Circular flow of signed graphs
Xuding Zhu
Zhejiang Normal University
2013.7
Budapest
G: a graph
A circulation on G
An orientatio n together with a
mapping f: E(G)  R,
G: a graph
1
2
1
1
1
An orientatio n together with a
mapping f: E(G)  R,  0
R
1
2
3
A circulation on G
1
2
G: a graph
y
-1
x
2
1
1
1
An orientatio n together with a
mapping f: E(G)  R,  0
R
1
2
3
A circulation on G
1
2
G: a graph
y
1
x
2
1
1
1
An orientatio n together with a
mapping f: E(G)  R,  0
R
1
2
3
A circulation on G
1
2
G: a graph
1
0
2
0
A circulation on G
An orientatio n together with a
mapping f: E(G)  R,
1
1
1
1
1 2
1 0
2
2
3
1
The boundary of f
f : V  R
f (v) 

eE  ( v )
f (e) 
 f (e)
eE  ( v )
 f (v)   f (e)   f (e)  0.
vV
eE
If f  0, then f is a flow
eE
1
0
2
0
1
1
1
1
1 2
1 0
2
2
3
1
A circulation on G
The boundary of f
An orientatio n together with a
f (v) : V  R
mapping f: E(G)  R,
f (v) 

eE  ( v )
f (e) 
 f (e)
eE  ( v )
 f (v)   f (e)   f (e)  0.
vV
eE
If f  0, then f is a flow
eE
A Γ  flow
1
 : an abelian group
0
2
0
A Γ A circulation on G
1
1
1
1
1 2
1 0
2
2
3
1
The boundary of f
An orientatio n together with a
f (v) : V  R
mapping f: E(G)  R,
f (v) 

eE  ( v )
f (e) 
 f (e)
eE  ( v )
If f  0, then f is a flow
1
0
2
0
1
1
1
1
1 2
1 0
2
2
3
1
A circulation on G
The boundary of f
An orientatio n together with a
f (v) : V  R
mapping f: E(G)  R,
f (v) 

eE  ( v )
f (e) 
 f (e)
eE  ( v )
If f  0, then f is a flow
1
0
2
0
0
1
2
0
1
1
1 0
1
2
3
0
A circulation on G
The boundary of f
An orientatio n together with a
f (v) : V  R
mapping f: E(G)  R,
f (v) 

eE  ( v )
f (e) 
 f (e)
eE  ( v )
If f  0, then f is a flow
If 1 | f (e) | r  1 for every e
then f is a circular r - flow
The circular flow number of G
Φc (G)  min r : G admits a circular r-flow 
A circulation on G
The boundary of f
An orientatio n together with a
f (v) : V  R
mapping f: E(G)  R,
f (v) 

eE  ( v )
f (e) 
 f (e)
eE  ( v )
1
1.5
x
1
1.5
1
A circular 2.5 - flow
y
Assume f is a circular r-flow
X
[X, X ] is a cut
flow from X to X
X
If a cut has exactly 2k  1 edges
assume | E[X  X] | k
| E[X  X ] | k  1
 flow from X to X
flow from X to X  (r  1)k
 flow from X to X  k  1
1
r  2
k
A graph with an edge cut of size 2k  1 has
 c (G )  2 
1
k
1
1.5
x
y
1
1.5
1
A circular 2.5 - flow
 c (G)  2.5
A graph with an edge cut of size 2k  1 has
 c (G )  2 
1
k
1
(
2

)  flow conjecture
Jaegerk Conjecture
[1981] :
A 4k-edge connected graph has
1
 c (G )  2 
k
tight, if true
k  1 case  3  flow conjecture
k  2 case  5  flow conjecture
A graph with an edge cut of size 2k  1 has
 c (G )  2 
1
k
Theorem
[Zhu,
2013]
Thomassen
[Lovasz-Thomassen-Wu-Zhang,
[2012]
2013]
Conjecture
12k 
6143k)ksigned have
graphs have
(8k 2 (10
) edge connected graphs
1
 c (G )  2 
k
A signed graph G









A signed graph G
a positive edge
a negative edge
An orientation of a signed edge
x
y
x
y
a positive edge
a negative edge
An orientation of a signed edge
x
y
x
y
x
y
a positive edge
a negative edge
An orientation of a signed edge
x
y
x
y
x
y
a positive edge
x
y
x
y
a negative edge
An orientation of a signed edge
y
x
y
x
y
x
y
x
y
x
y
x
a positive edge
a negative edge
An orientation of a signed edge
x
x
x
e
e
e


y e  E ( x)  E ( y )


x
y e  E ( x)  E ( y )
x
y
x
a positive edge
e
e
e
e  E  ( x)  E  ( y )
y
e  E  ( x)  E  ( y )
y
y
a negative edge
A signed graph G
1
2
3
A circulation on G
An orientatio n together with a
mapping f: E(G)  R,
A signed graph G
1
2
3
3
1
2 3
4
A circulation on G
An orientatio n together with a
mapping f: E(G)  R,
1
1
A signed graph G
1
0
3
1
3
1
1
2
1
0
0
2 3
1
4
0
The boundary of f
A circulation on G
An orientatio n together with a
mapping f: E(G)  R,
f : V  R
f (v) 

eE  ( v )
f (e) 
 f (e)
eE  ( v )
 f (v)  0.
If f  0, then f is a flow
vV
1
0
3
1
3
1
1
2
1
0
0
2 3
1
4
0
A circulation on G
The boundary of f
An orientatio n together with a
f (v) : V  R
mapping f: E(G)  R,
f (v) 

eE  ( v )
f (e) 
 f (e)
eE  ( v )
1
0
3
0
2
0
1
2
1
0
0
2 3
1
4
0
If f  0, then f is a flow
If 1 | f(e) | r - 1 for every e
f is a circular r - flow
The circular flow number of G
Φc (G)  min r : G admits a circular r-flow 
A circulation on G
The boundary of f
An orientatio n together with a
f (v) : V  R
mapping f: E(G)  R,
f (v) 

eE  ( v )
f (e) 
 f (e)
eE  ( v )
A signed graph G
1
2
3
2
1
1
2 3
Flip at a vertex x
x
1
change signs
of edges incident
to x
4
A flow on G
An orientatio n together with a
mapping f: E(G)  R,

eE  ( v )
f (e) 
 f (e)
eE  ( v )
A signed graph G
1
2
3
2
1
1
2 3
Flip at a vertex x
x
1
change signs
of edges incident
to x
4
A flow on G
An orientatio n together with a
mapping f: E(G)  R,

eE  ( v )
f (e) 
 f (e)
eE  ( v )
A signed graph G
1
2
3
2
1
1
2 3
Flip at a vertex x
x
1
change signs
of edges incident
to x
4
A flow on G
An orientatio n together with a
mapping f: E(G)  R,

eE  ( v )
f (e) 
 f (e)
eE  ( v )
A signed graph G
1
2
3
2
1
1
2 3
Flip at a vertex x
x
1
4
A flow on G
An orientatio n together with a
mapping f: E(G)  R,

eE  ( v )
f (e) 
change signs
of edges incident
to x
 f (e)
eE  ( v )
A signed graph G
1
2
3
2
1
Flip at a vertex x
1
2 3
x
1
change signs
of edges incident
to x
4
A flow on G
An orientatio n together with a
mapping f: E(G)  R,

eE  ( v )
f (e) 
 f (e)
eE  ( v )
A signed graph G
1
2
3
2
1
1
2 3
Flip at a vertex x
x
1
change signs
of edges incident
to x
4
A flow on G
An orientatio n together with a
mapping f: E(G)  R,

eE  ( v )
f (e) 
 f (e)
eE  ( v )
Change the directions
of `half’ edges incident to x
1
2
3
2
1
1
2 3
Flip at a vertex x
x
1
change signs
of edges incident
to x
4
A flow on G
An orientatio n together with a
mapping f: E(G)  R,

eE  ( v )
f (e) 
 f (e)
eE  ( v )
Change the directions
of `half’ edges incident to x
1
2
3
2
1
1
2 3
Flip at a vertex x
x
1
change signs
of edges incident
to x
4
A flow on G
An orientatio n together with a
mapping f: E(G)  R,

eE  ( v )
f (e) 
 f (e)
eE  ( v )
Change the directions
of `half’ edges incident to x
The flow remains
a flow
1
2
3
2
1
1
2 3
Flip at a vertex x
x
1
change signs
of edges incident
to x
4
A flow on G
An orientatio n together with a
mapping f: E(G)  R,

eE  ( v )
f (e) 
 f (e)
eE  ( v )
G  G'
G can be obtained from G’ by
a sequence of flippings
Fliping at vertices in X
change the sign of edges in
G  G'
E[ X , X ]
signs of edges in G and G '
disagrees on E[X,X ] for some X
Observatio n
This
is a source
( f(e)  0)
is a sink
This
 f(e)   f(e)
If f is a circular r - flow, then
e is a sink
e is a source
Assume G has exactly 2k  1 negative edges
# source edges # sink edges
 f(e)   f(e)  (r  1)k
1
r  2
e is a sink
e is a source
k
A graph
an 2edge
of sizeedges
2k  1 has
A signed graph
withhaving
exactly
k  1cut
negative
1
 c (G )  2 
k
k 1 
Theorem [Zhu, 2013]
essentiall y ( 2k  1 )-unbalanced
(12k 614k)k-edge connected graphs
signed have
graphs have
1
 c (G )  2 
k
One technical requirement is missing
A signed graph G is essentiall y ( 2k  1 )-unbalanced
if any G'  G either has an even number of negative edges
or at least 2k  1 negative edges
Z 2k 1)
flow nn f is special if
An integer A
(2k
circulatio
flow
1 --circulatio
 c (G )  2 
f (e) k, k  1
1
k
G has a special integer (2k  1) - flow
g(e) 
f (e)
1
is a circular ( 2  )  flow
k
k
Theorem [Loavsz-Thomassen-Wu-Zhang, 2013]
Corollary
6k-edge connected graphs have
1
 c (G )  2 
k
G has a special Z 2k 1 - circulatio n
f with
f  0
Theorem [Loavsz-Thomassen-Wu-Zhang, 2013]
For any β: V  Z 2 k 1 with
  (e)  0
in Z 2 k 1
eE
G has a specail Z 2k 1 - circulatio n f with
f  β
Theorem [Zhu, 2013]
essentiall y ( 2k  1 )-unbalanced
(12k 614k)k-edge connected graphs
signed have
graphs have
a special integer (2k1 1) - flow
 c (G )  2 
k
Lemma 1. (12k  1)  edge connected essentiall y (2k  1) - unbalanced
graphs have a special Z 2 k 1 - flow
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
12k 1
If |Q| is even,
then # source edges  # sink edges
f (e)  k for all e  Q
If |Q| is odd,
then # source edges  # sink edges  1
As G is essentiall y ( 2k  1 ) - unbalanced , # sink edges  k
f (e)  k for all e  Q, except tha t k sink edges e have f(e)  k  1
f : Q  k , k  1

vV
f (v)  0
By LTWZ - Theorem ,
G[ R] has a special Z 2k1  circulatio n g
with g  f
f  g is a special Z 2 k 1  flow in G
Theorem [Zhu, 2013]
essentiall y ( 2k  1 )-unbalanced
(12k 614k)k-edge connected graphs
signed have
graphs have
a special integer (2k1 1) - flow
 c (G )  2 
k
Lemma 1. (12k  1)  edge connected essentiall y (2k  1) - unbalanced
graphs have a special Z 2 k 1 - flow
To prove Theorem above, we need
(12k  1)  edge connected essentiall y (2k  1) - unbalanced
(2k  1) - flow
graphs have a special Zinteger
2 k 1 - flow
If Gsigned
is a graph,graphs
then
For
special Z2k1 - flow  special (2k  1) - flow
NWZ special Z3 - flow
NWZ special (2k  1) - flow
If G is a graph, then
special Z2k1 - flow  special (2k  1) - flow
Assume f  0
u, v, f (u )  0, f (v)  0,
there is a directed path from u to v
G
f (v)  0
f (v)  0
q'
q
If G is a graph, then
special Z2k1 - flow  special (2k  1) - flow
Assume f  0
u, v, f (u )  0, f (v)  0,
there is a directed path from u to v
f (v)  0
f (v)  0
q'
q
If G is a graph, then
special Z2k1 - flow  special (2k  1) - flow
u, v, f (u )  0, f (v)  0,
there is a directed path from u to v
f (v)  0
f (v)  0
q'
q
If G is a graph, then
special Z2k1 - flow  special (2k  1) - flow
u, v, f (u )  0, f (v)  0,
there is a directed path from u to v
f (v)  0
f (v)  0
(2k  1)
q ' - q'
q
(2k  1) - q
If G is a graph, then
special Z2k1 - flow  special (2k  1) - flow
u, v, f (u )  0, f (v)  0,
there is a directed path from u to v
f (v)  0
f (v)  0
(2k  1)
q ' - q'
(2k  1) - q
If G is a graph, then
special Z2k1 - flow  special (2k  1) - flow
u, v, f (u )  0, f (v)  0,
there is a directed path from u to v
X
f (v)  0
If such a path
does not exist
G
vertices can be reached by
f (v)  0
a directed path from
a vertex u with f(u)  0
If G is a graph, then
special Z2k1 - flow  special (2k  1) - flow
u, v, f (u )  0, f (v)  0,
there is a directed path from u to v
X
f (v)  0
If such a path
does not exist
G
vertices can be reached by
f (v)  0
a directed path from
a vertex u with f(u)  0
 f (v)   f (e)   f (e)  0
vX
eE [ X  X ]
eE [ X  X ]
u, v, f (u )  0, f (v)  0,
there is a directed path from u to v
X
f (v)  0
X
G
vertices can be reached by
f (v)  0
a directed path from
a vertex u with f(u)  0
 f (v)   f (e)   f (e)  0
vX
eE [ X  X ]
For a signed graph
eE [ X  X ]
Such a path
may not exist
u, v, f (u )  0, f (v)  0,
there is a directed path from u to v
X
f (v)  0
X
G
vertices can be reached by
f (v)  0
a directed path from
a vertex u with f(u)  0
 f (v)   f (e)   f (e)  0
vX
eE [ X  X ]
For a signed graph
eE [ X  X ]
Such a path
may not exist
u, v, f (u )  0, f (v)  0,
there is a directed path from u to v
X
f (v)  0
many source edges in X
G
X
f (v)  0
many sink edges in X
f : a special (2k  1) - circulatio n in G[Q]
f ( X )   f (v)
vX
( X )  k | ER [ X , X ] | f ( X )
f is balanced if ( X )  k  2
X
for any X
X
f ( X )   f (v)
vX
( X ) | E[ X , X ] | f ( X )
f is balanced if ( X )  k  2
for any X
Lemma 2 There exists a special balanced Z 2 k 1  flow
Lemma 3 A special balanced Z 2 k 1  flow can be mdoified to
a special (2k  1) - flow
The same proof as for ordinary graph
Lemma 3 A special balanced Z 2 k 1  flow can be mdoified to
a special (2k  1) - flow
Lemma There exists a special balanced Z 2 k 1  flow
Lemma 2 There exists a special balanced Z 2 k 1  flow
G[R] are 6k-edge connected.
By Williams-Tutte Theorem
G[R] contains 3k edge-disjoint spanning trees
T1 , T2 ,, T3k
Lemma There exists a special balanced Z 2 k 1  flow
T1  G[Q] is connected
T2 contains a parity subgraph F of T1  G[Q]
T1  G[Q]  F is eulerian
C : an eulerian cycle
orient the negative edges on C alternatel y sink or source
By Williams-Tutte Theorem
G[R] contains 3k edge-disjoint spanning trees
T1 , T2 ,, T3k
Lemma 2 There exists a special balanced Z 2 k 1  flow
C : an eulerian cycle
Lemma 2 There exists a special balanced Z 2 k 1  flow
C : an eulerian cycle
Thank you