xuding zhu

Xuding Zhu

Zhejiang Normal University

2013.7Budapest

Circular flow of signed graphs

G: a graph

A circulation on G

, mapping

a with n togetherorientatioAn

Rf: E(G)

A circulation on G

1

11

122

3

121

, mapping


Rf: E(G) 0R

G: a graph

A circulation on G

-1

11

122

3

121

, mapping


Rf: E(G) 0R

G: a graph

x

y

A circulation on G

1

11

122

3

121

, mapping


Rf: E(G) 0R

G: a graphy

x

A circulation on G

1

11

122

3

121

)( )(

)()()(vEe vEe

efefvf, mapping


Rf: E(G)

The boundary of f

RVf :

1

2

1

0

0

0

G: a graph

A circulation on G

1

11

122

3

121

)( )(

)()()(vEe vEe

efefvf, mapping


Rf: E(G)

The boundary of f

RVvf :)(

1

2

1

0

0

0

flow a is then ,0 If ff .0)()()( EeEeVv

efefvf

A circulation on G

1

11

122

3

121

)( )(

)()()(vEe vEe

efefvf, mapping


Rf: E(G)

The boundary of f

RVvf :)(

1

2

1

0

0

0

flow a is then ,0 If ff .0)()()( EeEeVv

efefvf

groupabelian an :

ΓA

flowΓ A

A circulation on G

)( )(

)()()(vEe vEe

efefvf, mapping


Rf: E(G)

The boundary of f

RVvf :)(

flow a is then ,0 If ff

1

11

122

3

121

1

2

1

0

0

0

A circulation on G

)( )(

)()()(vEe vEe

efefvf, mapping


Rf: E(G)

The boundary of f

RVvf :)(


1

12

112

3

121

0

0

0

0

0

0

A circulation on G

)( )(

)()()(vEe vEe

efefvf, mapping


Rf: E(G)

The boundary of f

RVvf :)(

flow-circular a is then rf

G ofnumber flowcircular The

flowcircular a admits :min r-Gr (G)Φc


every for 1|)(|1 If eref

1

1

1

5.1

5.1

flow-5.2circular A

x y

cut a is ]X[X,

X XX toX from flow

X toX from flow

edges 12exactly hascut a If k

1|]X[|

k |]XE[X| assume

kXE

kr )1(X toX from flow

1kX toX from flow

kr

12

flowcircular a is Assume r-f

kG

k

c

12)(

has 12 size ofcut edgean graph withA

1

1

1

5.1

5.1

flow-5.2circular A

5.2)( Gc

x y

kG

k

c

12)(


kG

k

c

12)(


kG

k-

c

12)(

hasgraph connected edge4A

:[1981] ConjectureJaeger conjecture flow1

2 )k

(

conjecture flow3 case 1 k

conjecture flow5 case 2 k

trueif tight,

ConjectureThomassen [2012]

kG

k-

c

12)(

have graphs connected edge4

)3108( 2 kk

Theorem [Lovasz-Thomassen-Wu-Zhang, 2013]

k6

Theorem [Zhu, 2013]

)112( k have graphs signed

A signed graph G

A signed graph G

a positive edge a negative edge

An orientation of a signed edge


x

x

y

y



x

x

x

y

y

y



x

x

x

y

y

y

x

x

y

y



x

x

x

y

y

y

y

y

y

x

x

x



x ye

x ye

)()( yExEe x ye

x ye

x ye

x ye

)()( yExEe

)()( yExEe

)()( yExEe

A signed graph G

1 23

A circulation on G

, mapping


Rf: E(G)

A signed graph G

1 23

3

4

12 13

1

A circulation on G

, mapping


Rf: E(G)

A signed graph G

1 23

3

4

12 13

1

A circulation on G

, mapping


Rf: E(G)

)( )(

)()()(vEe vEe

efefvf

The boundary of f

RVf :

0

00

0

1

1

A circulation on G

)( )(

)()()(vEe vEe

efefvf, mapping


Rf: E(G)

The boundary of f

RVvf :)(

flow a is then ,0 If ff .0)(

vfVv

1 23

3

4

12 13

1

0

00

0

1

1

1 23

2

4

12 13

1

0

00

0

0

0

A circulation on G

)( )(

)()()(vEe vEe

efefvf, mapping


Rf: E(G)

The boundary of f

RVvf :)(

flow-circular a is rf

G ofnumber flowcircular The

flowcircular a admits :min r-Gr (G)Φc


every for 1-r|f(e)|1 If e

A signed graph G

A flow on G

)( )(

)()(vEe vEe

efef, mapping


Rf: E(G)

1 23

2

4

12 13

1

Flip at a vertex x

change signs of edges incidentto x

x

A signed graph G

A flow on G

)( )(

)()(vEe vEe

efef, mapping


Rf: E(G)

1 23

2

4

12 13

Flip at a vertex x


x1

1

3

A signed graph G

A flow on G

)( )(

)()(vEe vEe

efef, mapping


Rf: E(G)

1 2

2

4

12 3

Flip at a vertex x


x1

A signed graph G

A flow on G

)( )(

)()(vEe vEe

efef, mapping


Rf: E(G)

1 23

2

4

12 13

Flip at a vertex x


x1

A flow on G

)( )(

)()(vEe vEe

efef, mapping


Rf: E(G)

1 23

2

4

12 13

Flip at a vertex x


x1

Change the directions of `half’ edges incident to x

A flow on G

)( )(

)()(vEe vEe

efef, mapping


Rf: E(G)

1 23

2

4

12 13

Flip at a vertex x


x

Change the directions of `half’ edges incident to x

The flow remainsa flow 1

G can be obtained from G’ by a sequence of flippings

'GG

Fliping at vertices in X

change the sign of edges in ],[ XXE

'GG X]XE[X,

GG

somefor on disagrees

' and in edges of signs

kG

k

c

12)(

has 12 size ofcut edgean havinggraph A

nObservatio

This source a is

This sink a is)0( f(e)

source a is sink a is

then flow, -rcircular a is Ifee

f(e)f(e)f

edges negative 12exactly graph with signedA k

edges negative 12exactly has Assume kG

edgessink #edges source#

krf(e)f(e)kee

)1(1source a is sink a is

k

r1

2

kG

k-

c

12)(


k6

Theorem [Zhu, 2013]


One technical requirement is missing

edges negative 12least at or

edges negative ofnumber even an haseither any if

unbalanced12y essentiall is graph signedA

k

GG'

)-k(G

unbalanced12y essentiall )-k(

if special is n circulatio-A 12 fZ k 1kk,(e) f

kGc

12)(

flow ncirculatio-1)(2kinteger An flow

flow-1)(2kinteger special a has G

flow1

2circular a is (e)

)k

(k

fg(e)

kG

k-

c

12)(


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

Ee

kZβ: V

0(e)

with any For 12

withn circulatio- specail a has 12 fZG k βf

with

n circulatio- Zspecial a has 12k

f

G

0 f

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

Corollary

12in kZ

kG

k-

c

12)(


k6

Theorem [Zhu, 2013]



flow-1)(2kinteger special a

Lemma 1. connected edge)112( k unbalanced-1)(2ky essentiall flow- special a have graphs 12 kZ

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

112 k

even, is If |Q|

Qeke f allfor )(

edgessink # edges source#then

odd, is If |Q| 1 edgessink # edges source#then

1 have edgessink t except tha , allfor )( kf(e)e kQeke f

k)k(G edgessink # ,unbalanced-12y essentiall is As

1,: kkQ f 0)(

vfVv

,Theorem-LTWZBy

GZf k in flow special a is g 12

fg

gRG

with

n circulatio Zspecial a has ][ 12k

kG

k-

c

12)(


k6

Theorem [Zhu, 2013]



flow-1)(2kinteger special a

Lemma 1. connected edge)112( k unbalanced-1)(2ky essentiall flow- special a have graphs 12 kZ

To prove Theorem above, we need

connected edge)112( k unbalanced-1)(2ky essentiall flow- special a have graphs 12 kZ flow-1)2k(integer

then graph, a is If G

flow-1)(2k sflow- Zs 12k pecialpecial

For signed graphs

flow- Zspecial NWZ 3

flow-1)(2k special NWZ


0)( vf 0)( vf

q ' q


vu to frompath directed a is there

,0)(,0)( ,, vfufvu

G

0 Assume f


0)( vf 0)( vf

q ' q



,0)(,0)( ,, vfufvu0 Assume f


0)( vf 0)( vf

q ' q



,0)(,0)( ,, vfufvu


0)( vf 0)( vf

q ' q



,0)(,0)( ,, vfufvu

' -1)(2k qq -1)(2k


0)( vf 0)( vf

' q



,0)(,0)( ,, vfufvu

' -1)(2k qq -1)(2k


0)( vf 0)( vf



,0)(,0)( ,, vfufvu

G

If such a path does not exist

0 with vertex a

frompath directed a

by reached becan vertices

f(u)u

X

0)( vf 0)( vf

G

0 with vertex a

frompath directed a


f(u)u

XX

][][

)()()(XXEeXXEeXv

efefvf 0


,0)(,0)( ,, vfufvu

0)( vf 0)( vf

G

0 with vertex a

frompath directed a


f(u)u

XX

][][

)()()(XXEeXXEeXv

efefvf 0


,0)(,0)( ,, vfufvu

For a signed graph

Such a path may not exist

0)( vf 0)( vf

G

Xin edgessink many

XX

][][

)()()(XXEeXXEeXv

efefvf 0


,0)(,0)( ,, vfufvu

For a signed graph

Such a path may not exist

Xin edges sourcemany

X X

G[Q]in n circulatio-1)(2k special a : f

)(|],[|)( XfXXEkX R

2)( if balanced is kXf Xany for

Xv

vfXf )()(

Xv

vfXf )()(

)(|],[|)( XfXXEX

2)( if balanced is kXf Xany for

flow balanced special a exists There 2 Lemma 12 kZ

flow-1)(2k pecial a

tomdoified becan flow balanced specialA 3 Lemma 12

s

Z k

flow-1)(2k pecial a

tomdoified becan flow balanced specialA 3 Lemma 12

s

Z k

The same proof as for ordinary graph

flow balanced special a exists There Lemma 12 kZ


G[R] are 6k-edge connected.

By Williams-Tutte Theorem

G[R] contains 3k edge-disjoint spanning trees

kTTT 321 ,,,

By Williams-Tutte Theorem

G[R] contains 3k edge-disjoint spanning trees

kTTT 321 ,,,

connected is ][1 QGT

][ of Fsubgraph parity a contains 12 QGTT

eulerian is ][1 FQGT

cycleeulerian an :C

sourceor sink y alternatel Con edges negative orient the

flow balanced special a exists There Lemma 12 kZ


cycleeulerian an :C

Thank you

xuding zhu

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