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ELSEVIER

Discre te Mathemat ics 187 (1998) 137-149

DIS RETE

M THEM TICS

On l is t edge co l or i ngs o f s ubcubi c graphs

M a r t i n J u v a n 1, B o j a n M o h a r * ,1 , R i s t e S k r e k o v s k i 1

Department of Mathematics University of Ljubljana Jadranska 19 1111 Ljubljana Slovenia

Received 11 December 1995; revised 15 August 1997; accepted 25 August 1997

Abstract

In this paper w e study list edge-colorings of graphs with sm all maxim al degree. In particular,

we sho w that simple subcubic gra ph s are 10/3-edge choosable . T he precise meaning o f this

statement is that no m atter how w e prescribe arbitrary lists o f three colors on ed ges of a subgraph

H of G such that

A(H)~<

2, and prescribe lists of four colors on

E(G)\E(H),

the subcubic graph

G w ill have an edg e-coloring with the given c olors. Several consequences follow from this result.

(~) 1 998 Elsevier S cience B .V . All rights reserved

Keywords:

Coloring; List coloring; Edge coloring; Cubic graph

1 Introduct ion

Al l g raphs i n t h i s pape r a r e und i r ec t ed and f i n i t e . They have no l oops bu t t hey m ay

con t a i n m u l t i p l e edges and edges w i t h on l y one end , ca l l ed

halfedges.

A g raph i s

simple

i f it h as n o h a l f e d g e s a n d n o m u l t i p le e d g e s . T h e m a x i m a l d e g r e e o f G i s d e n o te d b y

A(G).

A g raph i s

subcubic

i f A ( G ) ~ < 3 . A

l ist assignment

of G i s a func t i on L wh i ch

as s i gns t o each e dge e E

E (G ) a list L(e) C_N .

The e l em en t s o f t he l is t

L(e)

a re ca l l ed

admissible colors

fo r t he edge e . An

L-edge-colorin9

i s a f u n c ti o n 2 : E ( G ) - - + N s u c h

that

2( e ) EL( e )

f o r

e E E ( G )

and s uch t ha t fo r any pa i r o f ad j acen t edges

e , f

in

G , 2 ( e ) ~ ) , ( f ) . I f G a d m i t s a n L - e d g e - c o l o r in g , i t i s

L-edge-colorable.

F o r k E N ,

t he g raph i s

k-edoe-choosable

i f i t i s L -edge-co l o rab l e fo r eve ry l i s t a s s i gnm en t L wi t h

IL(e)l ~ k

f o r e a c h e E E ( G ) .

L i s t co l o r i ngs were i n t roduced by Vi z i ng [5 ] and i ndependen t l y by E rd r s e t a l .

[1 ] . P robab l y , t he m os t we l l -known con j ec t u re abou t l i s t co l o r i ngs i s t he fo l l owi ng

con j ec t u re abou t l i s t - edge-ch rom at i c num bers ( s ee [4, P rob l em 12 .20 ] ). I t s ta t e s tha t

e v e r y ( m u l t i) g r a p h G i s z ( G ) - e d g e - c h o o s a b l e , w h e r e

x (G)

i s t he u s ua l ch rom at i c

i ndex o f G . In 1979 Di n i t z pos ed a ques t i on abou t a gene ra l i za t i on o f La t i n s qua res

* Corresp onding author.

1 Supported part ial ly by the M inistry of Science an d Techn ology o f Slovenia, R esearch Project J1-7036.

0012-365X/98/ 19.00 Cop yright (~) 1998 Elsevier Science B.V. All r ights reserved

P I I S 0 0 1 2 - 3 6 5 X ( 9 7 ) 0 0 2 3 0 - 6



38 ~ Juvan et al./D iscret e Mathematics 187 1998) 137-149

wh i ch i s equ i va l en t t o t he a s s e r t ion t ha t ev e ry com pl e t e b i pa r t i t e g raph Kn, n is n -edge-

c h o o s a b l e . T h i s p r o b l e m b e c a m e k n o w n a s t h e D i n i t z c o n j e c t u r e a n d r e s i s t e d p r o o f s

up t o 1995 w hen Ga l v i n [2 ] p roved t he con j ec t u re i n t he a f fi rm a t ive . More gene ra l l y ,

Ga l v i n e s t ab l i s hed t ha t eve ry b i pa r t it e (m u l t i )g raph G i s A(G ) -edge-c hoos a b l e . An o t he r

rece nt resu l t abo ut l i s t -edge-c hrom at ic numb ers i s a resu l t o f H~iggkvis t and Janssen [3] ,

w h o p r o v e d t h a t e v e r y s i m p l e g r a p h w i t h m a x i m a l d e g r e e A i s ( A + ( 9 ( A 2 / 3 1 x ~ A ) ) -

edge-choos ab l e .

In t h is pape r w e s t udy l i s t edge c o l o r i ngs o f g raphs wi t h s m a l l m ax i m al deg ree . In

p a r ti c u la r , w e s h o w t h a t s im p l e s u b c u b ic g r a p h s a r e ~ - e d g e - c h o o s a b l e . T h e p r e c i s e

m e an i ng o f t h is s t a t em en t i s tha t no m a t t e r how we p res c r i be a rb i t r a ry li s ts o f t h ree

co l o r s on edges o f a s ubg raph H o f G s uch t ha t A(H)~ < 2 , and p res c r i be l is ts o f fou r

co l o r s on E ( G ) \ E ( H ) , t he s ubcub i c g raph G wi l l have an e dge-co l o r i ng wi t h the g i ven

co l o r s . S om e cons eq uence s o f t h is r e s u l t a r e a l so p res en t ed .

2 Coloring paths and cycles with halfedges

L e t G b e a g r a p h a n d H a s u b g ra p h o f G . E a c h e d g e e E E ( G ) \ E ( H ) w i t h b o t h

ends in H i s a

c h o r d

o f H .

Le t G be a g rap h and S i ts s e t o f ha l f edges . I f z : S - - + S i s an i nvo l u t i on , t hen we

say that s E S i s z - f ree i f z ( s ) = s , a n d z -cons t r a i ned o t he r w is e . L e t s a n d z ( s ) C s

be a z - cons t r a i ned pa ir . I f L i s a l is t a s s i gnm en t and 2 an L -ed ge-co l o r i ng o f G , we

say that 2 i s res idual ly d i s t inct a t s ( and a t z ( s ) ) i f [ L ( s ) \ { 2 ( e ) , 2 ( f ) } U L ( z ( s ) ) \

{ 2 ( e ) , 2 ( f ) } [ > ~ 3 w h e n e v e r e , f and e , f a re edges o f G ad j acen t t o s and z ( s ) ,

r e s pec t i ve l y .

L e m m a 2 . 1 . L e t G be a s ubc ub i c g r aph o f o r der n >~2 t h a t i s c o m p o s e d o f a H a m i l t o n

pa t h H , a s e t S o f ha l f edges, and a s e t D o f chords. Supp os e t ha t a an d z a r e

i nvo l u ti ons o f S s uch t ha t no a -cons t r a i ne d ha l fedge i s z - cons t r a i ned Supp os e a l s o

t ha t no a - o r z - cons t r a i ned ha l f edge i s i nc i den t w i th an endver t ex o f H and t ha t

t her e is no cho r d j o i n i ng t he endver t ices o f H . L e t L be a l i s t a s s i gnm en t s uch t ha t

4, e c D, or e E S i s z -cons t ra ined ,

]L(e) l t> 3 , e C E ( H ) , o r e E S i s a - c o n st r a in e d , ( 1 )

2, e E S i s z - f ree an d a- f ree .

Mo r eover , i f each endve r t ex o f H i s i nc i den t w i t h t w o ha l fedges , t hen a t l ea s t one

end ver tex i s incident wi th ha l fedges s , s ~ such that [L(s )UL(s~)[>~3. Then G has an

L -edge-co l o r i ng 2 s uch t ha t f o r each pa i r o f d i s t i nc t ha l f edges s , s ~ w i t h a ( s ) -=-s

w e h a v e 2 ( s ) ~ 2 ( s ) an d s uch t ha t f o r each z -cons t r a i ned ha l f edge s , 2 i s r e s idua l l y

d is t inct a t s .

P roo f . S i nce no cho rd i s ad j acen t t o a cons t r a i ned ha l f edge , m u l t i p l e edges t ha t a r e i n

D c a n b e r e m o v e d a n d c o l o r e d a t t h e e n d . T h e r e f o r e , w e m a y a s s u m e t h a t G c o n t a i n s

n o m u l ti p le e d g e s . W e m a y a l s o a s s u m e t h a t G h a s o n l y v e r t ic e s o f d e g r e e 3 ( b y



M. Juvan et al. / Discrete Ma thematics 187 (1 998 ) 137-149 139

a d d i n g a d d i t io n a l h a l f e d g e s w i t h a r b i tr a r y l is t s o f t w o n e w c o l o r s i f n e c e s s a r y ) , a n d

t h a t w e h a v e e q u a l i t i e s i n ( 1 ) .

W e e n u m e r a t e t h e v e r t i c e s o f H a s V l , .. . , v n a s t h e y a p p e a r o n H a n d d e n o t e b y

ei E E ( H )

t h e e d g e j o i n i n g

vi

a n d

vi+l

(1

<~ i<n) .

B y o u r a s s u m p t i o n s , w e m a y a c h i e v e

t h a t vn i s n o t a n e n d v e r t e x i n c i d e n t w i t h t w o h a l f e d g e s w i t h t h e s a m e p a i r o f a d m i s s i b l e

c o l o r s . F o r i = 1 . . . . n , l e t

si

b e t h e c h o r d o r th e h a l f e d g e a d j a c e n t t o

vi,

a n d l e t s ' a n d

s'n b e t h e a d d i t i o n a l e d g e s a d j a c e n t t o V l a n d Vn, r e s p e c t i v e l y .

S u p p o s e f ir s t t h a t al l h a l f e d g e s a r e z - f re e a n d a - f r e e . W e s t a rt c o l o r in g e d g e s o f

G a t v e r t e x v l . I f b o t h S l a n d s ' a r e h a l fe d g e s , t h e n l e t 2 (S l ) b e a n a r b i t r a ry c o l o r

f r o m

L ( s l ) ,

l e t 2 ( s ] ) b e a c o l o r f r o m

L ( s 1 ) \ { ) ] , s 1 ) } ,

a n d l e t 2 ( e l ) b e a c o l o r f r o m

L ( e l ) \ { 2 ( S l ) , 2 (s 11 )} . I f o n e o f s l o r s/1 i s a h a l f e d g e a n d t h e o t h e r o n e i s a c h o r d

( s a y s l i s a h a l fe d g e a n d s ~ i s a c h o r d ) , t h e n w e c o l o r s i w i t h a c o l o r f r o m

L ( s l )

a n d e l w i t h a c o l o r f r o m

L (e l

) \ { 2 ( s l ) } . W e s h a l l c o l o r s~ w h e n e n c o u n t e r e d f o r t h e

s e c o n d t i m e a n d t h e n w e s h a ll r e g a r d i t a s a h a l f e d g e w i t h a l is t o f t w o c o l o r s f r o m

L(s~ ) \ { 2 ( s l ), 2 ( e l ) } . I f b o t h s l a n d s ] a r e c h o r d s , t h e n w e c o l o r e l w i t h a c o l o r f r o m

L(el ) .

L e t v k a n d v k, b e t h e o t h e r e n d s o f s l a n d

s l,

r e s p e c t i v e l y . W e m a y a s s u m e t h a t

k < U . W e w i l l c o l o r S l w h e n e n c o u n t e r e d a t vk , a n d a f t e r th a t w e w i l l t r e at s~ in t h e

s a m e w a y a s d e s c ri b e d a b o v e .

I n a g e n e r a l s t ep i, l < i < n , w e a s s u m e t h a t w e h a v e c o l o re d e i- 1 . I f si i s a

h a l f e d g e , l e t 2 ( s i ) b e a c o l o r f r o m

L ( s i ) \ { 2 ( e i - l ) }

a n d l e t 2 ( e i ) b e a c o l o r f r o m

L ( e i ) \ { 2 ( e i - 1 ) , 2 ( s i ) } .

O t h e r w i s e ,

si

i s a c h o r d . I f

si

i s i n c i d e n t w i t h

vn

a n d v , i s i n -

c i d e n t w i t h a h a l f e d g e , s a y s , , t h e n l e t 2 ( e i ) b e a c o l o r f r o m

L ( e i ) \ { 2 ( e i - 1 ) }

s u c h

t h a t t h e r e e x i s t t w o c o l o r s p , q E L ( s i ) \ { 2 ( e l -1 ) , 2 ( e l ) } s u c h t h a t { p , q } L(sn) . O t h e r -

w i s e , c o l o r e i a r b i t r a r i l y w i t h a c o l o r f r o m

L (e i ) \ { )~ (e i -1

) } a n d c h o o s e { p , q } C_

L ( s , ) \

{ 2 ( ei _ 1 ) , 2 ( e i ) } . F r o m n o w o n , w e s h a l l r e g a r d

si

a s a h a l f e d g e i n c i d e n t w i t h t h e o t h e r

e n d v e r t e x h a v i n g a s th e a d m i s s i b l e c o l o r s t h e p a i r

{ p , q } .

A f t e r w e h a v e c o l o r e d e , _ l, c o l o r s , a n d sin w i t h d i s t in c t c o l o r s f r o m L ( s , ) \ { 2 ( e , _ 1) }

a n d

L ( s l , ) \ { 2 ( e , _

1 } , r e s p e c t i v e l y . N o t e t h a t s u c h c o l o r s e x i s t s in c e

IL (s, ) U L(s~,)l >~3.

T h i s g i v e s a n L - e d g e - c o l o r i n g o f G .

I f so m e h a l f e d g e s a r e a - o r z - c o n s tr a in e d , w e c a n a p p l y t h e s a m e m e t h o d a s a b o v e .

O b s e r v e t h a t a f te r c o l o r in g t h e f i rs t h a l f e d g e o f a a - c o n s t r a i n e d p a i r , th e s e c o n d

h a l f e d g e s b e h a v e s l ik e a a - f r e e h a l f e d g e s i n c e it h a s ( a t le a s t ) t w o a d m i s s i b l e c o l o r s

l e ft . S i m i l a r t e c h n i q u e i s u s e d f o r z - c o n s t r a in e d h a l f e d g e s w i t h t h e d i f f er e n c e t h a t f o r

t h e s e c o n d h a l f e d g e s o f t h e p a i r w e c h o o s e a p a i r o f c o l o r s f r o m

L ( s )

t h a t i s d i s -

j o i n t f r o m t h e r e s i d u u m a t

z (s ) .

T h i s a s s u r e s t h a t 2 w i l l b e r e s i d u a l l y d i s t i n c t a t s

a n d z(s) . []

T h e n e x t l e m m a s h o w s a re s u lt r e la t e d to L e m m a 2 .1 i n ca s e o f c y c le s i n s te a d o f

p a t h s . A l t h o u g h s i m i l a r in n a tu r e , i ts p r o o f is m u c h m o r e i n v o l v e d t h a n t h e p r o o f o f

L e m m a 2 . 1 .

L e m m a 2 . 2 .

Le t G be a sub cubic 9raph o f order n >~3 com pos ed o f a H am i l ton cyc le

H, a se t S o f hal fedoes , and a se t D o f chords o f H. Suppo se that z is an involu t ion



140

M. Juvan et al./Discrete Mathematics 187 1998) 137-149

o f S s uch t ha t t her e i s a t m os t one z - cons t r a i ne d pa i r o f ha l fedges . L e t L be a l i s t

a s s i onmen t s uch t ha t

4 , e E D, or e C S i s z -cons t ra in ed ,

[L(e)l~> 3, e E E H ) ,

2, e E S is z-free.

( 2 )

T hen G has an L -edoe-co l o r i n9 2 t ha t i s r e s i dua l l y d i s t i nc t a t each z -cons t r a i ned

ha l f edge un l es s z = id , D = 0, H i s an od d cyc le , each ver t ex o f G has a ha l fedoe , a nd

t her e a r e co l o r s a ,b , c s uch t ha t

L ( e ) =

{ a , b , c } f o r e a ch e E E H ) a n d

L ( e ) =

{ a , b }

f o r each e E S .

Pr o o f . S i n c e mu l t i p l e e d g e s c a n b e r e mo v e d a n d c o l o r e d a t t h e e n d , w e a s s u me t h a t

t h e r e a r e n o n e. W e m a y a s s u m e t h a t G h a s o n l y v e r t i c e s o f d e g r e e 3 ( s i n c e o t h e r w i s e

w e c a n a d d h a l f e d g e s w i t h a r b i tr a r y li s ts o f t w o n e w c o l o r s ) . W e m a y a s w e l l a s s u me

tha t we have equal i t i es in (2 ) .

Sup pose f i r st t ha t D ~ 0 o r z ~ id. For the f i r s t subcase , suppose tha t G i s a cu -

b i c g r a p h w i t h o u t z - f r e e h a l f e d g e s a n d t h a t a ll e d g e s e o n H h a v e t h e s a me l is t

L ( e ) =

{ a , b , c }

o f c o l o r s . I t is e a s y t o s e e t h a t t h e r e e x i s t s a n L - e d g e - c o l o r i n g o f H

which i s res idua l ly d i s t inc t a t z -cons t ra ined ha l fedges . Clear ly , each chord has an ad -

mi s s i b l e c o l o r d i s t in c t f r o m a , b , c t h a t c a n b e u s e d t o o b t a i n a n L - e d g e - c o l o r i n g o f G .

Otherwise , l e t V l ,V2 vn b e t h e v e r ti c e s o f G a s th e y a p p e a r o n H . F o r i = 1 . . . . n ,

d e n o t e b y

ei

t h e e d g e

vivi+1 E H )

( i n d e x i + 1 t a k e n mo d u l o n ) a n d b y

si

t h e c h o r d

o r t h e h a l f e d g e i n c i d e n t w i t h vi. S i n ce D ~ 0 o r z ~ / d , w e c a n a s s u m e t h a t Vn is

inc iden t wi th a chord o r a z -cons t ra ined ha l fedge and tha t e i ther v l i s i nc iden t wi th a

z - f r e e h a lf e d g e ( i f S c o n t a i n s a z - f r e e h a l f e d g e ) , o r w e h a v e L e l ) ~ L e n ) . Su p p o s e

t h a t t he o t h e r e n d v e r t e x o f t h e c h o r d a t

Vn

is

Vm

(1 < m < n - 1 ). S imi la r ly , i f sn is a

z-cons t ra ined ha l fedge , l e t Vm (1 <<.m<n) b e t h e e n d v e r t e x o f z s , ) . I f v l i s i nc iden t

wi th a chord , l e t vk be the o ther end o f th i s chord . I f s t i s a ha l fedge , l e t ok be

t h e e n d o f z ( s l ) . I f Sn i s z - c o n s tr a i n e d , i t ma y h a p p e n t h a t m = 1 . H o w e v e r , w e c a n

a l w a y s a c h i e v e ( b y p o s s i b l y r e v e r s i n g t h e o r i e n t a ti o n o f th e c y c l e , l e a v i n g Vl f i x e d ) t h a t

m > l .

W e w i l l c o n s t r u c t a n L - e d g e - c o l o r i n g 2 b y c o l o r i n g e d g e s o f G o n e a f t e r a n o t h e r i n

the fo l lowing o rder : e b s 2 ) , e 2 , s 3 ) . . . . , e n , S l w h e r e t h e n o t a t i o n s i ) m e a n s t h a t w e d o

n o t c o l o r

s i )

i f i t i s a chord and i t s o ther end i s e i ther v o r

vj

( j > i ) . T h e e x c ep t io n

t o t h is r u l e i s t h e c h o r d s k w h e n k < m .

W e c o l o r e l a s f o ll o w s : i f s l i s a z -f r e e h a lf e d g e , l e t 2 ( e l ) b e a n y c o l o r f r o m

L e l ) \ L S l ) . T h i s i s p o s s ib l e si n ce I L ( e l ) l = 3 a n d I L ( S l ) t = 2 . O t h e r w is e , l e t 2 ( e l )

b e a n e l e m e n t f r o m

L e l ) \ L e , ) .

N o t e t h a t t h i s i s p o s s i b l e b y o u r a s s u mp t i o n t h a t

L e l ) ~ L e ~ ) , w h e n s l i s a c h o r d o r a z - c o n s t r a i n e d h a l f e d g e . I n a g e n e r a l s t e p i > 1

w e a s s u m e t h a t w e h a v e c h o s e n a c o l o r 2 e l - 1 a n d w e c o l o r s i ) a n d ei. W e d i s t i n g u i sh

s e v e n c a s e s :

( 1 )

i f [ { k , m , n } .

In th i s case , i f

s i E S

i s z - f r e e , l e t 2 ( s i ) b e a c o l o r f r o m

L s i ) \

{ J , e i - 1 ) }

a n d l e t 2 ( e i ) b e a c o l o r f r o m

L e i ) \ { 2 e i - l ) , 2 s i ) } .

I f

si 6 D

o r

s i E S



M. Juvan et al. / Discrete Ma thematics 187 ( 199 8) 137-149

141

i s z -cons t ra ined , l e t )~(ei) b e a c o l o r f r o m L ( e i ) \ { 2 ( e i - l ) } . Now, the l i s t L ( s i )

c o n t a i n s t w o e l e m e n t s , s a y p , q , d i s ti n c t f r o m

2(e i -1 )

a n d

2 ( e i ) .

I f

s i E D ,

w e s h a l l

c o l o r

si

w h e n e n c o u n t e r e d f o r t h e s e c o n d t i me a n d w e s h a l l r e g a r d i t a t t h a t t i me

a s a z - f r e e h a l f e d g e w i t h a d m i s s i b le p a i r o f c o l o r s { p , q } . I f

si E S

i s z -cons t ra ined ,

t h e n w e c h o o s e ) ~ ( s i ) = p and w e sha l l cons ider z s i ) as a z - f ree ha l fedg e wi th a

p a i r {~ ,/ 3} o f a d m i s s ib l e c o l o r s f r o m

L ( z ( s i ) ) \ { p , q } .

We say tha t the pa i r {~ , /3}

is

f o r c e d

b y ) . e i - l ) a n d

);(el) .

N o t e t h a t s u c h a c h o i c e e n s u r e s t h a t 2 w i l l b e

res idua l ly d i s t inc t a t

si

and z s i ) .

2 ) i = k a n d l < k < m . I n t hi s c a se s l = s k i s a c h o rd o r z s k ) = S l . I f s k E D , w e

c o l o r s k a r b i t r a r i l y w i t h a c o l o r f r o m

L ( s k ) \ { 2 ( e l ) , 2 ( e k - 1 ) }

a n d c o l o r

ek

wi th a

c o l o r f r o m

L ( e k ) \ { 2 ( e k - 1 ) , ) ~ ( s k ) } .

O t h e r w i s e , w e c o l o r

ei

a n d d e t e r mi n e p , q a s

i n 1 ) . T h e n w e c o l o r s l w i t h a c o l o r f r o m

L ( s l ) \ { p , q , A ( e l ) }

a n d c o l o r

si

w i t h

an admiss ib le co lo r . As in 1 ) , t h i s cho ice ensure s res idua l d i s t inc tness .

3 )

i = m

a n d

k < m .

N o t e t h a t

L ( e n )

con ta ins two d i s t inc t co lo rs

a , b

such tha t the

s o f a r c o n s t r u c t e d L - e d g e - c o l o r i n g 2 c a n b e e x t e n d e d t o e n b y u s i n g e i t h e r o f

t h e s e t w o c o l o r s . Mo r e o v e r , b y s e l e c t in g a n y o f a , b a s a c o l o r o f e n , w e c a n

e x t e n d t h e c o l o r i n g a l s o t o S l i f S l h a s n o t y e t b e e n c o l o r e d . I f Sm E D , let d

b e a c o l o r i n L ( s ~ ) \ { a , b , 2 ( e m _ l ) } . N o w , w e c o l o r e m b y u s i n g a c o l o r f r o m

L ( e m ) \ { 2 ( e m - l ) , d } .

W e s h a l l r e g a r d

sn

as a ha l fedge a t v , wi th the l i s t o f co lo rs

{ d , r } C L ( s , ) \ { 2 ( e m _ l ) ,

2 era)} . I f

Sm

i s z -cons t ra ined , we co lo r

em

a n d

Sm

so tha t

the fo rced pa i r {~ , 13} o f co lo rs fo r Sn i s d i s tinc t f ro m the pa i r {a , b} . W e sha l l

r e g a r d sn as a z - f ree ha l fedg e w i th the l i s t o f co lo rs {~, /~}.

4 )

i = m

a n d

k > m .

C o l o r

em

w i t h a c o l o r f r o m

L ( e m ) \ { 2 ( e m - 1 ) } .

I f

Sm E D ,

let

x , y

b e t w o c o l o r s f r o m

L ( s m )

d i s ti n c t f r o m 2 e m- 1 ) a n d 2 e r a) . I f

S m E S ,

w e c o l o r i t

b y a n a v a i l a b l e c o l o r , a n d l e t

{ x , y }

b e a p a i r f o r c e d b y 2 e r a - t ) a n d 2 e r a) . W e

shal l now rega rd s , as a z - f ree ha l fedge a t vn wi th the l i s t o f co lo rs L ( s , ) = { x , y } .

5 )

i = k

a n d

k > m .

L e t p b e a c o l o r f r o m

L ( e ~ ) \ L ( s ~ ) .

N o t e t h a t w e r e g a r d s ,

a f t e r S t e p 4 ) a s a h a l f e d g e a n d h e n c e I L s .) l = 2 . ) I f

S k E D ,

c h o o s e 2 Sk ) f r o m

L ( s k ) \ { A ( e l ) , p , 2 ( e k _ l ) }

a n d l e t 2 e k ) b e a c o l o r f r o m

L ( e t ) \ { 2 ( e k - 1 ) , 2 ( S k ) } .

I f

s t E S , w e c a n c h o o s e 2 e t ) E L ( e k ) \ { 2 ( e t - )} such tha t the fo rced pa i r {~ , /3} on

S l c o n ta i n s a c o l o r q d is t in c t f r o m p a n d 2 e l ) . W e c o l o r s l b y q a n d c o l o r s t

a rb i t ra r i ly .

6 )

i = n

a n d

k < m .

Fi r s t c a s e i s w h e n

s m C D ,

L e t

a , b , d , a n d r

b e c o l o r s f r o m

St e p 3 ) . I f 2 e , _ 1 = d , c o l o r s , w i t h r . O t h e r w i s e l e t 2 s , ) = d . S i n c e d ~ {a , b } ,

w e c a n c o l o r e , u s i n g a c o l o r f r o m {a , b } \ { 2 e , _ l ) , 2 s , ) } . I f s l i s a h a l f e d g e , w e

c a n c o l o r s l b y a c o l o r f r o m

L ( S l ) \ { 2 ( e , ) } ,

s ince

2 ( e l ) q l L ( S l ) .

T h i s g i v e s a n

L - e d g e - c o l o r i n g o f t h e e n t ir e g r a p h G . T h e s e c o n d c a s e i s w h e n

Sm

a n d s , f o r m th e

or ig ina l z -cons t ra ined pa i r . Le t a , b , ~ , ~ b e c o l o r s f r o m 3 ) . S i n c e { a , b } ¢ {~,/3},

w e c a n c o l o r s , a n d

en

b y t h e i r a d mi s s i b l e c o l o r s d i s t i n c t f r o m

2(en-1 ) .

If k = 1 ,

w e a l s o c o l o r s l a n d t h u s o b t a i n a n L - e d g e - c o l o r i n g o f G .

7 ) T h e l a s t p o s s i b i l i t y i s w h e n

i = n

a n d

k > m .

L e t

x , y ,

a n d p b e t h e c o l o r s d e f i n e d

i n S t e ps 4 ) a n d 5 ) . I f 2 e , _ l ) ¢ p , l e t 2 s n ) b e a c o l o r f r o m { x , y } \ { 2 ( e n - 1 ) }

a n d l e t 2 e , ) = p . O t h e r w is e , l et 2 e , ) b e a c o lo r f r o m

L ( e , ) \ { p , A ( s i ) } ,

a n d



42

M. Juvan et al. /D iscrete Ma thema tics 187 1998) 137-149

l e t 2 s n ) b e a c o l o r f r o m { x , y } \ { 2 e n ) } . A g a i n , w e o b t a i n a n L - e d g e - c o l o r i n g

o f G .

Su p p o s e n o w t h a t D = 0 a n d z =

id .

W e w i l l u s e a s i m i l a r c o l o r i n g p r o c e d u r e a s

a b o v e . L e t u s c h o o s e v l s u c h t h a t L s 2 ) ~

L e l ) .

I f such a ch o ice i s no t poss ib le , l e t v l

b e s u c h t h a t L S l ) ~ L s 2 ) o r L e l ) ~ L e ~ ) . I f a l so th i s ru le ca nno t be sa t i sf i ed , G i s

a s e x c l u d e d b y o u r l e mma e x c e p t t h a t i t s l e n g t h ma y b e e v e n . H o w e v e r , i n t h a t c a s e

i t c a n e a s i l y b e L - c o l o r e d . )

L e t u s s t a r t c o l o r i n g a t t h e v e r t e x v l . Co l o r e l w i t h a c o l o r f r o m

L e l ) \ L S l )

a n d

proceed to the ver t ex v2 .

A t v e r t e x

v i 2 < - . . i < : n ) ,

we c o l o r

si

w i t h a c o l o r f r o m

L s i ) \ { 2 e i _ l ) }

a n d

e i

w i t h

a c o l o r f r o m

L e i ) \ { 2 e i - 1 ) , 2 s i ) }

a n d t h e n p r o c e e d t o t h e n e x t v e r t e x . A r r i v i n g a t

Vn,

i t r e ma i n s t o c o l o r

s n , e , ,

a n d S l . B y o u r c h o i c e o f 2 e l) , e v e r y L - e d g e - c o lo r i n g

o f Sn a n d e n c a n b e e x t e n d e d t o s l . So , a n o b s t r u c t i o n c a n o c c u r o n l y w h e n c o l -

o r ing the edge e~. Suppose tha t

L s ~ ) = { c , d } , L s l

) = {a,

b } , x

= 2 e l ) , y = 2 s2), and

z = 2 e 2 ) . I f w e c a n n o t c o lo r

en,

w e h a v e :

2 e , _ l ) C { c , d }

s a y 2 e ~ _ 1 ) = c ) a nd

L e n ) = { c , d , x } .

I f

Z e l

) ~ {x , y , z} , w e re co lo r e l by us ing a co lo r in

L e l ) \ { x , y , z } ,

a n d s et 2 s ~ ) = d , 2 e ~ ) = x . S in c e x f [ L s l ) , t h e r e i s a l s o a n a v a i l a b l e c o l o r f o r

Sl . Therefo re , L e l ) = { x , y , z } . I f L s 2 ) ~ { y , z } , we reco lo r : 2 s2 ) E L s 2 ) \ { y , z } ,

2 e l ) = y , 2 en ) = x , 2 s~ ) = d , and 2 s l ) E

L s~

) \ { y } .

T h e r e f o r e ,

L s 2 ) = { y , z } C _ L e l) .

Then Vl was no t se l ec ted accord ing to the f i r s t

ru le , and hence a l so L S l ) C L e , ) a n d L S l ) C_ L e l ) . T h i s i mp l i e s t h a t L s l ) = { y , z }

a n d L e n ) = { x , y , z } , whic h con t rad ic t s our cho ic e o f Vl. [ ]

I f th e r e a r e mo r e t h a n t w o z - c o n s t r a i n e d h a l f e d g e s , L e m m a 2 . 2 c a n b e s t r e n g t h e n e d

as fo l lows .

L e m m a 2 . 3 .

L e t G b e a s u b c u b i c g r a p h o f o r d e r n > 13 c o m p o s e d o f a H a m i l t o n

c y c l e H , a s e t S o f h a l f ed g e s , a n d a s e t D o f c h o r d s o f H . S u p p o s e t h a t z ~ i d i s

a n i n v o l u t io n o f S , a n d l e t s o b e a z - c o n s t r a i n e d h a l f e d oe . L e t L b e a l i s t a s s i g n m e n t

s u ch t h a t

4 , e E D , o r e E S i s z - co n s t r a i n ed ,

IZ e)l~> 3,

e E E H ) ,

3 )

2 , e C S i s z - f ree .

T h e n G h a s a n L - e d g e - c o l o ri n g 2 t h a t i s r e si d u a l ly d i s ti n c t a t e a c h z - c o n s t r a i n e d

h a l f e d g e d i s t i n c t f r o m s o a n d z S o ). I f t h e r e e x i s t s a z - f r e e h a l f e d g e , w e c a n a l s o

a ch i eve t h a t 2 i s re s i d u a l l y d i s t i n c t a t s o a n d z s o ) .

Pr o o f . W e ma y a s s u m e t h a t t h e r e a r e mo r e t h a n t w o z - c o n s t r a i n e d h a l f e d g e s o t h e r w i s e

L e m m a 2 .2 a p p l i e s ) . I f th e r e i s a z - f r e e h a l f e d g e , t h e p r o o f o f L e m m a 2 .2 c h o o s e s t h e

c a s e w h e r e S l i s a z - f r e e h a l f e d g e a n d

Sn

i s e i ther in D o r z -cons t ra ined . Now, the

p r o o f o f L e m m a 2 . 2 y i e l d s th e r e s u l t o f L e m m a 2 .3 . I f t h e re a r e n o z - f r e e h a lf e d g e s,

w e c h a n g e z s o t h a t so a n d Z s o ) b e c o m e z - f r ee a n d th e a b o v e a r g u m e n t s ap p l y . [ ]



M. Juvan et al./Disc rete Mathem atics 187 (1 998) 137--149 143

3 . Co lor in g su b c u b ic gr ap h s

L e t Y b e a g r a p h o f o r d e r 4 c o m p o s e d o f a c o p y o f Ka,3 t o g e t h e r w i t h a p a i r o f

pa ra l l e l edges be t ween t wo ve r t i ces i n t he l a rge r b i pa r t i t i on c l a s s ( s ee F i g . 1 ) .

L e m m a 3 . 1 . L e t L be a l i s t a s s i gnmen t o f Y w h i ch a s s i gns t o each edge o f Y a t l ea s t

a s many co l o r s a s i nd i ca t ed by t he number s i n F i g . l ( a ) . T hen Y can be L -co l o r ed .

L e m m a 3 . 2 . L e t L be a l i s t a s s i gnmen t o f Y w h i ch a s s i gns t o each edge o f Y a t l ea s t

a s many co l o r s a s i nd i ca t ed by t he number s i n F i g .

l (b ) .

T hen Y can be L -co l o r e d

unless the admiss ib le co lors are as shown in Fig . l ( c ) .

P roo f s o f Lem m as 3 .1 and 3 .2 a re s t ra i gh t fo rward and a re l e f t t o t he r eade r a s an

eas y exe rc i s e .

Le t G be a s ubcub i c g raph w i t h t he s e t S o f ha l f edges and l e t F C E ( G ) b e a n e d g e

s e t s uch t ha t each ve r t ex o f G o f deg ree 3 i s i nc i den t w i t h e i t he r a ha l f edge o r an

edge f ro m F . Le t L be a l is t a s s i gnm en t fo r G s uch tha t

4 e E F

IL (e)[> ~ 3 , e E E ( G ) \ ( F U S ) , ( 4 )

2 , e E S .

S uppos e t ha t G con t a i n s a s ubg raph I7 i s om orph i c t o Y . Deno t e by uo t he ve r t ex

o f d e g r e e 1 in I7 and l e t e0 the e dge o f 17 i nc i den t w i t h u0 . Lem m as 3 .1 and 3 .2

s how t ha t t he re i s a t m os t o ne co l o r co E L ( e o ) s uch t ha t 17 canno t be / ~ -co l o red w here

/~(e0) = {c o} a nd [ , ( e ) = L ( e ) f o r e E E ( 1 7 ) \ { e 0 } . L e t G b e t h e g r a p h o b ta i n e d f r o m G

by r ep l ac i ng I by a ha l f edge ~ i nc i den t w i t h uo , wh ere t he adm i s s i b l e co l o r s fo r

a r e L ( e o ) \ { c o } i f Co exi s t s , and L ( e o ) o t he rwi s e . Lem m as 3 .1 and 3 .2 gua ran t ee t ha t

G c a n b e L - c o l o r e d i f a n d o n l y i f G can be L -co l o red . R epea t i ng t he above r educ t i on ,

we can ach i eve t ha t t he ob t a i ned g raph con t a i n s no s ubg raphs i s om orph i c t o Y . S uch

a g raph i s ca l l ed r educed . Not i ce t ha t s i m p l e g raphs a re a l ways r educed .

G i v e n a r e d u c e d s u b c u b i c g r a p h G a n d F C_ E (G ) as above , t he s ubg raph H = G - F

of G i s a d i s j o in t un i on o f pa t h s (pos s i b l y o f l eng t h 0 ) and cyc l es w i t h ha l f edges ,

4 3 abe

Ca) Co) (e)

Fig. 1. G raph Y and the ex ceptional ist assignment.



1 44 M . J u v a n e t a l . / D i s c r e t e M a t h e m a t i c s 1 8 7 1 9 9 8 ) 1 3 7 - 1 4 9

ca l l ed pa th componen t s and cycle components of H , r e s pec tive ly . A pa th com ponen t

w i th one ve r t ex i s ca l l ed trivial. A non- t r iv ia l pa th com ponen t Q o f H i s b a d i f e a c h

o f i t s endver t i ces i s inc iden t w i th tw o ha l f edges and each pa i r o f thes e ha l f edges has

the s ame l i st o f tw o adm is s ib le co lor s . I f t h i s happens a t one e nd o f Q on ly and the

o the r end o f Q i s no t inc iden t w i th tw o ha l f edges hav ing d i s t inc t pa i rs o f admis s ib le

colors , then Q is potent ia l ly bad. S imi la r ly , a cyc le compon en t Q i s b a d i f i t is an o dd

cycle w hose edges a l l have the same l is t of three adm iss ib le colors , say {a , b , c} , a l l

ha l f edges o f Q have the s ame pa i r o f admis s ib le co lo r s con ta ined in { a , b , c} , an d each

ver t ex o f Q i s inc iden t w i th a ha l f edge . I f w e r ep lace the cond i t ion tha t a l l ve r ti ces o f

Q a r e inc iden t w i th ha l f edges and r equ i r e tha t Q con ta in s a t l eas t tw o ha l f edges and

a ve r t ex no t inc iden t w i th a ha l f edge , then w e s ay tha t Q i s potent ia l ly bad. A tr iv ia l

pa th componen t i s

b a d

i f i t con ta in s th r ee ha l f edges w i th the s ame p a i r o f admis s ib le

co lo rs on each o f them. I t i s po ten t ia l l y bad i f it has p r ec i s e ly tw o ha l f edges e , f and

[L(e ) U L ( f ) ] = 2 .

P ropos i t ion 3 .3 .

Le t G be a reduced subcubic 9raph and le t S ,F ,H, and the l i s t as-

s ignmen t L be as above, l f H has no bad com ponen t s and has a t mos t one po ten t ia l l y

bad component, then G is L-edge-colorable.

P roof . We may as s ume tha t w e have equa l i t i e s in (4 ) . We may a l s o a s s ume tha t a l l

ve r t ices have deg ree 3 b y ad d ing add i t iona l ha lf edges w i th new co lo r s i f neces sa ry .

M o r e o v e r, w e m a y a s s u m e t h a t n o t w o e n d v e rt i ce s o f d i st in c t p a t h c o m p o n e n t s o f / - /

a r e c o n n e c te d b y a n e d g e f r o m F ; o t h e rw i s e w e c a n r e m o v e s u c h a n e d g e f r o m F .

Thes e changes can be done s o tha t no bad componen t s occu r and no new po ten t i a l ly

bad comp onen t s a r is e ( excep t tha t the po ten t i a l ly bad co mpo nen t ma y change in to a

l a rge r pa th ) . S imi la r ly , i f an edge e E F jo in s endver t i ces o f the s ame pa th : w e can

select three colors f rom L e ) t o be the new l i s t and r emove e f rom F , s o tha t the

pa th componen t changes in to a cyc le componen t w h ich i s ne i the r bad no r po ten t i a l ly

bad.

T h e p r o o f p r o c ee d s b y i n d u c ti o n o n t h e n u m b e r o f c o m p o n e n t s o f H , t h e b a se

of induc t ion being the em pty graph. Fo r the induct ive s tep we shal l f ir st se lect a

com ponen t Q o f H . Le t Q F be the s e t o f edges in F w i th one endver t ex in Q and

the o ther in

V G ) \V Q ) .

Let Q be the g r aph ob ta ined f rom Q by add ing a l l edges

f rom F w i th bo th endver t i ces in Q a nd by r ep lac ing each edge e E QF by a ha l f edge

~ . We s ha ll a s s ign to Y a l i s t L ( g ) C

L e )

and then L-edge-co lo r Q . M oreover , s ome

ha l f edges o f Q w i l l be a - o r z - cons t r a ined in o rde r to avo id bad and po ten t i a l ly bad

compo nen t s in the r emain ing g raph G . T he g raph G I is ob ta ined f rom G b y r em ov ing

V Q ) and r ep lac ing each edge uv E QF, u E V Gt ) , v E V Q) , by a ha l f edge inc iden t

with u who se l is t of adm iss ib le colors is L u v ) w i thou t the co lo r s u s ed w hen co lo r ing

the edges o f Q inc iden t w i th v . A dd i t iona l ly , i f Q i s a pa th componen t and v i t s

endver t ex inc iden t w i th tw o edges e ,e ~ f rom

QF,

t hen e and e ~ becom e ha l f edges

in G w i th ( a t l eas t ) th ree adm is s ib le co lo rs , bu t w e m us t r equ ir e tha t they r ece ive

d i s t inc t co lo r s w he n co lo r ing G . There fo re , w e r ega rd them as a - cons t r a ined in G ~.



M. Juvan et al. / Discrete Mathematics 187 1998 ) 137-149

145

When s e l ec t i ng Q we wi l l t ake ca re s o t ha t fo r each a -cons t r a i ned pa i r a t l eas t one o f

t he ha l f edg es w i l l be on a pa t h co m p one n t o f G ~. There fo re , cyc l e com p onen t s w i l l no t

con t a i n a -co ns t r a i ned pa i r s o f ha l f edges . S i nce ends o f d is t i nc t pa t h com pone n t s a r e

no t ad j acen t , a - cons t r a i ne d edges a re no t i nc i den t w i t h endve r t i ces o f pa t h co m po nen t s

i n G . I f e and e t a r e i n t he s am e pa t h c om pone n t o f G r , t hen L em m a 2 .1 wi l l t ake

ca re t ha t t hey wi l l r ece i ve d i s t i nc t co l o r s . I f t hey a re i n d i s t i nc t com ponen t s , one o f

t hem wi l l becom e a - f r ee w i t h t wo adm i s s i b l e co l o r s l e f t a f t e r co l o r i ng t he o t he r one .

T h e n a n L - e d g e - c o l o r i n g o f G , o b t a i n e d b y t h e i n d u c ti o n h y p o t h e si s , a n d t h e c o l o r i n g

o f 2 g i v e r is e t o a n L - e d g e - c o lo r i n g o f G ( w h e r e t h e e d g e s i n

QF

r e c e i v e c o l o rs f r o m

t he co l o r i ng o f GP ) .

I t r e m a i n s t o s h o w h o w t o s e l e c t Q , h o w t o d e t e r m i n e a a n d z o n Q , a n d h o w t o

co l o r 2 s uch t ha t G has a t m o s t one po t en t i a l l y bad pa t h o r cyc l e com po nen t .

I f G con t a i n s a po t en t i a l l y bad c yc l e com po nen t , we s e l ec t th i s com p one n t a s Q . I f

t w o e d g e s e , f o f

QF

l ead to t he s am e endv er t ex o f a non - t r i v i a l pa t h com pone n t Q t ,

t h e n J a n d f a r e z - c o n s tr a i n ed a n d L ( 4 ) - - L ( e ) ,

L f ) - - - - L f ) .

I f e l . . . .

,ek

(k>~ 2) a re

e d g e s f r o m

QF

l e a d in g to th e s a m e c y c l e c o m p o n e n t Q w h e r e Q h a s n o h al f ed g e s ,

t hen w e l e t 41 ,42 be z -cons t r a i ned wi t h adm i s s i b l e co l o r s a s abo ve and fo r i = 3 . . . . k ,

we l e t 4i be z - f r ee ha l f edges w i t h a pa i r

L 4i)C_L ei)

of adm i s s i b le co l o r s . W e do

t h e s a m e a s a b o v e a l s o i n t h e c a s e w h e n t w o o r t h r e e e d g e s o f

QF

l ead to a t r iv ia l

pa t h com pone n t Q . S uch cho i ces i n a l l o f the above cas es ens u re t ha t i n G t t he

c o m p o n e n t Q ~ w i l l n o t b e b a d o r p o t e n t ia l l y b a d w h e n e v e r u n d e r t h e c o l o r i n g o f Q ,

t he z -cons t r a i ned ed ges a re r e s i dua l l y d i s t inc t . I f e E

QF

l e a d s t o a c y c l e c o m p o n e n t

w i t h a t l e a s t o n e h a l f e d g e , s a y f , t h e n w e c h o o s e L ( 4 ) t o b e a 2 - e l e m e n t s u b s e t

o f

L e )

whi ch i s d i s j o i n t f rom

L f ) .

Thi s cho i ce gua ran t ees t ha t R wi l l no t becom e a

po t en t i a l l y bad co m p one n t i n G ~. S i m i l a r ly , i f e l eads t o an en d o f a pa t h c om pon en t

wh i ch has a ha l f edge f a t the s am e ve r t ex . In o t he r cas es, L (4 ) i s an a rb it r a ry 2 -

s ubs e t o f

L e) .

I f 2 i s no t t he odd cyc l e obs t ruc t ion f rom Le m m a 2 .2 , t hen i t can be

L - e d g e - c o l o r e d b y L e m m a 2 . 2 o r 2 .3 s o th a t n o b a d o r p o t e n t ia l l y b a d c o m p o n e n t i s

i n t roduce d i n G ( s i nce Q con t a i n s ha l f edges ) . I f 2 i s an odd cyc l e obs truc t i on , t hen

a l l ha l f edges a re z - f r ee . S i nce Q i s no t bad ( i t i s on l y po t en t i a l l y bad ) ,

QF -¢ O.

B y

chang i ng t he l is t o f an edge 4 , e E

QF, 2

b e c o m e s c o l o r a b l e . T h e c o n s t r u c t io n o f L

and z gua ran t ees tha t i n G on l y t he com po nen t con t a i n i ng t he endver t ex o f e no t i n

Q m a y b e c o m e p o t e n t i a l l y b a d .

S uppos e nex t t ha t G con t a i n s a cyc l e com ponen t Q wh i ch has a t l eas t one a -

cons t r a i ned ha l f edge e0 . No t e t ha t IL (e0 ) l ~ >3 . Then we app l y t he s am e m et hod as

above and s e l ec t a pa i r o f adm i s s i b l e co l o r s f rom

L eo)

s uch tha t 2 i s no t an odd

cyc l e obs t ruc t i on . B y L em m a 2 .3 we can co l o r 2 s uch tha t t he co l o r ing i s r e s i dua l l y

d i s t inc t a t a l l z - cons t r a i ned pa ir s and , a s be fo re , we s ee t ha t no new po t en t i a l l y bad

com ponen t s a r i s e .

I f G has a po t en t i a l l y bad t r i v i a l pa t h com pone n t Q , we c o l o r it s ha l f edges and

r e m o v e Q . C l e a r ly , G h a s a t m o s t o n e p o t e n t ia l l y b a d c o m p o n e n t .

S uppos e no w t ha t G has a non - t r i v i a l pa t h com pone n t R wh i ch i s po t en t i a l l y bad , L e t

v be t he e ndve r t ex o f R w h i ch i s no t i nc i den t w i t h t wo ha l f edges hav i ng t he s am e pa i r



146 M. Juvan et al. /Discrete Mathematics 187 (1998) 137-149

of adm i s s i b le co l o r s . L e t e , e t be t he ha l f edg es o r edge s o f

E ( G ) \ E ( R )

i nc i den t w i t h

v . I f e , e E F l ead , r e s pec t i ve l y , t o cy c l e com pone n t s Q , QI (po s s i b l y Q = QI ) , t he n w e

wi l l choos e Q t o be c o l o red nex t . (O t he rw i s e e , e I m i gh t be com e a a -co ns t r a i ned pa i r

wi t h bo t h ha l f edges be l ong i ng t o cyc l e com ponen t s . ) Q , z and adm i s s i b l e co l o r s fo r 2

a re de t e rm i ned as above . I f Q ~ QI , t hen 4 i s a z - f r ee ha l f edge i n Q . S i nce L (4 ) can be

c h o s e n s o t h a t 0 i s n o t a n o d d c y c l e o b st r u ct io n , n o n e w p o t e n t ia l l y b a d c o m p o n e n t

i s i n t roduced i n G . M oreov er , R r em a i ns (o n l y ) p o t en t i a l l y bad i n G I . I f Q = QI , t hen

4 and 41 a re z - cons t r a i ned . S i nce G i s r educed , t he o rde r o f Q i s a t l eas t 3 . There fo re ,

Le m m a 2 .3 (o r Lem m a 2 .2 i f 4 , 4 i s t he on l y z -cons t r a i ned pa i r i n Q) s hows t ha t t he re

i s a co lo r ing o f Q tha t i s res idua l ly d i s t inct a t 4 , 4I . Hence , R i s no l onger po t en t i a l l y

b a d i n G , b u t w e m a y o b t a in a n e w p o t e n t i a ll y b a d c o m p o n e n t i n G I d u e t o t h e

fac t t ha t t he co l o r i ng i s no t r e s i dua l l y d i s ti nc t a t one o f t he z -cons t r a i ned pa i r s. ( I f a

c o m p o n e n t b e c a m e b a d , i t w o u l d b e a c y c l e c o m p o n e n t , s a y ~ ) , a n d t h e r e w o u l d b e

a t l eas t t h ree edges be t we en Q and Q . O ne o f t hem w ou l d g i ve r i s e to a ha l f edge i n

2 , hence , we cou l d have t aken ca re o f a l l z - cons t r a i ned pa ir s , a con t r ad i c t i on . ) The

l as t cas e i s when e o r e I is a ha l f edge o r o ne o f t hem l eads t o a pa t h com ponen t .

( R e c a l l t h a t w e h a v e a s s u m e d a t t h e b e g i n n i n g o f t he p r o o f th a t n o n e o f e , e I l e a d

t o a n e n d v e r t e x o f a p a t h c o m p o n e n t d i s ti n c t fr o m R . ) T h e n w e s e l e c t

Q = R .

W e

det e rm i ne z and l is t s o f adm i s s i b l e co l o r s on ha l f edges 4 , e E Q F , as i n t he cas e o f

c y c l e s . N o t e t h a t s o m e p a i r s o f h a l f e d g e s o f Q m a y b e a - c o n s tr a i n e d . I f Q h a s t h e

s am e pa i r o f adm i s s i b l e co l o r s a l so a t ha l f edges i nc i den t w i t h v , we cha nge on e o f t he

pa i r s . ( In s uch a cas e , i n G I a new po t en t i a l l y bad c om pon en t m a y a r i s e . ) To c o l o r 2

we a pp l y Lem m a 2.1 wh i ch a l s o takes ca re o f a - cons t r a i ned pa i r s i n Q . No t e t ha t no

a - c o n s t r a in e d h a l f e d g e e o f Q h a s i t s m a t e a ( e ) i n a c y c l e c o m p o n e n t ( b y o u r c h o i c e

o f Q ) , a n d t h a t a ( e ) i s n o t i n c id e n t w i th a n e n d o f a p a th c o m p o n e n t . T h e r e f o r e , t h e

c h a n g e o f a ( e ) i n to a a - f r e e h a l f e d g e i n

G 1

does no t r e s u l t i n a new po t en t i a l l y bad

c o m p o n e n t .

I f G h a s n o p o t e n t i a l ly b a d c o m p o n e n t s , w e l e t Q b e a c y c l e c o m p o n e n t i f p o ss i b le .

Th i s cho i ce gua ran t ees t ha t a t leas t one edge o f each a -con s t r a i ned pa i r occu r s i n a

pa t h c om pone n t o f G I . I f t he re a re no c yc l e com ponen t s , w e l e t Q be a non - t r i v i a l

pa t h com ponen t , i f pos s i b le . O t he rw i s e Q i s any ( t r i v i a l ) pa t h com ponen t . Th i s cho i c e

ens u res t ha t i n G t he re a re no t h ree ha l f edges whos e co l o r s need t o be m u t ua l l y d i s t i nc t

b e c a u s e o f a c o m m o n e n d v e r t e x i n Q . I f Q i s a c y c l e ( p a t h ) c o m p o n e n t , w e p r o c e e d

a s i n th e c a s e w h e n Q w a s a p o t e n t i a l ly b a d c y c l e ( p a t h ) c o m p o n e n t . I f w e s u c c e e d to

co l o r Q s o t ha t t he co l o r i ng i s r e s i dua l l y d i s t i nc t a t each z -cons t r a i ned ha l f edge , t hen

n o b a d o r p o t e n t ia l l y b a d c o m p o n e n t s o c c u r . O t h e r w i se w e g e t a t m o s t o n e p o t e n t i a l ly

bad co m po nen t . (W e s ee t ha t no com pon en t i n G / is bad i n t he s am e w ay as abov e .

T h e o n l y e x c e p t i o n is t h e g r a p h o b t a i n e d f r o m th e g r a p h Y b y r e m o v i n g t he v e r t e x o f

deg ree 1 o f Y and r ep l ac i ng t he ad j acen t edge by a ha l f edge i nc i den t w i t h t he o t he r

end . Th i s g raph i s r educed and has t o be checked s epa ra t e l y u s i ng Lem m as 3 .1 and

3 .2 . ) Th i s com pl e t e s t he p roo f . [ ]

The fo l l owi ng t heo rem i s a s t r a i gh t fo rward con s equen ce o f P ropos i t i on 3 .3 .



M. Juvan et al./ Discrete Ma thematics 187 1998 ) 137-149

147

a b e d ~

~ / a b e d

pqrs

a b c

abc

ab ed x ,x ~ /

Fig. 2. Theorem 3.4 cannot be extended to non-reduced graphs.

T h e o r e m 3 . 4 .

L e t G be a r educed s ubcub i c g r aph w i t hou t ha l f edges , H a s ubgr aph o f

G s uch t ha t A H )<~ 2 , and L a li s t a s s i gnme n t o f G s uch t ha t

] L e ) l

>/3

f o r e E E H )

a n d

[L(e) l > /4

f o r e c E G ) \ E H ) . T h e n G is L - e d ge - c ol o ra b l e.

N o t e t h a t T h e o r e m 3 . 4 d o e s n o t h o l d i f w e o m i t t h e a s s u m p t i o n t h a t G i s r e d u c e d

( s ee F i g . 2 ) .

A n o t h e r c o n s e q u e n c e o f P r o p o si t io n 3 .3 i s :

C o r o l l a r y 3 . 5 .

Ever y s ubcub i c g r aph i s 4 -edge-choos ab l e , and t her e i s a l i near t i me

a l gor i t hm t ha t f o r ever y s ubcub i c g r aph G and a l is t a s s i gnme n t L w i t h I L e ) l

>14

e E E G ) ) r e t u r ns an L -edge-co l o r i ng .

P roof L e t G p b e a r e d u c e d g r a p h o b t a i n e d f r o m G b y t h e r e d u c t io n . ( O b v i o u s l y ,

t h e r e d u c t i o n c a n b e p e r f o r m e d i n l in e a r t i m e . ) W e f i rs t fi n d a c o l le c t i o n o f m a x i m a l

p a t h s a n d c y c l e s i n G ( b y a s i m p l e s e a r c h ) c o v e r i n g a l l v e r t ic e s o f G . T h e n w e l e t

F b e t h e s e t o f ed g es t h a t a r e n o t co n t a i n ed i n t h e s e p a t h s an d cy c l e s . F i n a l l y , w e

a p p l y t h e c o n s t r u c t io n o f a n L - e d g e - c o l o ri n g f r o m t h e p r o o f o f P r o p o s i t io n 3 . 3 ( a n d

L e m m a s 2 . 1 - 2 . 3 ) . I t i s e a s y t o c h e c k th a t e a c h o f th e s e s t e p s c a n b e a c c o m p l i s h e d i n

l inear t ime. [ ]

L e t u s r e m a r k t h a t 4 - e d g e - c h o o s a b i l i t y o f s u b c u b i c g r a p h s a l s o f o l l o w s f r o m t h e l i s t

v e r s i o n o f B r o o k s T h e o r e m [ 5 ,1 ] .

4 Som e applications

P r o p o s i t io n 3 .3 c a n b e u s e d t o g e t a s i m p l e p r o o f o f 5 - e d g e - c h o o s a b i li t y f o r a l ar g e

c l a s s o f 4 - r eg u l a r g r ap h s .

C o r o l l a r y 4 . 1 .

L e t G b e a g r a p h w i t h

A ( G ) ~ < 4

tha t conta ins two d i s jo in t 1- factors .

Then G i s 5-edge-choosable .



148 M. Juvan et al./Discrete Mathematics 187 (1998 ) 137-149

Pr o o f . L e t L b e a l i s t a s s i g n me n t w i t h I L ( e ) I > ~5 f o r e v e r y

e E E ( G ) .

Si n c e e a c h

h a l f e d g e i s a d j a c e n t t o a t mo s t t h r e e o t h e r e d g e s , h a l f e d g e s c a n b e r e mo v e d a n d c o l -

o red a t t he end . I f M1, M2 are d i s jo in t 1 - fac to rs o f G , deno te b y H the i r un ion .

T h e n H i s a u n i o n o f d i s jo i n t e v e n c y c l e s C1 . . . . .

Ct.

F o r i - - 1 , . . . , 1, c o n s id e r t he

c y c l e

Ci

a n d l e t e l , . . . , e 2 k b e t h e e d g e s o f

Ci

i n t h e s a me o r d e r a s t h e y a p p e a r o n

Ci. L e t E i = { e l , e 3 , e 5 . . . . . e 2 k - t } . W e 2 - c o l o r E i as fo l lows . I f L ( e ) = L ( f ) f o r e v e r y

e , f E E(C i) , t h e n w e c h o o s e a E L ( e l ) a n d p u t 2 ( e ) = a f o r e v e r y e E Ei. O t h e r w i s e ,

w e m a y a s s u m e t h a t L ( e l ) ~ L ( e 2 k ) . T a k e 2 ( e l ) E L( e l ) \L ( e 2 k ) . F o r j = I . . . . . k - 1,

let

A j

b e a 4 - e l e m e n t s u b s e t o f

L ( e 2 j ) \ { 2 ( e 2 j - 1 ) } .

T h e n t a k e 2 ( e 2 j + l ) E L ( e 2 j + I ) \ A j .

Co n s i d e r t h e s u b c u b i c g r a p h G = G - U l = 1Ei. D e f i n e a li s t a s s i g n me n t U o n E ( G I )

a s L ( e ) = L ( e ) \ { a , b } , w h e r e a a n d b a r e t h e c o l o r s u s e d o n t h e a l r e a d y c o l o r e d e d g e s

o f G i n c i d e n t w i t h e . O b s e r v e t h a t

I L ( e ) l / > 3

f o r e v e r y e E E ( G I ) a n d t h a t

I L ( e ) ]

>~4

f o r e v e r y

e E E ( G ) f q H .

Si n c e

F = E ( G ~ ) M H

i s a 1 - f a c t o r o f G , t h e r e d u c e d g r a p h

o b t a i n e d f r o m G b y t h e r e d u c t i o n h a s n o b a d o r p o t e n t i a l l y b a d c o mp o n e n t s . H e n c e ,

P r o p o s i t i o n 3.3 c a n b e u s e d to g e t a n L - e d g e - c o l o r i n g , a n d w e a r e d o n e . [ ]

T h e s e c o n d a p p l i c a t i o n c o n c e r n s 4 - e d g e - c o l o r i n g s o f c u b i c g r a p h s s u c h t h a t t h e f o u r t h

c o l o r i s n o t u s e d t o o o f t e n . N o t e t h a t i n e v e r y 4 - e d g e - c o l o r i n g o f t h e Pe t e r s e n g r a p h ,

each co lo r i s used a t l eas t tw ice . Therefo re , t here a re a rb i t ra r i ly l a rge cub ic g raphs

G w h e r e e a c h c o l o r o f a 4 - e d g e - c o l o r i n g i s u s e d a t l e a s t 21E(G)I/15 t imes . Tr iv i a l ly ,

u n d e r e v e r y c o l o r i n g , t h e r e i s a c o l o r u s e d o n a t mo s t ]E(G)I /4 e d g e s . Be l o w w e g i v e

a s l i g h t i mp r o v e me n t . Re c a l l t h a t t h e d o m i n a t i o n n u m b e r d ( G ) o f G i s t h e mi n i ma l

c a r d i n a l it y o f a v e r t e x s e t U s u c h t h a t e a c h v e r t e x o f G i s e it h e r in U o r a d j a c e n t t o

a v e r t e x o f U .

Corollary 4 .2 .

L et G be a subcub ic 9raph . Then G has a 4 -edge-co lorin9 such tha t

o n e o f t h e c o lo rs i s u se d a t mo s t d ( G) t ime s .

P r o o f . L e t U c_ V ( G ) b e a d o mi n a t i n g s e t w i t h d ( G ) v e r t ic e s . D e n o t e b y F t h e s e t o f

edge s inc iden t wi th ver t i ces in U , and l e t L be a l is t as s ign m ent wi th

L( e ) = -

{ 1,2 , 3 , 4}

i f e E F , a n d L ( e ) = { 1 , 2 , 3 } o t h er w i se . I t is e a s y t o s e e t h a t a f t e r t h e re d u c ti o n e a c h

h a l f e d g e o f th e o b t a i n e d g r a p h s ti ll h a s a t l e a s t th r e e a d mi s s i b l e c o l o r s . T h e r e f o r e ,

P r o p o s i t i o n 3 . 3 c a n b e a p p l i e d t o g e t a n L - e d g e - c o l o r i n g w h e r e c o l o r 4 i s u s e d a t mo s t

d ( G )

t imes . [ ]

I t i s easy to see tha t every subcub ic g raph G (wi thou t i so la t ed ver t i ces ) sa t i s f i es

IV (G )I/4 <~d(G ) <~ IV ( G) I /2 . N o t e t h a t b e i n g c l o s e t o t h e l o w e r b o u n d , Co r o l l a r y 4 . 2

y i e l d s a b o u n d o f IE(G)I /6 w h i c h i s n o t f a r f r o m 2 I E ( G) I /1 5 .

cknowledgements

T h e a u t h o r s g r a t e f u l l y a c k n o w l e d g e t h e c o n s t r u c t i v e c r it i c i s m o f t h e a n o n y mo u s

referee .



M. Juvan et al./Discrete Mathematics 187 1998) 137-149

149

eferences

[1] P. Erd6s, A.L. Rubin, H. Taylor, Choosability in graphs, Congr. Num er. 26 1979) 125-157.

[2] F. Galvin, The list chromatic index o f a bipartite multigraph, J. Combin. Theory S er. B 63 1995)

153-158.

[3] R. H~iggkv ist, J. Janssen, New bounds on the list-chromatic index o f the complete graph and other sim ple

graphs, Combin. Probab. Comput. 6 1997) 295 -313.

[4] T.R. Jensen, B. Tort, Graph Coloring Problems, Wiley, New York, 1995.

[5] V.G. Vizing, Colouring the vertices of a graph in prescribed colours, Metody Diskret. Analiz. 29 1976)

3-10 in Russian).

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