Xuding Zhu Zhejiang Normal University Budapest Circular flow of signed graphs

G: a graph A circulation on G

G: a graph

A circulation on G G: a graph

A circulation on G G: a graph

A circulation on G The boundary of f G: a graph

A circulation on G The boundary of f

A circulation on G The boundary of f

A circulation on G The boundary of f

A circulation on G The boundary of f

A circulation on G The boundary of f

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

A signed graph G

a positive edge a negative edge

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

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

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

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

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

A signed graph G A circulation on G

A signed graph G A circulation on G

A signed graph G A circulation on G The boundary of f

A circulation on G The boundary of f

A circulation on G The boundary of f

A signed graph G A flow on G Flip at a vertex x change signs of edges incident to x x

A signed graph G A flow on G Flip at a vertex x change signs of edges incident to x x 1

A signed graph G A flow on G Flip at a vertex x change signs of edges incident to x x 1

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

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

A signed graph G A flow on G Flip at a vertex x change signs of edges incident to x x1

A flow on G Flip at a vertex x change signs of edges incident to x x1 Change the directions of `half ’ edges incident to x

A flow on G Flip at a vertex x change signs of edges incident to x x1 Change the directions of `half ’ edges incident to x

A flow on G 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

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

Theorem [Zhu, 2013] One technical requirement is missing

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

Theorem [Zhu, 2013] Lemma 1.

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

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

For signed graphs

G

G If such a path does not exist

G If such a path does not exist

G

G For a signed graph Such a path may not exist

G For a signed graph Such a path may not exist

The same proof as for ordinary graph

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

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

Thank you