1. Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs

2. G: a graph A circulation on G

3. 1 1 1 1 2 2 3 1 2 1 G: a graph

4. A circulation on G 1 1 1 2 2 3 1 2 1 G: a graph

5. A circulation on G 1 1 1 1 2 2 3 1 2 1 G: a graph

6. A circulation on G 1 1 1 1 2 2 3 1 2 1 The boundary of f G: a graph

7. A circulation on G 1 1 1 1 2 2 3 1 2 1 The boundary of f

8. A circulation on G 1 1 1 1 2 2 3 1 2 1 The boundary of f

9. A circulation on G The boundary of f 1 1 1 1 2 2 3 1 2 1

10. A circulation on G The boundary of f 1 1 2 1 1 2 3 1 2 1

11. A circulation on G The boundary of f

17. Conjecture Thomassen [2012]Theorem [Lovasz-Thomassen-Wu-Zhang, 2013] Theorem [Zhu, 2013]

18. A signed graph G

19. a positive edge a negative edge

20. An orientation of a signed edge a positive edge a negative edge x x y y

21. An orientation of a signed edge a positive edge a negative edge x x x y y y

22. An orientation of a signed edge a positive edge a negative edge x x x y y y x x y y

23. An orientation of a signed edge a positive edge a negative edge x x x y y y y y y x x x

24. An orientation of a signed edge a positive edge a negative edge x y e x y e x y e x y e x y e x y e

25. A signed graph G 1 2 3 A circulation on G

26. A signed graph G 1 2 3 3 4 1 2 1 3 1 A circulation on G

27. A signed graph G 1 2 3 3 4 1 2 1 3 1 A circulation on G The boundary of f

28. A circulation on G The boundary of f 1 2 3 3 4 1 2 1 3 1

29. 1 2 3 2 4 1 2 1 3 1

30. A circulation on G The boundary of f

31. A signed graph G A flow on G 1 2 3 2 4 1 2 1 3 1 Flip at a vertex x change signs of edges incident to x x

32. A signed graph G A flow on G 1 2 3 2 4 1 2 1 3 Flip at a vertex x change signs of edges incident to x x 1

33. A signed graph G A flow on G 1 2 3 2 4 1 2 1 3 Flip at a vertex x change signs of edges incident to x x 1

34. 1 3 A signed graph G A flow on G 1 2 2 4 1 2 3 Flip at a vertex x change signs of edges incident to x x 1

35. 1 3 A signed graph G A flow on G 1 2 2 4 1 2 3 Flip at a vertex x change signs of edges incident to x x 1

36. A signed graph G A flow on G 1 2 3 2 4 1 2 1 3 Flip at a vertex x change signs of edges incident to x x1

37. A flow on G 1 2 3 2 4 1 2 1 3 Flip at a vertex x change signs of edges incident to x x1 Change the directions of `half ’ edges incident to x

38. A flow on G 1 2 3 2 4 1 2 1 3 Flip at a vertex x change signs of edges incident to x x1 Change the directions of `half ’ edges incident to x

39. A flow on G 1 2 3 2 4 1 2 1 3 Flip at a vertex x change signs of edges incident to x x Change the directions of `half ’ edges incident to x The flow remains a flow 1

40. G can be obtained from G ’ by a sequence of flippings Fliping at vertices in X change the sign of edges in

42. Theorem [Zhu, 2013] One technical requirement is missing

44. Theorem [Loavsz-Thomassen-Wu-Zhang, 2013] Corollary

45. Theorem [Zhu, 2013] Lemma 1.

46. Proof Assume G is (12k-1)-edge connected essentially (2k+1)-unbalanced Assume G has the least number of negative edges among its equivalent signed graphs Q: negative edges of G R: positive edges of G G[R] is 6k-edge connected

49. Theorem [Zhu, 2013] Lemma 1. To prove Theorem above, we need

50. For signed graphs

52. G

57. G If such a path does not exist

58. G If such a path does not exist

59. G

60. G For a signed graph Such a path may not exist

61. G For a signed graph Such a path may not exist

64. The same proof as for ordinary graph

66. G[R] are 6k-edge connected. By Williams-Tutte Theorem G[R] contains 3k edge-disjoint spanning trees

67. By Williams-Tutte Theorem G[R] contains 3k edge-disjoint spanning trees

70. Thank you