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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
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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
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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
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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
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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
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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 . [ ]
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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.
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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 ~.
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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
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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 .
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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 .
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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 .
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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.
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3-10 in Russian).