A Lemma on Cycle Decompositions

Annalsof Discrete Mathematics 27 (1985) 227-232 0 Elsevier Science Publishers B.V. (North-Holland)

221

A LEMMA ON CYCLE DECOMPOSITIONS

Roland H a g g k v i s t P i l g r i m s v ' a g e n 54B 5-12648 H a g g e r s t e n SWEDEN

Some r e s u l t s a b o u t decomposing v a r i o u s g r a p h s i n t o e v e n l e n g t h c y c l e s are g i v e n .

$1.

MAIN RESULTS Let

b e a g r a p h ( w i t h o u t l o o p s ) and

G

r e p e t i t i v e ) of ( u n l a b e l l e d ) graphs. subgraph isomorphic w i t h where

E(G)

Each

1

@ G

Gi

@

where

M d i v i d e s G).

(M,M,..

of

.. , m ,

. ,M)

...,Gm

is isomorphic with G

pack

G

.

I

=

m

has a

G

IE(Hi)

C

i=l

i s s a i d t o pack

I

if

G

G

(we o f t e n w r i t e t h i s

for

Hi

i

=

1,2,. ..,m.

G(2) x'

i s s a i d t o h a v e a n M-decomposition

T h i s i s sometimes w r i t t e n i s o b t a i n e d from

, x" ,

G

MIG

if (read

by r e p l a c i n g e a c h

joining every v e r t e x i n

{x'

, x''}

to

w i t h e d g e s o f t h e same m u l t i p l i c i t y a s t h e m u l t i p l i -

[ x , y ] ; no o t h e r e d g e s a r e p r e s e n t .

c i t y of

L

IE(G)

An euen l i s t i s a l i s t w h e r e e a c h e n t r y o c c u r s

F i n a l l y , t h e graph {y' ,y"}

G1,G2,

a list(possib1y

i s proper ( f o r G ) i f

L

and m o r e o v e r

The list

G.

The g r a p h

b y a p a i r of v e r t i c e s

every v e r t e x i n

The l i s t

1,2,.

Gi

2 is a l a b e l l e d graph.

the proper l i s t x

=

u n i o n of t h e g r a p h s

... CH Gm)

a n e v e n number of times.

vertex

,i

denotes t h e edge-set

is t h e edge-disjoint

G = G

H.

...,Hm )

L = (H1,H2,

S e e F i g u r e 1.

Figure 1 The p u r p o s e o f t h i s n o t e i s t o g i v e t h e f o l l o w i n g lemma w h i c h p e r t a i n s t o t h e decomposition of LEMMA.

Let

2-regular graph on

where

G'

=

G"

=H.

G(2)

i n t o even c y c l e s .

be a path o r a c y c l e w i t h

G

2n

n

edges and let

v e r t i c e s w i t h a27 components euen. Therefore,

H ~ G ( Z.)

Then

H

be a

G(2) = G ' tt3 G"

228

R. Haggkvist Proof.

If

i s a path, then

G

d i s j o i n t cycles with lengths Put

m

j

=

C i=l

Let

G

j

ni

mo

and

has length

and

{x;

Gi

[ X ~ , X ~ , . . . , X ~where +~] [xm ,x +l,...,x i-1 mi-l

t h e segment

1 = 1,2,

ni,

...,m .

Note t h a t

b e t h e subgraph of

"

d i s t i n c t ) and edges

,XI

i-1 mi..1

independent edges between and f i n a l l y

Gi(2)

-1'

IJ {x;.-~.x; 1

[x;

H

c o n s i s t s of

G

m

m

C ni = n . i=1

1.

=

Gi

+l~x~i-l+l~ i-1

...,m.-2

and n o t e , moreover, t h a t

r e s p e c t i v e l y , where

{x! 3

[x;.-~, 1

+1]

, x'.'} 3 1

XI'

i

I

i

, [x;

with v e r t i c e s

"

{xi i

'

of

m ] i

{x;

G

if

G.

G

is

Clearly

is a cycle

1u

i-1

( a l l t h e s e v e r t i c e s are

,x" +1] i-1 mi-l

and

xn+l = x1

xm = x if i-1 mi

m = l . Let

"

# V(G")

2n1,2n 2 , . . . , 2 n m ,

be the path o r cycle

a c y c l e and denote by Gi

V(G')

Assume w i t h o u t l o s s of g e n e r a l i t y t h a t

could b e a 2-cycle.

t o g e t h e r w i t h any p a i r of

{X;+~,X;+~} f o r

j = mi-l+l,mi-2>+2,

[ x i -l,x" ] ( s e e F i g u r e 2 ) . i mi

Figure 2

It is clear that G . (2) 1

is a n o t h e r

Gi

i s a c y c l e of l e n g t h

2ni-cycle

G'!

2n.

( s e e F i g u r e 3).

Figure 3

whose edge-induced complement i n

229

Cycle Decompositions A l s o , w e h a v e t h a t t h e g r a p h s G;, m G' = G; i s isomorphic with H i=l

i = 1,2,

...,m

are p a i r w i s e d i s j o i n t , m G" = U G;. Moreover i= 1

and s o i s

and t h e lemma i s proved. o

G(2) = G ' fB G"

The above o b s e r v a t i o n h a s some immediate c o r o l l a r i e s . COROLLARY 1. Let

be a graph w i t h a Hamilton decomposition ( t h a t i s ,

G

Then any proper even l i s t

a decomposition i n t o edge-disjoint Hamilton c y c l e s ) . of b i p a r t i t e 2-factors packs Proof.

A 2-factor i n

w e may assume t h a t Let

G. 2

1,2,

=

... @

Gm

...,m .

2n

G(2)

and

i s t h e proper l i s t of such m , Hm) v e r t i c e s , where of c o u r s e n is t h e o r d e r o f

b e a Hamilton d e c o m p o s i t i o n o f

By t h e lemma,

... H3 Gm (2)

G(2) = G1(2) Q G2(2) @

whence

is a 2 - r e g u l a r s p a n n i n g subgraph of

G(2)

g r a p h s on

G1 CH G2 @

Cn , i

.

L = (H1,H1,H2,H 2 , . . . , H

2-regular b i p a r t i t e Gi

G(2)

Gi(2) = H I = H'

1

ct)

ct)

H" tt, 1

Hy

Hi

t h a t is,

G ,

where

d HY

H!

rr

HY

Hi

rr

... W Hm' tt, H C .

W

The f o l l o w i n g r e s u l t i s immediate.

Ang proper even l i s t o f b i p a r t i t e 2-factors packs

COROLLARY 2.

KZn

where

-

F

denotes t h e complete graph on

Any proper even l i s t of 2-factors packs

COROLLARY 3 .

Proof, K2n+1

both

K 4 n i 2 - F = K 2n+lC2)

Now and

and

K4n+2

2n v e r t i c e s minus a 1-factor

K4n,4n

=

K4n, 4n

K2n,2n(2)

.

-

F.

'

*

Moreover,

have Hamilton d e c o m p o s i t i o n s . o

K2n, 2n

A l i t t l e less o b v i o u s , b u t s t i l l immediate from t h e lemma i s t h e n e x t r e s u l t .

Every proper even List o f even c y c l e s without a 4n-cycZe

PROPOSITION 1.

- F where

K2n+2

packs

2

<

mi

~

dnd

2n+l

for

Ho H3 H1 W is,

Ho

...,M,,M,)

2mi

=

is a 1-factor.

I n f a c t w e s h a l l p r o v e s l i g h t l y more.

Proof.

L = (M1,M1,M2,M2,

IV(Mi)I

F

b e t h e s t a n d a r d Hamilton d e c o m p o s i t i o n of

has v e r t e x s e t

U {O,l,.

{m}

. . ,2n-1)

{ [ O , ~ ] , [ a , n ] , [ n , n + l ] l U{[i,2n-i],[i,2n-i+1] modulo

2n

and

Ho

and

.

... Hn-l

o b t a i n e d from

Namely t h a t any e v e n l i s t

of b i p a r t i t e 2 - r e g u l a r g r a p h s Mi w i t h m m 1 4 m . = 1 21E(Mi)I = (4n$2) - 2 n - 1 , where mi # 2n i=l i=l i = 1,2,.. , m , packs K4n+2 - F . To see t h i s , l e t

by a d d i n g m+i

=

-.

anti-clockwise r o t a t i o n of

i

,...,n-11

with

*

Hi

t o e a c h v e r t e x where a d d i t i o n i s p e r f o r m e d

I n o t h e r words, i

and edge s e t

: i = 1,2

H = K 2n+1

H.

s t e p s i n Figure 4.

is o b t a i n e d from

Ho

by a n

That

230

R. Haggkvisl 0

' *n-*2 n

Figure 4

We o r i e n t

Hi

such t h a t

Assume now t h a t t h e l a s t

2k

t h e rest have o r d e r a t most from

H

T = [V1,V2,

2n-1

...,V ( Z n f l ) (n-k-1)

direction, that is,

m

, 0 , 1 , .. .

H1 i n t h e forward d i r e c t i o n .

Hn-k-l the required property. in

T

segment of

S

Case 2. {n,n+l,n-1,.

5

.,

x1

=

p

x

P

=

T

and t r a v e r s e

Ho

i n t h e forward

m

The t o u r

again, then continue T

has

.

Hn-k-l

followed by a

There i s no l o s s of g e n e r a l i t y

Ho

ending i n

followed by a

p = 2q

.

Then

V(S) =

II

{m}

.,n+q-l,n-q+l)

u

In t h i s case a l l v e r t i c e s a r e d i s t i n c t

x~~ and t h e r e f o r e i s odd, s a y S

,

G.

m .

i s even, s a y

a l l v e r t i c e s are d i s t i n c t and

'

S = [ x ~ , x ~ , . . . , x ~be ~ ]a segment of l e n g t h

...,n+q+2,n-q+l}. p

'(2n+l)(n-k-l) = 1 ' r + i 5 (2n+l) (n-k-1) in

x ~ = ~ I}m l U t n , n + l , n - l , n + 2 , n - 2 , . .

Assume

' ' Hn-k has an

I t i s s t r a i g h t - f o r w a r d t o check t h a t

(2n+l) (n-k-1))

. . ,n+q-1,n-q+l,n+q}

,

*

(2n-l)-cycle

u n t i l we r e a c h

c o n s i s t s of a segment of

Assume

{ 1 , 2 , 0 , 3,2n-1,

except t h a t

rti

So assume t h a t

H1.

Case 1.

u

m

where

-

and t h a t

and s o o n u n t i l f i n a l l y we have t r a v e r s e d

Indeed, l e t

(since

U txp-1,xp-2,..

U

'

. .. ,veil

which does n o t c o n s i s t of a segment of Ho

t o assume t h a t segment of

[vi ,vi+l,

Begin a t

i n t h e forward d i r e c t i o n of

4n+2

Hn-l , H n 4 9 We s h a l l see t h a t G

G.

induces a p a t h o r a

has t h e following description.

have o r d e r

i 2 n-1.

Delete t h e e d g e s of

w i t h t h e p r o p e r t y t h a t any segment of l e n g t h a t most

L

e n t r i e s i n the list 4n-2.

and c o n s i d e r t h e remaining graph

eulerian tour

2n-1

i s i n t h e forward d i r e c t i o n ,

(m,i)

S

is a cycle i n p = 2q+l

.

Then

G

of l e n g t h

V(S) =

{m}

U {1,2,0,3,2n-1,.

. . ,n-q,n+q+2} .

i s a path i n

of l e n g t h

G

2n-1.

2n-1.

I: In t h i s case

23 1

Cycle Decompositioas It i s now e a s y t o see t h a t where e a c h

G l , G 2 , - * * , Gm-k Gi

h a s a decomposition i n t o g r a p h s

G

is a path o r cycle with

Gi

m

be t h e graph induced by t h e edges of t h e T-segment

v

j where

c

pj =

G1 @ G2 d

i=l

mi

.. . @ Gm

ensures t h a t

po = 1 .

and where

H(2)

GWi

Hence

H = G d H1I-k

for i = 1,2,

= Hn-i-l

i s packed by

since

L

(Mi,Mi)

@

e d g e s - simply l e t

i

Pi-1

,v

Hn-k+l

...,k .

packs

Pi-1

+l,..,,v

' * '

@

@

pi

Hn-l =

The lemma now

Gi(2)

.

An analogous r e s u l t now f o l l o w s .

PROPOSITION 2.

811-6,

lengths

8n-4

L = (M1,M1,M2,M2,

1

=

811-2

or

packs

K4n,4n *

As i n t h e proof of P r o p o s i t i o n 1 w e show t h a t any even l i s t

Proof.

IV(Mi)

Any proper even l i s t o f even c y c l e s Without c y c l e s of

2mi

...,Mm ,M m )

of b i p a r t i t e 2 - r e g u l a r graphs m Z 4mi = 1 (E(Mi) = 1611' where 2 i=l i=l m

and

...,m-k,

i = 1,2,

and

Ho Q HI d

Indeed, l e t

m

i

=

I

4n

.. . @ Hn-l

I

for

i = O,l,. . . , n - 1 .

y2i+l.

reach

x1

Now t r a v e r s e

Hn-k-l

t h a t any segment 411-4

from

is traversed,

v . , v i+l,...,vr+i

i s a p a t h o r a (4n-4)-cycle

,...,x~~

and

4n,4n

*

Y ~ , Y ~ , . . . , Y ~and ~

( w i t h i n d i c e s reduced modulo

x1

L e t t h e forward

i s t r a v e r s e d from

x

i n t h e forward d i r e c t i o n u n t i l w e

H1

G.

1

i n t h e forward d i r e c t i o n and s o on

Again we n o t e t h a t with

in

... @ Hn-k-l.

T

r+i 5 4nk(n-k-l)

has t h e property of l e n g t h a t most

The v e r i f i c a t i o n i s l e f t t o t h e

I t i s now c l e a r from t h e proof of C o r o l l a r y 3 t h a t

reader.

H(2) 1 K $2.

Ho

again, then continue along

until finally

for

Consider t h e f o l l o w i n g e u l e r i a n t o u r

] of G = H @ H1 @ T = [ v v 0 1' 2 * * '"4nk(n-k-l) d i r e c t i o n of Hi be t h e one where t h e edge

to

5 411-4

...

..

for

Imi

with

i = m-k+l, m packs K4n,4n = K2n,2n(2) b e t h e s t a n d a r d Hamilton decomposition o f

H = K 2n,2n *' t h a t i s , H . h a s v e r t i c e s x1,x2 edges [ x j ,yj+2i] , [ x j ,yj+2i+l] : j = 1,2,. ,2n

2n)}

Mi

L

packs

*

REMARKS AND WILD CONJECTURES

C o r o l l a r i e s 2 and 3 a r e of i n t e r e s t i n c o n n e c t i o n w i t h t h e famous Oberwolf a c h problem (see [ l ] ) which a s k s f o r t h e d e t e r m i n a t i o n of t h o s e 2 - f a c t o r s which decompose

K2n+l.

The r e l a t e d near-Oberwolfach problem [ 2 ] a s k s f o r t h o s e

2 - f a c t o r s which decompose K2n - F ; above we showed t h a t a l l b i p a r t i t e ones do

so i f K2n,2n,

n

i s odd.

Similarly, i f

n

i s e v e n , t h e n a l l 2 - f a c t o r s decompose

whence t h e b i p a r t i t e analogue of t h e Oberwolfach problem i s completely

solved i n h a l f t h e cases.

R. Haggkvist

232

P r o p o s i t i o n s 1 and 2 s u p p o r t t h e c o n j e c t u r e s t h a t any p r o p e r l i s t of c y c l e s p a c k s K - F a n d K 2 n , 2 n , r e s p e c t i v e l y . F o r a s u r v e y on t h e s u b j e c t o f c y c l e 2n d e c o m p o s i t i o n s s e e D. S o t t e a n [ 3 ] . The a r e a is f u l l of c o n j e c t u r e s and r a t h e r empty on g e n e r a l r e s u l t s , a l t h o u g h s p e c i f i c d e c o m p o s i t i o n s c a n b e found i n t h e literature.

I t i s t e m p t i n g t o p u t f o r w a r d two w i l d c o n j e c t u r e s which b o t h a r e

t o t a l l y o u t of r ea c h a t p r e s e n t , b u t where n a t u r a l s p e c i a l c a s e s probabl y can be t r e a t e d . C o n j e c t u r e 1:

Every p r o p e r l i s t o f c y c l e s p a c k s graph on

Conjecture 2:

n

where

G

is an e u l e r i a n 3n

G

4 .

v e r t i c e s e a c h o f d e g r e e more t h a n

Any p r o p e r l i s t o f 2 - f a c t o r s p a c k s 12m g r a p h on n < __ v e r t i c e s .

G

where

G

is a n 2 m - r e g u l a r

5

Any c o u n t e r e x a m p l e s t o t h e above c o n j e c t u r e s would b e most welcome, o r f o r t h a t m a t t e r t o t h e b i p a r t i t e a n a lo g u e s where bipartition

(S,T) w i t h

Note t h a t t h e

4C3

G

now is assumed t o b e b i p a r t i t e w i t h

I S 1 = IT1 = n .

d o e s n o t decompose

K12

- F

(see [2])

and t h e r e

e x i s t s a n e u l e r i a n g r a p h ( f o u n d by Kon Graham) w i t h minimum d e g r e e edges without triangle-decomposition

( s e e Nash-Williams

-

and

3k

[4]).

REFERENCES [ l ] P. H e l l , A. K o t z i g and A. ROSA, Some r e s u l t s on t h e O b e r w o l f a c h p r o b l e m , A e q u a t i o n e s Math. 1 2 ( 1 9 7 5 ) , 1-5.

[ 2 ] C. Huang, A . K o t z i g and A. Ross, On a v a r i a t i o n o f t h e O b e r w o l f a c h p r o b l e m , D i s c r e t e Math. 27 ( 1 9 7 9 ) , 261-278. [ 3 ] Dominique S o t t e g n , D e c o m p o s i t i o n s d e g r a p h e s e t h y p e r g r a p h e s , t h e s e s L ' u n i v e r s i t ; Paris-Sud (1980).

[ 4 ] C. S t . J . A . N a s h - W i l l i a m s , P r o b l e m p. 1 1 7 9 , C o m b i n a t o r i a l M a t h e m a t i c s a n d i t s A p p l i c a t i o n s 111, e d . Erd:s (1970).

e t a l , C o l l o q u i a Math. SOC. J . B o l y a i 4