xuding zhu zhejiang normal university 2013.7 budapest circular flow of signed graphs
DESCRIPTION
G: a graphTRANSCRIPT
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Xuding Zhu
Zhejiang Normal University2013.7
Budapest
Circular flow of signed graphs
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G: a graph
A circulation on G
, mapping a with n togetherorientatioAn
Rf: E(G)
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A circulation on G
11
11
223
121
, mapping a with n togetherorientatioAn
Rf: E(G) 0R
G: a graph
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A circulation on G
-11
11
223
121
, mapping a with n togetherorientatioAn
Rf: E(G) 0R
G: a graph
x
y
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A circulation on G
11
11
223
121
, mapping a with n togetherorientatioAn
Rf: E(G) 0R
G: a graph y
x
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A circulation on G
11
11
223
121
)( )(
)()()(vEe vEe
efefvf, mapping a with n togetherorientatioAn
Rf: E(G)
The boundary of f RVf :
1
2
1
0
0
0
G: a graph
![Page 7: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/7.jpg)
A circulation on G
11
11
223
121
)( )(
)()()(vEe vEe
efefvf, mapping a with n togetherorientatioAn
Rf: E(G)
The boundary of f RVvf :)(
1
2
1
0
0
0
flow a is then ,0 If ff .0)()()( EeEeVv
efefvf
![Page 8: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/8.jpg)
A circulation on G
11
11
223
121
)( )(
)()()(vEe vEe
efefvf, mapping a with n togetherorientatioAn
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
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A circulation on G
)( )(
)()()(vEe vEe
efefvf, mapping a with n togetherorientatioAn
Rf: E(G)
The boundary of f RVvf :)(
flow a is then ,0 If ff
11
11
223
121
1
2
1
0
0
0
![Page 10: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/10.jpg)
A circulation on G
)( )(
)()()(vEe vEe
efefvf, mapping a with n togetherorientatioAn
Rf: E(G)
The boundary of f RVvf :)(
flow a is then ,0 If ff
11
21
123
121
0
0
0
0
0
0
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A circulation on G
)( )(
)()()(vEe vEe
efefvf, mapping a with n togetherorientatioAn
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
flow a is then ,0 If ff
every for 1|)(|1 If eref
![Page 12: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/12.jpg)
1
1
15.1
5.1
flow-5.2circular A
x y
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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
![Page 14: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/14.jpg)
kG
k
c12)(
has 12 size ofcut edgean graph withA
![Page 15: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/15.jpg)
1
1
15.1
5.1
flow-5.2circular A
5.2)( Gc
x y
kG
k
c12)(
has 12 size ofcut edgean graph withA
![Page 16: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/16.jpg)
kG
k
c12)(
has 12 size ofcut edgean graph withA
kG
k-
c12)(
hasgraph connected edge4A
:[1981] ConjectureJaeger conjecture flow12 )k
(
conjecture flow3 case 1 k
conjecture flow5 case 2 k
trueif tight,
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ConjectureThomassen [2012]
kG
k-
c12)(
have graphs connected edge4
)3108( 2 kk
Theorem [Lovasz-Thomassen-Wu-Zhang, 2013]
k6
Theorem [Zhu, 2013]
)112( k have graphs signed
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A signed graph G
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A signed graph G
a positive edge a negative edge
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An orientation of a signed edge
a positive edge a negative edgex
x
y
y
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An orientation of a signed edge
a positive edge a negative edgex
xx
yyy
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An orientation of a signed edge
a positive edge a negative edgex
xx
yyy
x
x
y
y
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An orientation of a signed edge
a positive edge a negative edgex
xx
yyy
yyy
x
xx
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An orientation of a signed edge
a positive edge a negative edgex ye x ye
)()( yExEe x ye
x yex ye
x ye)()( yExEe
)()( yExEe
)()( yExEe
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A signed graph G
1 23
A circulation on G
, mapping a with n togetherorientatioAn
Rf: E(G)
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A signed graph G
1 23
3
4
12 13
1
A circulation on G
, mapping a with n togetherorientatioAn
Rf: E(G)
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A signed graph G
1 23
3
4
12 13
1
A circulation on G
, mapping a with n togetherorientatioAn
Rf: E(G)
)( )(
)()()(vEe vEe
efefvf
The boundary of f RVf :
0
00
0
1
1
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A circulation on G
)( )(
)()()(vEe vEe
efefvf, mapping a with n togetherorientatioAn
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
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1 23
2
4
12 13
1
0
00
0
0
0
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A circulation on G
)( )(
)()()(vEe vEe
efefvf, mapping a with n togetherorientatioAn
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
flow a is then ,0 If ff
every for 1-r|f(e)|1 If e
![Page 31: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/31.jpg)
A signed graph G
A flow on G
)( )(
)()(vEe vEe
efef, mapping
a with n togetherorientatioAn Rf: E(G)
1 23
2
4
12 13
1
Flip at a vertex x
change signs of edges incidentto x
x
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A signed graph G
A flow on G
)( )(
)()(vEe vEe
efef, mapping
a with n togetherorientatioAn Rf: E(G)
1 23
2
4
12 13
Flip at a vertex x
change signs of edges incidentto x
x1
![Page 33: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/33.jpg)
A signed graph G
A flow on G
)( )(
)()(vEe vEe
efef, mapping
a with n togetherorientatioAn Rf: E(G)
1 23
2
4
12 13
Flip at a vertex x
change signs of edges incidentto x
x1
![Page 34: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/34.jpg)
1
3
A signed graph G
A flow on G
)( )(
)()(vEe vEe
efef, mapping
a with n togetherorientatioAn Rf: E(G)
1 2
2
4
12 3
Flip at a vertex x
change signs of edges incidentto x
x1
![Page 35: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/35.jpg)
1
3
A signed graph G
A flow on G
)( )(
)()(vEe vEe
efef, mapping
a with n togetherorientatioAn Rf: E(G)
1 2
2
4
12 3
Flip at a vertex x
change signs of edges incidentto x
x1
![Page 36: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/36.jpg)
A signed graph G
A flow on G
)( )(
)()(vEe vEe
efef, mapping
a with n togetherorientatioAn Rf: E(G)
1 23
2
4
12 13
Flip at a vertex x
change signs of edges incidentto x
x1
![Page 37: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/37.jpg)
A flow on G
)( )(
)()(vEe vEe
efef, mapping
a with n togetherorientatioAn Rf: E(G)
1 23
2
4
12 13
Flip at a vertex x
change signs of edges incidentto x
x1
Change the directions of `half’ edges incident to x
![Page 38: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/38.jpg)
A flow on G
)( )(
)()(vEe vEe
efef, mapping
a with n togetherorientatioAn Rf: E(G)
1 23
2
4
12 13
Flip at a vertex x
change signs of edges incidentto x
x1
Change the directions of `half’ edges incident to x
![Page 39: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/39.jpg)
A flow on G
)( )(
)()(vEe vEe
efef, mapping
a with n togetherorientatioAn Rf: E(G)
1 23
2
4
12 13
Flip at a vertex x
change signs of edges incidentto x
x
Change the directions of `half’ edges incident to x
The flow remainsa flow 1
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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
![Page 41: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/41.jpg)
kG
k
c12)(
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
r 12
![Page 42: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/42.jpg)
kG
k-
c12)(
have graphs connected edge4
k6
Theorem [Zhu, 2013]
)112( k have graphs signed
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(
![Page 43: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/43.jpg)
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
flow12circular a is (e) )k
(k
fg(e)
![Page 44: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/44.jpg)
kG
k-
c12)(
have graphs connected edge6
Theorem [Loavsz-Thomassen-Wu-Zhang, 2013]
Ee
kZβ: V
0(e)
with any For 12
withn circulatio- specail a has 12 fZG k βf
withn circulatio- Zspecial a has 12k
fG
0 f
Theorem [Loavsz-Thomassen-Wu-Zhang, 2013]
Corollary
12in kZ
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kG
k-
c12)(
have graphs connected edge4
k6
Theorem [Zhu, 2013]
)112( k have graphs signed
unbalanced12y essentiall )-k(
flow-1)(2kinteger special a
Lemma 1. connected edge)112( k unbalanced-1)(2ky essentiall flow- special a have graphs 12 kZ
![Page 46: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/46.jpg)
Proof Assume G is (12k-1)-edge connected
essentially (2k+1)-unbalancedAssume 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 connected112 k
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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
![Page 48: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/48.jpg)
,Theorem-LTWZBy
GZf k in flow special a is g 12
fg gRG
with n circulatio Zspecial a has ][ 12k
![Page 49: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/49.jpg)
kG
k-
c12)(
have graphs connected edge4
k6
Theorem [Zhu, 2013]
)112( k have graphs signed
unbalanced12y essentiall )-k(
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
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then graph, a is If G
flow-1)(2k sflow- Zs 12k pecialpecial
For signed graphs
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flow- Zspecial NWZ 3
flow-1)(2k special NWZ
![Page 52: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/52.jpg)
then graph, a is If G
0)( vf 0)( vf
q ' q
flow-1)(2k sflow- Zs 12k pecialpecial
vu to frompath directed a is there,0)(,0)( ,, vfufvu
G
0 Assume f
![Page 53: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/53.jpg)
then graph, a is If G
0)( vf 0)( vf
q ' q
flow-1)(2k sflow- Zs 12k pecialpecial
vu to frompath directed a is there,0)(,0)( ,, vfufvu0 Assume f
![Page 54: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/54.jpg)
then graph, a is If G
0)( vf 0)( vf
q ' q
flow-1)(2k sflow- Zs 12k pecialpecial
vu to frompath directed a is there,0)(,0)( ,, vfufvu
![Page 55: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/55.jpg)
then graph, a is If G
0)( vf 0)( vf
q ' q
flow-1)(2k sflow- Zs 12k pecialpecial
vu to frompath directed a is there,0)(,0)( ,, vfufvu
' -1)(2k qq -1)(2k
![Page 56: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/56.jpg)
then graph, a is If G
0)( vf 0)( vf
' q
flow-1)(2k sflow- Zs 12k pecialpecial
vu to frompath directed a is there,0)(,0)( ,, vfufvu
' -1)(2k qq -1)(2k
![Page 57: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/57.jpg)
then graph, a is If G
0)( vf 0)( vf
flow-1)(2k sflow- Zs 12k pecialpecial
vu to frompath directed a is there,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
![Page 58: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/58.jpg)
then graph, a is If G
0)( vf 0)( vf
flow-1)(2k sflow- Zs 12k pecialpecial
vu to frompath directed a is there,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
![Page 59: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/59.jpg)
0)( vf 0)( vfG
0 with vertex a frompath directed a
by reached becan vertices
f(u)u
XX
][][
)()()(XXEeXXEeXv
efefvf 0
vu to frompath directed a is there,0)(,0)( ,, vfufvu
![Page 60: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/60.jpg)
0)( vf 0)( vfG
0 with vertex a frompath directed a
by reached becan vertices
f(u)u
XX
][][
)()()(XXEeXXEeXv
efefvf 0
vu to frompath directed a is there,0)(,0)( ,, vfufvu
For a signed graph
Such a path may not exist
![Page 61: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/61.jpg)
0)( vf 0)( vfG
Xin edgessink many
XX
][][
)()()(XXEeXXEeXv
efefvf 0
vu to frompath directed a is there,0)(,0)( ,, vfufvu
For a signed graph
Such a path may not exist
Xin edges sourcemany
![Page 62: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/62.jpg)
X X
G[Q]in n circulatio-1)(2k special a : f
)(|],[|)( XfXXEkX R
2)( if balanced is kXf Xany for
Xv
vfXf )()(
![Page 63: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/63.jpg)
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
sZ k
![Page 64: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/64.jpg)
flow-1)(2k pecial a tomdoified becan flow balanced specialA 3 Lemma 12
sZ k
The same proof as for ordinary graph
![Page 65: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/65.jpg)
flow balanced special a exists There Lemma 12 kZ
![Page 66: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/66.jpg)
flow balanced special a exists There 2 Lemma 12 kZ
G[R] are 6k-edge connected.
By Williams-Tutte Theorem
G[R] contains 3k edge-disjoint spanning treeskTTT 321 ,,,
![Page 67: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/67.jpg)
By Williams-Tutte Theorem
G[R] contains 3k edge-disjoint spanning treeskTTT 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
![Page 68: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/68.jpg)
flow balanced special a exists There 2 Lemma 12 kZ
cycleeulerian an :C
![Page 69: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/69.jpg)
flow balanced special a exists There 2 Lemma 12 kZ
cycleeulerian an :C
![Page 70: Xuding Zhu Zhejiang Normal University 2013.7 Budapest Circular flow of signed graphs](https://reader036.vdocuments.net/reader036/viewer/2022062317/5a4d1b147f8b9ab059990c0a/html5/thumbnails/70.jpg)
Thank you