A circulation on G
-1
11
122
3
121
, mapping
a with n togetherorientatioAn
Rf: E(G) 0R
G: a graph
x
y
A circulation on G
1
11
122
3
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
A circulation on G
1
11
122
3
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
A circulation on G
1
11
122
3
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
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
1
11
122
3
121
1
2
1
0
0
0
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
1
12
112
3
121
0
0
0
0
0
0
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
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
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
An orientation of a signed edge
a positive edge a negative edge
x ye
x ye
)()( yExEe x ye
x ye
x ye
x ye
)()( yExEe
)()( yExEe
)()( yExEe
A signed graph G
1 23
3
4
12 13
1
A circulation on G
, mapping
a with n togetherorientatioAn
Rf: E(G)
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
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
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
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
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
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
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
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
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
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
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
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
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)(
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(
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)(
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
with
n circulatio- Zspecial a has 12k
f
G
0 f
Theorem [Loavsz-Thomassen-Wu-Zhang, 2013]
Corollary
12in kZ
kG
k-
c
12)(
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
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
kG
k-
c
12)(
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
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
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
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
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
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
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
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
0)( vf 0)( vf
G
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
0)( vf 0)( vf
G
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
0)( vf 0)( vf
G
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
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 2 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