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Xuding Zhu

Circular flow of signed graphs. Xuding Zhu. Zhejiang Normal University. 2013.7 Budapest. G: a graph. A circulation on G. G: a graph. 1. 1. 1. 1. 1. 2. 1. 2. 2. 3. A circulation on G. G: a graph. -1. 1. 1. 1. 1. 2. 1. 2. 2. 3. A circulation on G. G: a graph. 1.

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Xuding Zhu

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

  2. G: a graph A circulation on G

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

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

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

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

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

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

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

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

  11. The boundary of f A circulation on G

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

  13. A signed graph G

  14. A signed graph G a negative edge a positive edge

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

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

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

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

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

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

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

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

  23. 1 2 3 1 1 3 1 2 3 4 The boundary of f A circulation on G

  24. 1 2 3 1 1 2 1 2 3 4

  25. The boundary of f A circulation on G

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

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

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

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

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

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

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

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

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

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

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

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

  38. Theorem [Zhu, 2013] Lemma 1.

  39. 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

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

  41. For signed graphs

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