Decompositions of graphs into
closed trails of even sizes
Sylwia Cichacz
AGH University of Science and Technology, Kraków, Poland
Part 1
Definition
Part 2
Decompositions of pseudographs
Part 3
Decompositions of complete bipartite digraphs
and even complete bipartite multigraphs
Part 4
Problem
Definition
Decomposition
• graph G of size ||G||
• sequence (t1,...,tp)
p
•  ti
|| G ||
ti
G
i 1
• there is a closed trail of length
in
(for all 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.
Example
• graph G of size ||G||=12 • there are closed
trails of lengths
4,6,8 in G
• sequence (4,8)
(6,6)
(4,4,4)
G
Observation
If G is arbitrarily decomposable into closed trails,
then G is eulerian.
K4
K4 can not be edge-disjointly
decomposed into closed trails
of lengths (3,3).
• there is a closed trail of length 3 in K4
Decompositions
of pseudographs
Irregular coloring
{1,12}}
1
{1,1,1}
1
21
{1,1,2}
1
{1,1,{1}2,2,2}
c(G ) - irregular
13
2
number 1
{1}
{2,2}
2
2
2
G
1
{1}
2
{11,2}
1
{1,2,2
3}
{1,1}
2
c{1(,31G
})2
1
{1,2,2}
{2}
2
2
1
2
{1{1,2,2},2}
{2}
2
{2,2,2}
Results
• 2-regular graph G  Ct1   Ct p of size
p
G   ti  n
i 1
t1 , , t p - even
T.1. [M. Aigner, E. Triesch, Zs. Tuza, 1992]:
T.3. [S. C., J. Przybyło, M. Woźniak, 2005]:
c(G )  8n  (1)

c(G )  2n


T.2. [P. Wittmann, 1997]:
   
c(G )   2n  ,  2n   1
c(G ) 
c(G )  22nn 1,(1) 2n

Correspondence
G  C4  C6  C6
{
}
M
L66
c(G ) 
?
6
M 5  15  4  6  6
M 5  L5
Results
p
T.4. [P.N. Balister, 2001]. Let L   ti , ti  3.
i 1
Then we can write same subgraph of K n as an edge
disjoint union of circuits of lengths t1 ,, t p iff either:
n
n

• n is odd, L    or L     3 , or
 2
 2
n n
• n is even, L     .
 2 2
 s ' (G) - irregular number
for proper coloring
T.5. [P.N. Balister, B. Bollobás, R.H. Schelp,2002]
Let G be a 2-regular graph of order n. Then
 s ' (G)  2n  24
Results
• G  Ct1  Ct p
• k - even
( k  6)
• t1 , , t p - even
• ti  4
p

L.1. If
ti  Lk , then we can edge-disjointly
T.5. [M.i 1
Horňák, M. Woźniak, 2003]:
pack closed trails of lengths t1 , , t p into Lk .
The graph K a ,b is edge-disjointly decomposable
p
edge-disjointly
, q iff:
into
L.2.closed
If
then Lk 1 ,is
ti trails
 Lk of, lengths

q
i 1
a, b - even
•decomposable
 j  atrails
 b of lengths t1 , , t p .
•
into
closed
j 1
• there is a closed trail of length  j in K a ,b (for all j ).
Proof
L.2. If
p
t
i 1
i
 Lk , then Lk is edge-disjointly
decomposable into closed trails of lengths t1 , , t p .
Proof:
Lk

K 2 ,k  2

kL
k 2
2
L4
K 4 ,k  4
k 4
Lk 4
Proof
p
t
L.1. If
i 1
 Lk , then we can edge-disjointly
i
packp closed trails of lengths t1 , , t p into Lk .
t  L 2

L.2. If  t  L , then L is edge-disjointly
• t  L
decomposable into closed trails of lengths t , , t
•
i
ip1
i 1
i
p
i 1
k
i
k
k
k
L6
1
p.
Application
• G  Ct1  Ct p
•
t1 , , t p - even
p
• G   ti  n
• k - even
( k  6)
i 1
Lk 2
n
c(G )  k  1, k 
 2n  1,  2n 
 2n  ,  2n  1
Lk 1
n
c (G )  k 
Lk

2n 
Exception
c(C8 )  4   2n 
c(C4 C
 C4 )  5   2n   1
L4
Decompositions
of complete bipartite
digraphs
and even complete bipartite
multigraphs
Definitions

G - digraph obtained from graph G by replacing each
edge x, y  E (G ) by the pair of arcs xy and yx.
r
G - multigraph where each edge xy
occurs with multiplicity r.
Reminder
p
T.4. [P.N. Balister, 2001]. Let L   ti , ti  3.
i 1
Then we can write same subgraph of K n as an edge
p
n

disjoint
union
of circuits
of Assume
lengths t1n,
 3, t, p iff
ti either:
r  ,
T.8. [P.N.
Balister,
2003]
p
i 1
n
2




n
n




t

2
,
t

2
,



i
[P.N.
2003]
If 3 , ori
r Balister,
•ti nT.7.
is
odd,
or
L


L



K
2. Then
1
as iedge-disjoint
union
2

 ncan
2  be written
 2
then K n can be decomposed
as edge-disjoint
t1 ,, t p iff either
of closed trailsof
n lengths
n
Lor
   closed
. trails of lengths t ,, t ,
•a)nunion
of directed
risiseven,
even,
1
p
2
2
 
n
  all
 tiand
 ti  3.
theboth
caseodd
when
b)except
r and ninare
andn=6
ti  2
 2
Results
T.6. [M. Horňák, M. Woźniak, 2003]
The graph K a ,b is edge-disjointly decomposable

 , , q iff: decomposable
intoThe
closed
trails K
ofa ,blengths
T.9.
T.10.
Letdigraph
is edge-disjointly
p 1
q
.
,ra
b • ti - even
t p iff:
• t1t,i
•a,r b-closed
odd
•trails
a,b -of
even
lengths
•into
- even
•
r
a b
p  
j
i 1
The multigraph K a ,bj 1is edge-disjointly decomposable
- even
• ofti length
t
 2a  b in K (for all j ).
•
i
• there is a closed trail
a ,b
j
i 1
into closed trails of lengths
iff:
t1,, t p
•  ti  ab if a, b  4
ti 2
 ti 
 (ti  2)  ab if a=2 or b=2
• t 0 (mod
4)
t  2 (mod 4 )
i
i
Proof

Proof:
T.9.
The digraph K a ,b is edge-disjointly decomposable
t
,

,
t
• iawe
the number
the vertex
set
B
and will argue
nto
trails ofof
lengths
iff:
 fix
2closed
1
p

on induction on a p • K1,b
•  ti  2a  b
• ti - even
i 1

K a ,b

K1,b

K a1,b
Proof

T.9. The digraph K a ,b is edge-disjointly decomposable
into closed trails of lengths t1 ,, t p iff:
•t
p
i
- even
k
• t
i 1
i
 2a  b
k :  ti  2b
i 1
(t1,…tk) (tk+1,…tp)

K1,b

K a1,b
Proof

T.9. The digraph K a ,b is edge-disjointly decomposable
into closed trails of lengths t1 ,, t p iff:
•t
p
i
- even
• t
i 1
i
 2a  b
k
k 1
i 1
i 1
k :  t1  2b and

Tk '

TTkk' '
vw
 t1  2b
k 1
 ti t k '  2b
i 1
tk ' , tk ' '  2
tk ' , tk ' ' - even
w (t1 ,..., tk 1 , tk ' ) (tk ' , tk 1 ,..., t p )

K1,b

K1,b
tk  tk 'tk ' '
 
K aK
1,ba ,b

K a1,b
Results

Observation
r be even. decomposable
K a ,b Let
T.9.
The digraph11.
is edge-disjointly
r
t1 ,, tq iff:decomposable
The
is edge-disjointly
intomultigraph
closed trailsKof
a ,b lengths
p
into• closed
trails of• 
lengths
ti - even
ti  2at1,
b , t p iff:
i 1
• ti - even
p
•  ti  r  ab
i 1
Proof
Observation 11. Let r be even.
Proof:
r
K a ,b is edge-disjointly decomposable
The multigraph
r
• we consider K a,b as an edge-disjoint union
K of lengths t1 ,, t p iff:
K a ,b andtrails
of 2closed
into
r 2
a ,b
p
• ti - even induction
•  ton
 rr  ab
i
T.9. (digraphs)
i 1
Proof
T.10. Let
p
• r - odd • a,b - even •  ti  ra  b • ti - even
The multigraph
r
i 1
K a ,b is edge-disjointly decomposable
t1,, t p iff:
into
 closed trails of lengths
 ti  r K (ti  2)  ab if a=2 or b=2
• 1t 0((mod
2 4))
a ,b 4 )
t  2 (mod
rab
2
i
i
1) b
 •2 (2t(ir 
,6,2bif 6a),, bb  64
ab
ti 2
r
K 2,b
Proof
T.10. Let
p
• r - odd • a,b - even •  ti  ra  b • t i - even
i 1
The multigraph K a ,b is edge-disjointly decomposable
into closed trails of lengths t1 ,, t p iff:

ti   (ti  2)  ab if a=2 or b=2
r
•
•
ti 0 (mod 4 )

ti 2
ti  2 (mod 4 )
ti  ab if a, b  4

• we consider
of K a,b and
r
r 1
K a ,b as an edge-disjoint union
K a ,b
T.6. [M. Horňák, M. Woźniak] Ob.11. (for even multiplicity)
• Case 1. a=2 or b=2
• Case 2. a, b  4
Proof
T.10. Let
p
• r - odd • a,b - even •  ti  ra  b • t i - even
i 1
The multigraph K a ,b is edge-disjointly decomposable
into closed trails of lengths t1 ,, t p iff:

ti   (ti  2)  ab if a=2 or b=2
r
•
•
ti 0 (mod 4 )

ti 2
ti  2 (mod 4 )
ti  ab if a, b  4
• Case 1. a=2 or b=2 M1  i : t j  0(mod 4)  1,...,m
i
M 2  i : t ji  2  m  1,...,r
M 3  i : t ji  2(mod 4)  m  1,..., r
Let k  r be the smallest integer such that
m
k
 ti j   (ti j  2)  2
i 1
i r 1
t ' ji  t ji  2, t ' ' ji  2 for i=r+1,…,k
Proof
T.10. Let
p
• r - odd • a,b - even •  ti  ra  b • t i - even
i 1
The multigraph K a ,b is edge-disjointly decomposable
into closed trails of lengths t1 ,, t p iff:

ti   (ti  2)  ab if a=2 or b=2
r
•
•
ti 0 (mod 4 )

ti 2
ti  2 (mod 4 )
ti  ab if a, b  4
• Case 1. a=2 or b=2 M1  i : t j  0(mod 4)  1,...,m
i
M 2  i : t ji  2  m  1,...,r
M 3  i : t ji  2(mod 4)  m  1,..., r
t ' ji  t ji  2, t ' ' ji  2 for i=r+1,…,k
K 2 ,b
r 1
K 2 ,b
Proof
T.10. Let
p
• r - odd • a,b - even •  ti  ra  b • t i - even
i 1
The multigraph K a ,b is edge-disjointly decomposable
into closed trails of lengths t1 ,, t p iff:

ti   (ti  2)  ab if a=2 or b=2
r
•
•
ti 0 (mod 4 )

ti 2
ti  2 (mod 4 )
ti  ab if a, b  4
• Case 2. a, b  4
t1  ...  t p
k
Let k  p be the smallest integer such that  ti  ab
k 1
 ti  ab  2
i 1
i 1
k 1
tk  t 'k t ' 'k :  ti  t 'k  ab
i 1
t 'k  4, t ' 'k  0
(t1 ,..., tk 1 , t 'k ) (t ' 'k , tk 1,..., t p )
K a ,b
tk ' , tk ' ' - even
r
K a ,b
Proof
T.10. Let
p
• r - odd • a,b - even •  ti  ra  b • t i - even
i 1
The multigraph K a ,b is edge-disjointly decomposable
into closed trails of lengths t1 ,, t p iff:

ti   (ti  2)  ab if a=2 or b=2
r
•
•
ti 0 (mod 4 )

ti 2
ti  2 (mod 4 )
ti  ab if a, b  4
• Case 2. a, b  4
t1  ...  t p
k 1
 ti  ab  2 tk '  tk  2, t 'k 1  tk 1  2
i 1
Tk '
Tk 1 ' Tk
tk ' , tk ' ' - even
t 'k  6, t 'k 1  2
(t1 ,..., tk 1 , t 'k ) (t 'k 1 , tk 2 ,..., t p )
K a ,b
K a ,b
Tk 1
r 1
K a ,b
r 1
K a ,b
Problem
Problem
• sequence (t1,...,tp), ti  3.
ti  3
• graph Ln, n>2.
ti  3
0
1
t

4
i
f ti   
 ti 
  ti  5
2


ti  4
ti  5
ti  6
Problem
• sequence (t1,...,tp), ti  3.
• graph Ln, n>2.
3
ti  4
ti
0
Necessity:
1
p 
f ti  
t L
•  t
i  i  nt  5
i 1  
i
p  2 
•  f(t )  n
i 1
i
IT IS NOT ENOUGH?
Problem
• sequence (t1,...,tp), ti  3.
• graph Ln, n>2.
Example
L6  18, (3 , 3 , 6 , 6 )
Necessity:
4
p
•  ti
i 1
p
•
i 1
 Ln
 f (ti )  6
i 1
f (ti )  n
L6
Thank you very very much!!