edge = ⇒ Cross + turn . Example : CRM, October 2006 ▽ Olivier Bernardi - CRM – p.15/31

  5. The tour of a spanning tree Start from the root. Iterate the move : External half-edge = ⇒ Turn . Internal half-edge = ⇒ Cross + turn . Example : CRM, October 2006 ▽ Olivier Bernardi - CRM – p.15/31

  6. The tour of a spanning tree Start from the root. Iterate the move : External half-edge = ⇒ Turn . Internal half-edge = ⇒ Cross + turn . Example : CRM, October 2006 ▽ Olivier Bernardi - CRM – p.15/31

  7. The tour of a spanning tree Start from the root. Iterate the move : External half-edge = ⇒ Turn . Internal half-edge = ⇒ Cross + turn . Example : CRM, October 2006 ▽ Olivier Bernardi - CRM – p.15/31

  8. The tour of a spanning tree Start from the root. Iterate the move : External half-edge = ⇒ Turn . Internal half-edge = ⇒ Cross + turn . Example : CRM, October 2006 Olivier Bernardi - CRM – p.15/31

  9. Tour: graphical interpretation CRM, October 2006 Olivier Bernardi - CRM – p.16/31

  10. Activities of spanning trees CRM, October 2006 ▽ Olivier Bernardi - CRM – p.17/31

  11. Activities of spanning trees Consider the order of appearance of the edges around the tree. CRM, October 2006 ▽ Olivier Bernardi - CRM – p.17/31

  12. Activities of spanning trees 5 6 3 4 2 1 Consider the order of appearance of the edges around the tree. CRM, October 2006 ▽ Olivier Bernardi - CRM – p.17/31

  13. Activities of spanning trees 5 6 3 4 2 1 Consider the order of appearance of the edges around the tree. An internal/external edge is active if it is minimal in its fundamental cocycle/cycle. CRM, October 2006 ▽ Olivier Bernardi - CRM – p.17/31

  14. Activities of spanning trees 5 6 3 4 2 1 Consider the order of appearance of the edges around the tree. We denote by: • i ( A ) the number of internal active edges, • e ( A ) the number of external active edges. CRM, October 2006 Olivier Bernardi - CRM – p.17/31

  15. Tutte polynomial Theorem [B.] : The Tutte polynomial is equal to : � x i ( T ) y e ( T ) . T G ( x, y ) = T spanning tree CRM, October 2006 ▽ Olivier Bernardi - CRM – p.18/31

  16. Tutte polynomial Theorem [B.] : The Tutte polynomial is equal to : � x i ( T ) y e ( T ) . T G ( x, y ) = T spanning tree Example : T K 3 ( x, y ) = x 2 + x + y. x 2 T K 3 ( x, y ) = + + x y CRM, October 2006 ▽ Olivier Bernardi - CRM – p.18/31

  17. Tutte polynomial Theorem [B.] : The Tutte polynomial is equal to : � x i ( T ) y e ( T ) . T G ( x, y ) = T spanning tree Subgraph expansion [Brylawsky] : � ( x − 1) c ( S ) − 1 ( y − 1) | S | + c ( S ) −| V | . T G ( x, y ) = S ⊆ G CRM, October 2006 Olivier Bernardi - CRM – p.18/31

  18. From spanning trees to subgraphs CRM, October 2006 ▽ Olivier Bernardi - CRM – p.19/31

  19. From spanning trees to subgraphs CRM, October 2006 ▽ Olivier Bernardi - CRM – p.19/31

  20. From spanning trees to subgraphs T + T T − CRM, October 2006 Olivier Bernardi - CRM – p.19/31

  21. Partition of the subgraphs 2 E = � [ T − ; T + ] Proposition [B.] : T spanning tree CRM, October 2006 ▽ Olivier Bernardi - CRM – p.20/31

  22. Partition of the subgraphs 2 E = � [ T − ; T + ] Proposition [B.] : T spanning tree Example : CRM, October 2006 ▽ Olivier Bernardi - CRM – p.20/31

  23. Partition of the subgraphs 2 E = � [ T − ; T + ] Proposition [B.] : T spanning tree Example : CRM, October 2006 ▽ Olivier Bernardi - CRM – p.20/31

  24. Partition of the subgraphs 2 E = � [ T − ; T + ] Proposition [B.] : T spanning tree Proposition [B.] : ( x − 1) c ( S ) − 1 ( y − 1) | S | + c ( S ) −| V | = x i ( T ) y e ( T ) � S ∈ [ T − ,T + ] CRM, October 2006 Olivier Bernardi - CRM – p.20/31

  25. Bijection subgraphs ⇐ ⇒ orientations CRM, October 2006 Olivier Bernardi - CRM – p.21/31

  26. Φ : Subgraphs �→ Orientations For spanning trees: CRM, October 2006 ▽ Olivier Bernardi - CRM – p.22/31

  27. Φ : Subgraphs �→ Orientations For spanning trees: Internal edges are oriented from father to son. CRM, October 2006 ▽ Olivier Bernardi - CRM – p.22/31

  28. Φ : Subgraphs �→ Orientations For spanning trees: Internal edges are oriented from father to son. External edges are oriented in such a way their heads appear before their tails around the tree. CRM, October 2006 ▽ Olivier Bernardi - CRM – p.22/31

  29. Φ : Subgraphs �→ Orientations For spanning trees: Internal edges are oriented from father to son. External edges are oriented in such a way their heads appear before their tails around the tree. CRM, October 2006 Olivier Bernardi - CRM – p.22/31

  30. Activity et acyclicity Remark : The fundamental cycle of an external active edge is oriented. CRM, October 2006 ▽ Olivier Bernardi - CRM – p.23/31

  31. Activity et acyclicity Remark : The fundamental cycle of an external active edge is oriented. Hence, Φ( T ) acyclic = ⇒ T internal. CRM, October 2006 ▽ Olivier Bernardi - CRM – p.23/31

  32. Activity et acyclicity Remark : The fundamental cycle of an external active edge is oriented. Hence, Φ( T ) acyclic ⇐ ⇒ T internal. CRM, October 2006 Olivier Bernardi - CRM – p.23/31

  33. Activity et strong connectivity Remark : The fundamental cocycle of an internal active edge is oriented. CRM, October 2006 ▽ Olivier Bernardi - CRM – p.24/31

  34. Activity et strong connectivity Remark : The fundamental cocycle of an internal active edge is oriented. Hence, Φ( T ) strongly connected = ⇒ T external. CRM, October 2006 ▽ Olivier Bernardi - CRM – p.24/31

  35. Activity et strong connectivity Remark : The fundamental cocycle of an internal active edge is oriented. Hence, Φ( T ) strongly connected ⇐ ⇒ T external. CRM, October 2006 Olivier Bernardi - CRM – p.24/31

  36. Φ : Subgraphs �→ Orientations For any subgraph S : Φ ? CRM, October 2006 ▽ Olivier Bernardi - CRM – p.25/31

  37. Φ : Subgraphs �→ Orientations For any subgraph S : The subgraph S is in an interval [ T − , T + ] . Φ ? Φ CRM, October 2006 ▽ Olivier Bernardi - CRM – p.25/31

  38. Φ : Subgraphs �→ Orientations For any subgraph S : The subgraph S is in an interval [ T − , T + ] . Reverse the fundamental cycle/cocycle of the edges in T △ S . Φ Φ CRM, October 2006 Olivier Bernardi - CRM – p.25/31

  39. Φ : Subgraphs �→ Orientations Theorem [B.] : For any graph G , the mapping Φ establishes a bijection between subgraphs and orientations. CRM, October 2006 ▽ Olivier Bernardi - CRM – p.26/31

  40. Φ : Subgraphs �→ Orientations Theorem [B.] : For any graph G , the mapping Φ establishes a bijection between subgraphs and orientations. Example : CRM, October 2006 ▽ Olivier Bernardi - CRM – p.26/31

  41. Φ : Subgraphs �→ Orientations Theorem [B.] : For any graph G , the mapping Φ establishes a bijection between subgraphs and orientations. Example : CRM, October 2006 Olivier Bernardi - CRM – p.26/31

  42. Specializations Subgraphs Orientations general root- connected strongly connected general connectedexternal general mal general forest mini internal acyclic CRM, October 2006 Olivier Bernardi - CRM – p.27/31

  43. Specializations Connected: T G (1 , 2) CRM, October 2006 ▽ Olivier Bernardi - CRM – p.28/31

  44. Specializations Connected: T G (1 , 2) :Root-connected . CRM, October 2006 ▽ Olivier Bernardi - CRM – p.28/31

  45. Specializations External: T G (0 , 2) CRM, October 2006 ▽ Olivier Bernardi - CRM – p.28/31

  46. Specializations External: T G (0 , 2) :Strongly-connected . CRM, October 2006 ▽ Olivier Bernardi - CRM – p.28/31

  47. Specializations Forests: T G (2 , 1) CRM, October 2006 ▽ Olivier Bernardi - CRM – p.28/31

  48. Specializations Forests: T G (2 , 1) :Minimal ( ⇐ ⇒ score vectors). CRM, October 2006 ▽ Olivier Bernardi - CRM – p.28/31

  49. Specializations Internal: T G (2 , 0) CRM, October 2006 ▽ Olivier Bernardi - CRM – p.28/31

  50. Specializations Internal: T G (2 , 0) :Acyclic . CRM, October 2006 Olivier Bernardi - CRM – p.28/31

  51. Specializations Theorem [B.] : For any graph G , the mapping Φ induces a bijection between: CRM, October 2006 ▽ Olivier Bernardi - CRM – p.29/31

  52. Specializations Theorem [B.] : For any graph G , the mapping Φ induces a bijection between: • Connected subgraphs and root-connected orientations. T G (1 , 2) [Gioan 06] CRM, October 2006 ▽ Olivier Bernardi - CRM – p.29/31

  53. Specializations Theorem [B.] : For any graph G , the mapping Φ induces a bijection between: • Connected subgraphs and root-connected orientations. T G (1 , 2) [Gioan 06] • Forests and minimal orientations ( ⇐ ⇒ score vectors). T G (2 , 1) [Stanley 80] CRM, October 2006 ▽ Olivier Bernardi - CRM – p.29/31

  54. Specializations Theorem [B.] : For any graph G , the mapping Φ induces a bijection between: • Connected subgraphs and root-connected orientations. T G (1 , 2) [Gioan 06] • Forests and minimal orientations ( ⇐ ⇒ score vectors). T G (2 , 1) [Stanley 80] • Trees and minimal root-connected orientations ( ⇐ ⇒ root-connected score vectors). T G (1 , 1) [Gioan 06] CRM, October 2006 Olivier Bernardi - CRM – p.29/31

  55. Theorem [B.] : For any graph G , the mapping Φ induces a bijection between: • Internal subgraphs and acyclic orientations. T G (2 , 0) [Winder 66, Stanley 73, Gessel & Sagan 96] CRM, October 2006 ▽ Olivier Bernardi - CRM – p.30/31

  56. Theorem [B.] : For any graph G , the mapping Φ induces a bijection between: • Internal subgraphs and acyclic orientations. T G (2 , 0) [Winder 66, Stanley 73, Gessel & Sagan 96] • External subgraphs and strongly-connected orientations. T G (0 , 2) [Las Vergnas 84, Gioan & Las Vergnas 06] CRM, October 2006 ▽ Olivier Bernardi - CRM – p.30/31

  57. Theorem [B.] : For any graph G , the mapping Φ induces a bijection between: • Internal subgraphs and acyclic orientations. T G (2 , 0) [Winder 66, Stanley 73, Gessel & Sagan 96] • External subgraphs and strongly-connected orientations. T G (0 , 2) [Las Vergnas 84, Gioan & Las Vergnas 06] • Internal trees and acyclic root-connected orientations. T G (1 , 0) [Greene & Zaslavsky 83, Gessel & Sagan 96, Gebhard & Sagan 00] CRM, October 2006 ▽ Olivier Bernardi - CRM – p.30/31

  58. Theorem [B.] : For any graph G , the mapping Φ induces a bijection between: • Internal subgraphs and acyclic orientations. T G (2 , 0) [Winder 66, Stanley 73, Gessel & Sagan 96] • External subgraphs and strongly-connected orientations. T G (0 , 2) [Las Vergnas 84, Gioan & Las Vergnas 06] • Internal trees and acyclic root-connected orientations. T G (1 , 0) [Greene & Zaslavsky 83, Gessel & Sagan 96, Gebhard & Sagan 00] • External trees and minimal strongly-connected orientations ( ⇐ ⇒ strongly-connected score vectors). T G (0 , 1) [Gioan 06] CRM, October 2006 Olivier Bernardi - CRM – p.30/31

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