decompositions of graphs into closed trails of even sizes sylwia cichacz agh university of science...

Decompositions of graphs into closed trails of even sizes

Sylwia Cichacz

AGH University of Science and Technology, Kraków, Poland

Decompositions of pseudographs

Decompositions of complete bipartite digraphsand even complete bipartite multigraphs

Part 2

Part 3

Part 1

Part 4

Definition

Problem

Definition

• graph G of size ||G||

Decomposition

• sequence (t1,...,tp) ||||1

Gtp

ii

•

there is a closed trail of length in (for all ).it G i•

Df. 1. G is ,

arbitrarily decomposable into closed trails

iff G can be edge-disjointly decomposed into closed trails (T1,...,Tp) of lengths (t1,...,tp) resp.

• graph G of size ||G||=12

Example

• sequence (6,6) • sequence (4,4,4)

• there are closed trails of lengths 4,6,8 in G

G

• sequence (4,8)

Observation

If G is arbitrarily decomposable into closed trails,

then G is eulerian.

• there is a closed trail of length 3 in K4

K4 can not be edge-disjointly

decomposed into closed trails of lengths (3,3).

K4

Decompositions of pseudographs

Irregular coloring

1

1

1

22

2

}1,1,1{

}2,2,2{

}2,1{

1

}1,1{

}1{

2

}2,1{

2}2{

1}2,2,1{

1

}1,1{

2}2,2,1{

3

}3,1{

}3,2,1{

1

1

12

2 2}1,1,1{

}2,2,2{

1 1

2

2}1{

}2{

}2,1{

}1,1{

}2,2{

}2,2,1{

}2,1,1{

)(Gc

G

- irregularnumber

2)( Gc

Results

• 2-regular graph of size

ptt CCG 1

ntGp

ii

1

T.1. [M. Aigner, E. Triesch, Zs. Tuza, 1992]:

)1(8)( nGc

T.2. [P. Wittmann, 1997]:

)1(2)( nGc

ptt ,,1 - even

T.3. [S. C., J. Przybyło, M. Woźniak, 2005]:

nn 2,12 n2 12, n

)(Gc

)(Gc

nGc 2)(

Correspondence664 CCCG

)(Gc ?6

6M

5M

664155 M6

6L

5L

{ }

Results

2

nL• n is odd, or , or3

2

n

L

.22nn

L

• n is even, - irregular number for proper coloring

)(' Gs

,1

p

iitLT.4. [P.N. Balister, 2001]. Let .3it

nKThen we can write same subgraph of as an edge

ptt ,,1 disjoint union of circuits of lengths iff either:

T.5. [P.N. Balister, B. Bollobás, R.H. Schelp,2002] Let G be a 2-regular graph of order n. Then

242)(' nGs

Results

ptt CCG 1

• ptt ,,1 - even•

L.1. If , then we can edge-disjointlyk

p

ii Lt

1pack closed trails of lengths into .kLptt ,,1

L.2. If , then is edge-disjointlyk

p

ii Lt

1kL

decomposable into closed trails of lengths .ptt ,,1

k )6( k- even•

The graph is edge-disjointly decomposablebaK ,

into closed trails of lengths iff:q ,,1

T.5. [M. Horňák, M. Woźniak, 2003]:

ba, - even• baq

jj

1

•

j baK , j• there is a closed trail of length in (for all ).

4it•

ProofL.2. If , then is edge-disjointlyk

p

ii Lt

1kL

decomposable into closed trails of lengths .ptt ,,1

kL

2kL

4kL4L2,2 kK4,4 kK

Proof:

2k4k

Proof

L.1. If , then we can edge-disjointlyk

p

ii Lt

1pack closed trails of lengths into .kLptt ,,1

L.2. If , then is edge-disjointlyk

p

ii Lt

1kL

decomposable into closed trails of lengths .ptt ,,1 k

p

ii Lt

1

•

21

k

p

ii Lt•

6L

Application

ptt CCG 1

• ptt ,,1 - even•

kGc )(

2kLkL1kL

ntGp

ii

1

•

kkGc ,1)(

n n

nn 2,12 n2 12, n

n2

k )6( k- even•

Exception

4L

5)( 44 CCc

4)( 8 Cc 8C

44 CC

n2

12 n

Decompositions of complete bipartite digraphsand even complete bipartite

multigraphs

Definitions

- digraph obtained from graph G by replacing each edge by the pair of arcs xy and yx.)(, GEyx G�

- multigraph where each edge xy occurs with multiplicity r.Gr

Reminder

,1

p

iitLT.4. [P.N. Balister, 2001]. Let .3it

nKThen we can write same subgraph of as an edge

ptt ,,1 disjoint union of circuits of lengths iff either:

2

nL• n is odd, or , or3

2

n

L

.22nn

L

• n is even,

,2

21

p

ii

nt ,2it

nK�

,,,1 ptt

.3it

then can be decomposed as edge-disjoint

except in the case when n=6 and all

T.7. [P.N. Balister, 2003] If

union of directed closed trails of lengths

,2

,31

p

ii

nrtn

.2it

of closed trails of lengths iff either ptt ,,1 n

r KThen can be written as edge-disjoint union

T.8. [P.N. Balister, 2003] Assume

a) r is even, or

2 2it

i

ntb) r and n are both odd and

into closed trails of lengths iff:ptt ,,1 T.9. The digraph is edge-disjointly decomposablebaK ,

�

bratp

ii

1

•

Results

j baK , j• there is a closed trail of length in (for all ).

.

The graph is edge-disjointly decomposablebaK ,

into closed trails of lengths iff:q ,,1

T.6. [M. Horňák, M. Woźniak, 2003]

ba, - even• baq

jj

1

•

it - even• batp

ii

2

1•

T.10. Let

into closed trails of lengths iff:ptt ,,1 The multigraph is edge-disjointly decomposableba

r K ,

• r - odd • a,b - even it - even•

abtit

i 2

• if 4, baabtt

ii ti

ti

)4(mod2)4(mod0)2(• if a=2 or b=2

• we fix the number of the vertex set B and will argue on induction on a

Proof

Proof:

• bK ,1

�• 2ainto closed trails of lengths iff:ptt ,,1 T.9. The digraph is edge-disjointly decomposablebaK ,

�

it - even• batp

ii

2

1•

bK ,1

�baK ,

�baK ,1

�baK ,

�

Proofinto closed trails of lengths iff:ptt ,,1 T.9. The digraph is edge-disjointly decomposablebaK ,

�

it - even• batp

ii

2

1•

k

ii btk

12: (t1,…tk) (tk+1,…tp)

bK ,1

�baK ,1

�

Proof

bK ,1

�

into closed trails of lengths iff:ptt ,,1 T.9. The digraph is edge-disjointly decomposablebaK ,

�

it - even• batp

ii

2

1•

k

ibtk

11 2: and

1

11 2

k

ibt ''' kkk ttt

1

12'

k

iki btt 2'',' kk tt

'',' kk tt - even

)',,...,( 11 kk ttt ),...,,'( 1 pkk ttt

baK ,1

�

bK ,1

�baK ,1

�

'kT

''kT

v ww

baK ,

�

kT

into closed trails of lengths iff:qtt ,,1 T.9. The digraph is edge-disjointly decomposablebaK ,

�

it - even• batp

ii

2

1•

The multigraph is edge-disjointly decomposable

into closed trails of lengths iff:ptt ,,1

Observation 11. Let r be even.

bar K ,

Results

it - even• abrtp

ii

1•

Proof

The multigraph is edge-disjointly decomposable

into closed trails of lengths iff:ptt ,,1

Observation 11. Let r be even.

bar K ,

it - even• abrtp

ii

1

•

Proof:

• we consider as an edge-disjoint union of and

bar K ,

baK ,2

bar K ,

2

T.9. (digraphs) induction on r

T.10. Let

• a,b - even• r - odd bratp

ii

1• it - even•

into closed trails of lengths iff:ptt ,,1 The multigraph is edge-disjointly decomposableba

r K ,

abttii t

it

i )4(mod2)4(mod0

)2(• if a=2 or b=2

abtit

i 2

• if 4, ba

Proof

6),62,6,2( )1(

2 bbbr

barK ,)2( 2

1

rab

b

r K ,2

The multigraph is edge-disjointly decomposableba

r K ,

T.10. Let • a,b - even• r - odd bratp

ii

1• it - even•

into closed trails of lengths iff:ptt ,,1 abtt

ii ti

ti

)4(mod2)4(mod0)2(• if a=2 or b=2

abtit

i 2• if 4, ba

Proof

• we consider as an edge-disjoint union of and ba

r K ,1

bar K ,

baK ,

• Case 1. a=2 or b=2

• Case 2. 4, ba

T.6. [M. Horňák, M. Woźniak] Ob.11. (for even multiplicity)


r K ,


ii

1• it - even•


ii ti

ti

)4(mod2)4(mod0)2(• if a=2 or b=2

abtit

i 2• if 4, ba

Proof

• Case 1. a=2 or b=2 mtiMij

,...,1)4(mod0:1

rmtiMij

,...,12:2

rmtiMij

,...,1)4(mod2:3

Let be the smallest integer such thatrk 2)2(

11

k

rii

m

ii jj

tt

2'',2' iii jjj ttt for i=r+1,…,k


r K ,


ii

1• it - even•


ii ti

ti

)4(mod2)4(mod0)2(• if a=2 or b=2

abtit

i 2• if 4, ba

Proof

• Case 1. a=2 or b=2 mtiMij

,...,1)4(mod0:1

rmtiMij

,...,12:2

rmtiMij

,...,1)4(mod2:3

2'',2' iii jjj ttt for i=r+1,…,k

bK ,2b

r K ,21


r K ,


ii

1• it - even•


ii ti

ti

)4(mod2)4(mod0)2(• if a=2 or b=2

abtit

i 2• if 4, ba

Proof

ptt ...1

Let be the smallest integer such thatpk abtk

ii

1

1

12

k

ii abt '',' kk tt - even

)',,...,( 11 kk ttt ),...,,''( 1 pkk ttt

abttttt k

k

iikkk

':'''

1

1

0'',4' kk tt

baK , bar K ,

• Case 2. 4, ba


r K ,


ii

1• it - even•


ii ti

ti

)4(mod2)4(mod0)2(• if a=2 or b=2

abtit

i 2• if 4, ba

Proof

ptt ...1

1

12

k

ii abt

'',' kk tt - even

)',,...,( 11 kk ttt ),...,,'( 21 pkk ttt

2',6' 1 kk tt

baK ,ba

r K ,

1

• Case 2. 4, ba

2',2' 11 kkkk tttt

baK ,ba

r K ,1

'kT '1kT kT

1kT

Problem

Problem• sequence (t1,...,tp), .3it

• graph Ln, n>2.

itf

3it

4it

5it

6it

30 it41 it

52

ii t

t


• graph Ln, n>2.

itf

30 it41 it

52

ii t

t

Necessity:

p

ini Lt

1•

p

ii ntf

1)(•

IT IS NOT ENOUGH?


• graph Ln, n>2.

Necessity:

p

ini Lt

1•

p

ii ntf

1)(•

Example

,186 L (3 , 3 , 6 , 6 )

6)(4

1

iitf

6L

Thank you very very much!!

decompositions of graphs into closed trails of even sizes sylwia cichacz agh university of science...

Documents