Frugal Colouring of Graphs

HAL Id: inria-00144318

https://hal.inria.fr/inria-00144318v2

Submitted on 3 May 2007

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

Omid Amini, Louis Esperet, Jan van den Heuvel

To cite this version:

Omid Amini, Louis Esperet, Jan van den Heuvel. Frugal Colouring of Graphs. [Research Report]

RR-6178, INRIA. 2007, pp.12. �inria-00144318v2�

inria-00144318, version 2 - 3 May 2007

a p p o r t

d e r e c h e r c h e

IS

S

N

0

2

4

9

-6

3

9

9

IS

R

N

IN

R

IA

/R

R

--6

1

7

8

--F

R

+

E

N

G

Thème COM

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Frugal Colouring of Graphs

Omid Amini — Louis Esperet — Jan van den Heuvel

N° 6178

Unité de recherche INRIA Sophia Antipolis

2004, route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex (France)

Téléphone : +33 4 92 38 77 77 — Télécopie : +33 4 92 38 77 65

Omid Amini

∗

, Louis Esperet

†

,Jan van den Heuvel

‡

ThèmeCOMSystèmes ommuni ants ProjetsMASCOTTE

Rapportdere her he n°6178May200712pages

Abstra t: A

k

-frugal olouringofagraph

G

isaproper olouringoftheverti esof

G

su hthat no olour appears morethan

k

times in theneighbourhood of avertex. This type of olouring wasintrodu edbyHind, Molloyand Reedin 1997. Inthis paper,westudythefrugal hromati numberofplanargraphs,planargraphswithlargegirth,andouterplanargraphs,andrelatethis parameterwith severalwell-studied olourings, su h as olouringof thesquare, y li olouring, and

L(p, q)

-labelling. Wealsostudyfrugaledge- olouringsofmultigraphs.

Key-words: wavelengthassignment,graph olouring

Theresear hforthispaperwasdoneduringavisitofLEandJvdHtotheMas otteresear hgroupatINRIA Sophia-Antipolis.TheauthorsliketothankthemembersofMas ottefortheirhospitality.

JvdH'svisitwaspartlysupportedbyagrantfromtheBritishCoun il.

∗

ProjetMas otte,CNRS/INRIA/UNSA,INRIASophia-Antipolis,2004routedesLu iolesBP93,06902 Sophia-AntipolisCedex,Fran e. oaminisophia.inria.fr

†

LaBRI,UniversitéBordeaux1,Talen e,Fran e. esperetlabri.fr

‡

CentreforDis reteandAppli ableMathemati s,DepartmentofMathemati s,LondonS hoolofE onomi s, London,U.K.janmaths.lse.a .uk

Résumé : Une oloration

k

-frugaled'un graphe

G

est une oloration propre des sommets de

G

telle que haque ouleurapparaît auplus

k

fois au voisinagede haque sommet. Ce type de olorationaétéintroduitpourlapremièrefoisparHind. MoloyetReeden1997. Dans etarti le, on étudie le nombre hromatique frugal des graphes planairesen général, eux de maille assez grande, et les graphesplanairesextérieurs. Onrelie e paramètreàplusieursautres paramètres bien onnus ommela olorationdu arré, la oloration y lique,et les

L(p, q)−

étiquetages des graphesplanaires. Onétudieaussilaversionarête olorationfrugaledesmultigraphes.

1 Introdu tion

Mostof theterminologyand notationweusein this paperis standardand anbefoundin any textbook ongraphtheory (su has[1℄ or[4℄). All ourgraphsandmultigraphswill benite. A multigraph anhavemultipleedges;agraph issupposed tobesimple;loopsarenotallowed.

Foraninteger

k

≥ 1

,a

k

-frugal olouring ofagraph

G

isapropervertex olouringof

G

(i.e., adja ent verti esget adierent olour) su h that no olour appears more than

k

times in the neighbourhood ofanyvertex. The leastnumberof oloursin a

k

-frugal olouringof

G

is alled the

k

-frugal hromati number, denoted

χ

k

(G)

. Clearly,

χ

1

(G)

is the hromati number of the square of

G

;and for

k

atleast themaximumdegreeof

G

,

χ

k

(G)

is theusual hromati number of

G

.

A

k

-frugal edge olouring ofa multigraph

G

is a(possibly improper) olouringof the edges of

G

su h that no olour appears more than

k

times on the edges in ident with avertex. The leastnumberof oloursina

k

-frugaledge olouringof

G

,the

k

-frugaledge hromati number (or

k

-frugal hromati index),isdenotedby

χ

′

k

(G)

. Remarkthat for

k

= 1

wehave

χ

′

1

(G) = χ

′

(G)

, thenormal hromati indexof

G

.

When onsidering the possibility that ea h vertexor edge hasa list of available olours, we entertheareaoffrugal list(edge) olourings.

Frugal vertex olouringswere introdu ed byHind et al [13, 14℄, asatool towardsimproving results aboutthe total hromati number of a graph. One of their resultsis that a graph with largeenoughmaximumdegree

∆

hasa

(log

8

∆)

-frugal olouringusingatmost

∆+ 1

olours. They alsoshowthatthereexistgraphsforwhi ha

log ∆

log log ∆

-frugal olouring annotbea hievedusing only

O(∆)

olours.

Ouraiminthisnoteisto studysomeaspe tsoffrugal olouringsandfrugallist olouringsin theirownright. Intherst partwe onsider frugalvertex olouringsofplanargraphs. Weshow thatforplanargraphs,frugal olouringare loselyrelatedtoseveralotheraspe tsthathavebeen thetopi ofextensiveresear hthelast oupleofyears. Inparti ular,weexhibit lose onne tions with olouringthesquare, y li olourings,and

L(p, q)

-labellings.

Inthenal se tionwederivesomeresultsonfrugaledge olouringsofmultigraphsingeneral.

1.1 Further notation and denitions Given agraph

G

, thesquare of

G

, denoted

G

2

, is thegraphwith thesamevertexset as

G

and withanedgebetweenanytwodierentverti esthathavedistan eatmosttwoin

G

. Wealways assume that olours are integers, whi h allows us to talk about the distan e

|γ

1

− γ

2

|

of two olours

γ

1

, γ

2

.

The hromati numberof

G

,denoted

χ(G)

,istheminimumnumberof oloursrequiredsothat we anproperly olour its verti esusing those olours. A

t

-list assignment

L

on the verti es of agraphis afun tion whi hassignsto ea h vertex

v

ofthemultigraphalist

L(v)

of

t

pres ribed integers. Thelist hromati number or hoi enumber

ch

(G)

istheminimumvalue

t

, sothat for ea h

t

-list assignment onthe verti es, we annd a proper olouring in whi h ea h vertex gets assigneda olourfrom itsownprivatelist.

Weintrodu ed

k

-frugal olouring andthe

k

-frugal hromati number

χ

k

(G)

intheintrodu tory part. Inasimilarwaywe andene

k

-frugallist olouringandthe

k

-frugal hoi enumber

ch

k

(G)

.

Furtherdenitionsonedge olouringswill appearin thenalse tion.

2 Frugal Colouring of Planar Graphs

Inthenextfourse tionswe onsider

k

-frugal(list) olouringsofplanargraphs. Foralargepart, ourworkinthat areaisinspiredbyawell-known onje tureofWegneronthe hromati number of squaresof planargraphs. If

G

hasmaximum degree

∆

, thena vertex olouringof itssquare willneedatleast

∆ + 1

olours,but thegreedyalgorithmshowsitisalwayspossiblewith

∆

olours. Diameter two ages su h asthe 5- y le,thePetersengraphand theHoman-Singleton graph(see[1, page239℄)showthat thereexistgraphsthat infa t require

∆

2

_{+ 1}

olours. Forplanargraphs,Wegner onje turedthat farlessthan

∆

2

_{+ 1}

oloursshouldsu e. Conje ture 2.1 (Wegner[24℄)

Foraplanar graph

G

of maximumdegree

∆(G) ≥ 8

wehave

χ(G

2

_{) ≤}



3

2

∆(G) + 1

.

Wegneralso onje turedmaximumvaluesforthe hromati numberofthesquareofplanargraph withmaximumdegreelessthaneightandgaveexamplesshowinghisboundswouldbetight. For even

∆ ≥ 8

,theseexamplesaresket hedinFigure 1.

m

− 1

verti es

m

verti es

m

verti es

z

x

y

Figure1: Theplanargraphs

G

m

.

InspiredbyWegner'sConje ture,we onje turethefollowingboundsforthe

k

-frugal hromati numberofplanargraphs.

Conje ture 2.2

Forany integer

k

≥ 1

andplanar graph

G

with maximumdegree

∆(G) ≥ max { 2 k, 8 }

wehave

χ

k

(G) ≤









∆(G)−1

k

 + 3,

if

k

iseven;



3 ∆(G)−2

3 k−1

 + 3,

if

k

isodd.

Notethatthegraphs

G

m

inFigure1alsoshowthattheboundsinthis onje turearebestpossible. Thegraph

G

m

hasmaximum degree

2 m

. First onsidera

k

-frugal olouring with

k

= 2 ℓ

even. We anusethesame olouratmost

3

2

k

timesontheverti esof

G

m

,andevery olourthatappears exa tly

3

2

k

= 2 ℓ

timesmustappearexa tly

ℓ

timesonea hofthethreesetsof ommonneighbours of

x

and

y

,of

x

and

z

,andof

y

and

z

. Sowe antakeatmost

1

ℓ

(m − 1) =

1

k

(∆(G

m

) − 1)

olours thatareused

3

2

k

times. Thegraphthat remains anbe olouredusingjust three olours. If

k

= 2 ℓ + 1

isodd,thenea h olour anappearatmost

3 ℓ + 1 =

1

2

(3 k − 1)

times,andthe onlywaytousea oloursomanytimesisbyusingitontheverti esin

V

(G

m

) \ {x, y, z}

. Doing thisatmost

3 m−1

(3 k−1)/2

=

3 ∆(G)−2

3 k−1

times,weareleftwithagraphthat anbe olouredusingthree olours.

Wenextderivesomeupperbounds onthe

k

-frugal hromati numberof planargraphs. The rstoneis asimpleextensionof theapproa hfrom [11℄. Inthat paper, thefollowingstru tural lemmaisderived.

Lemma2.3 (Vanden Heuvel & M Guinness[11℄)

Let

G

be aplanar simple graph. Then there exists avertex

v

with

m

neighbours

v

1

, . . . , v

m

with

d(v

1

) ≤ · · · ≤ d(v

m

)

su hthat oneofthe followingholds: (i)

m

≤ 2

;

(ii)

m

= 3

with

d(v

1

) ≤ 11

;

(iii)

m

= 4

with

d(v

1

) ≤ 7

and

d(v

2

) ≤ 11

;

(iv)

m

= 5

with

d(v

1

) ≤ 6

,

d(v

2

) ≤ 7

,and

d(v

3

) ≤ 11

.

Van den Heuvel and M Guinness [11℄ use this stru tural lemma to prove that the hromati numberofthesquareofaplanargraphisatmost

2 ∆ + 25

. Makingsomeslight hangesin their proof,itisnotdi ultto obtainarstbound on

ch

k

(andhen eon

χ

k

)forplanargraphs. Theorem2.4

Forany planargraph

G

with

∆(G) ≥ 12

and integer

k

≥ 1

wehave

ch

k

(G) ≤



2 ∆(G)+19

k

 + 6

.

Proof Wewill provethat ifaplanargraphsatises

∆(G) ≤ C

forsome

C

≥ 12

,then

ch

k

(G) ≤



2 C+19

k

 + 6

. We useindu tion on thenumber ofverti es, notingthat the resultis obvious for small graphs. So let

G

be a graph with

|V (G)| > 1

, hoose

C

≥ 12

so that

∆(G) ≤ C

, and assumeea hvertex

v

hasalist

L(v)

of



2 C+19

k

 + 6

olours. Take

v, v

1

, . . . , v

m

asinLemma2.3. Contra tingtheedge

vv

1

toanewvertex

v

′

willresultinaplanargraph

G

′

in whi hallverti es ex ept

v

′

havedegreeat mostasmu hastheyhadin

G

,while

v

′

hasdegreeatmost

∆(G)

(for ase(i)) oratmost 12. (for the ases(ii)(iv)). Inparti ular wehavethat

∆(G

′

) ≤ C

. Ifwe give

v

′

thesamelistof oloursas

v

1

had(allverti esin

V

(G) \ {v, v

1

}

keeptheirlist),then,using indu tion,

G

′

hasa

k

-frugal olouring. Usingthesame olouringfor

G

,where

v

1

getsthe olour

v

′

hadin

G

′

,weobtaina

k

-frugal olouringof

G

withtheone de itthat

v

hasno olouryet. But thenumberof oloursforbiddenfor

v

arethe oloursonitsneighbours,andforea hneighbour

v

i

, the oloursthatalreadyappear

k

timesaround

v

i

. Sothenumberofforbidden oloursisatmost

m

+

m

P

i=1



d(v

i

)−1

k



. Usingtheknowledgefromthe ases(i)(iv),wegetthat

|L(v)| =



2 C+19

k

 + 6

isat leastonemorethanthis numberofforbidden olours, hen ewealways anndanallowed

olourfor

v

.

2

Inthenextse tionwewill obtain(asymptoti ally)betterresultsbasedon morere entwork on spe iallabellingsofplanargraphs.

3 Frugal Colouring and

L(p, q)

-Labelling

Let

dist

(u, v)

denote thedistan ebetweentwoverti es

u, v

in agraph. Forintegers

p, q

≥ 0

, an

L(p, q)

-labelling of

G

isanassignment

f

ofintegerstotheverti esof

G

su hthat:

• |f (u) − f (v)| ≥ p

,if

dist

(u, v) = 1

,and

• |f (u) − f (v)| ≥ q

,if

dist

(u, v) = 2

.

The

λ

p,q

-numberof

G

,denoted

λ

p,q

(G)

,isthesmallest

t

su hthatthereexistsan

L(p, q)

-labellings of

G

using labels from

1, 2, . . . , t

.

1

. Of ourse we an also onsider the list version of

L(p, q)

-labellings. Givenagraph

G

, thelist

λ

p,q

-number, denoted

λ

l

p,q

(G)

,is thesmallestinteger

t

su h that, for every

t

-list assignment

L

on the verti esof

G

, there exists an

L(p, q)

-labelling

f

su h that

f

(v) ∈ L(v)

foreveryvertex

v

.

Thefollowingisaneasyrelationbetweenfrugal olouringsand

L(p, q)

-labellings. Proposition 3.1

Forany graph

G

andinteger

k

≥ 1

wehave

χ

k

(G) ≤



1

k

λ

k,1

(G)



and

ch

k

(G) ≤



1

k

λ

l

k,1

(G)



. 1

Thedenitionof

λ

p,q

(G)

isnotuniforma rosstheliterature.Manyauthorsdeneitastheminimumdistan e between the largestand smallestlabel used, whi hgivesa

λ

-value oneless than withourdenition. We hose ourdenitionssin eitmeansthat

λ

1,1

(G) = χ(G

2

₎

,andsin eittsmorenaturalwiththe notionoflist

L(p, q)

-labellings.

Proof We only prove the se ond part, the rst one an be done in a similar way. Set

ℓ

=



1

k

λ

l

k,1

(G)



,andlet

L

bean

ℓ

-listassignmentontheverti esof

G

. Usingthatallelementsinthe listsareintegers,we andeneanewlistassignment

L

∗

bysetting

L

∗

_{(v) =}

S

x∈L(v)

{k x, k x + 1,

. . . , k x

+ k − 1}

. Then

L

∗

isa

(k ℓ)

-listassignment. Sin e

k ℓ

≥ λ

l

k,1

(G)

, thereexistsan

L(k, 1)

-labelling

f

∗

of

G

with

f

∗

_{(v) ∈ L}

∗

_(v)

forall verti es

v

. Dene anew labelling

f

of

G

bytaking

f

(v) =



₁

k

f

∗

_(v)



. Weimmediatelygetthat

f

(v) ∈ L(v)

forall

v

. Sin eadja entverti esre eived an

f

∗

-labelatleast

k

apart,their

f

-labelsaredierent. Also,allverti esinaneighbourhoodofa vertex

v

re eivedadierent

f

∗

-label. Sin ethemap

x

7→



1

k

x



mapsatmost

k

dierentintegers

x

to the same image, ea h

f

-label anappear at most

k

times in ea h neighbourhood. So

f

is a

k

-frugal olouringusing labels from ea h vertex'list. This provesthat

ch

k

(G) ≤ ℓ

, asrequired.

2

Wewill ombinethis propositionwiththefollowingre entresult. Theorem3.2 (Havetet al [9℄)

Forea h

ǫ >

0

,thereexistsaninteger

∆

ǫ

sothatthe followingholds. If

G

isaplanar graphwith maximumdegree

∆(G) ≥ ∆

ǫ

,and

L

isalist assignmentsothat ea h vertexgetsalist of atleast

(

3

2

+ ǫ) ∆(G)

integers,then we an nd aproper olouring of the squareof

G

using olours from thelists. Moreover,we antakethisproper olouringsothatthe oloursonadja entverti esof

G

dier byatleast

∆(G)

1/4

.

Intheterminologyweintrodu edearlier,animmediate orollaryisthefollowing. Corollary3.3

Fix

ǫ >

0

and an integer

k

≥ 1

. Then there existsan integer

∆

ǫ

sothat if

G

is a planar graph withmaximum degree

∆(G) ≥ max { ∆

ǫ

, k

4

_}

,then

λ

l

k,1

(G) ≤ (

3

2

+ ǫ) ∆(G)

.

Combining this withProposition3.1 givestheasymptoti ally best upperbound for

χ

k

and

ch

k

forplanargraphswe urrentlyhave.

Corollary 3.4

Fix

ǫ >

0

andan integer

k

≥ 1

. Then there existsaninteger

∆

ǫ,k

so thatif

G

isaplanar graph withmaximum degree

∆(G) ≥ ∆

ǫ,k

,then

ch

k

(G) ≤

(3+ǫ) ∆(G)

2 k

.

In[17℄,MolloyandSalavatipourprovedthatforanyplanargraph

G

,wehave

λ

k,1

(G) ≤



₅

3

∆(G)+

18 k + 60

. Together with Proposition 3.1, this renes the result of Proposition 2.4 and gives a better bound than Corollary3.4 for small values of

∆

. Note that this orollary only on erns frugal olouring,andnotfrugallist olouring.

Corollary 3.5

Forany planargraph

G

andinteger

k

≥ 1

,wehave

χ

k

(G) ≤



5 ∆(G)+180

3 k

 + 18

.

Proposition3.1hasanother orollaryforplanargraphsoflargegirththatwedes ribebelow. The girth ofagraphisthelengthofashortest y leinthegraph.

In[23℄,LihandWang provedthat forplanargraphsoflargegirththefollowingholds:

• λ

p,q

(G) ≤ (2 q − 1) ∆(G) + 6 p + 12 q − 8

forplanargraphsofgirthat leastsix,and

• λ

p,q

(G) ≤ (2 q − 1) ∆(G) + 6 p + 24 q − 14

forplanargraphsofgirthatleastve.

Furthermore,Dvo°áket al [5℄proved thefollowingtightbound for

(k, 1)

-labellingsof planar graphsofgirthatleastseven,andoflargedegree.

Theorem3.6 (Dvo°ák et al [5℄)

Let

G

beaplanar graphof girthatleastseven,andmaximumdegree

∆(G) ≥ 190 + 2 k

,for some integer

k

≥ 1

. Then wehave

λ

k,1

(G) ≤ ∆(G) + 2 k − 1

.

Adire t orollaryof theseresultsarethefollowingboundsforplanargraphswithlargegirth. Corollary3.7

Let

G

beaplanar graphwith girth

g

andmaximumdegree

∆(G)

. Forany integer

k

≥ 1

,we have

χ

k

(G) ≤

















∆(G)−1

k

 + 2,

if

g

≥ 7

and

∆(G) ≥ 190 + 2 k

;



∆(G)+4

k

 + 6,

if

g

≥ 6

;



∆(G)+10

k

 + 6,

if

g

≥ 5

.

4 Frugal Colouring of Outerplanar Graphs

Wenowprovea variantof Conje ture2.2 for outerplanar graphs (graphsthat anbedrawn in theplanesothatallverti esarelyingontheoutsidefa e). For

k

= 1

,i.e.,ifweare olouringthe squareofthegraph,HetheringtonandWoodall[10℄provedthebestpossibleboundforouterplanar graphs

G

:

ch

1

(G) ≤ ∆(G) + 2

if

∆(G) ≥ 3

, and

ch

1

(G) = ∆(G) + 1

if

∆(G) ≥ 6

.

Theorem4.1

For any integer

k

≥ 2

and any outerplanar graph

G

with maximum degree

∆(G) ≥ 3

, we have

χ

k

(G) ≤ ch

k

(G) ≤



∆(G)−1

_k

 + 3

.

Proof EsperetandO hem [6℄provedthatanyouterplanargraph ontainsavertex

u

su hthat oneofthefollowingholds: (i)

u

hasdegreeat mostone;(ii)

u

hasdegreetwoandisadja entto anothervertexof degreetwo; or(iii)

u

hasdegreetwoandits neighbours

v

and

w

are adja ent, and either

v

has degree three or

v

has degree four and its twoother neighbours (i.e., distin t from

u

and

w

)areadja ent.

Let

G

bea ounterexample to the theorem with minimum numberof verti es, and let

u

be a vertex of

G

having oneof the properties des ribed above. By minimality of

G

, there exists a

k

-frugal list olouring

c

of

G

− u

if the lists

L(v)

ontain at least



∆(G)−1

k

 + 3

olours. If

u

has property (i) or (ii), let

t

be the neighbour of

u

whose degreeis not ne essarily bounded by two. It is easy to see that at most

2 +



∆(G)−1

k



olours are forbidden for

u

: the olours of the neighbours of

u

and the olours appearing

k

times in the neighbourhood of

t

. If

u

has property (iii), at most

2 +



∆(G)−2

k



oloursare forbidden for

u

: the olours of the neighbours of

u

andthe oloursappearing

k

timesintheneighbourhoodof

w

. Notethatif

v

hasdegreefour, itstwootherneighboursareadja entandthe

k

-frugalityof

v

isrespe tedsin e

k

≥ 2

. Inall ases we found that at most



∆(G)−1

k

 + 2

olours are forbidden for

u

. If

u

has alist with onemore olour,we anextend

c

to a

k

-frugallist olouringof

G

, ontradi tingthe hoi eof

G

.

2

We anrenethisresultin the aseof2- onne ted outerplanargraphs,providedthat

∆

islarge enough.

Theorem4.2 For any integer

k

≥ 1

and any 2- onne ted outerplanar graph

G

with maximum degree

∆(G) ≥ 7

,wehave

ch

k

(G) ≤



∆(G)−2

k

 + 3

.

Proof InLihandWang[16℄itisprovedthatany2- onne tedouterplanargraphswithmaximum degree

∆ ≥ 7

ontainsavertex

u

ofdegreetwothathasatmost

∆ − 2

verti esatdistan eexa tly two.

Let

G

bea ounterexampleto thetheorem withminimumnumberofverti es,andlet

u

bea vertexof

G

havingthe property des ribed above,and let

v

and

w

be itsneighbours. Let

H

be

G

− u

iftheedge

vw

exists,or

G

− u + vw

otherwise. Byminimalityof

G

,thereisa

k

-frugallist olouring

c

of

H

ifalllists ontainatleast



∆(G)−2

k

 + 3

olours. Atmost



∆(G)−2

k

 + 2

oloursare forbiddenfor

u

: the oloursof

v

and

w

,andthe oloursappearing

k

timesintheirneighbourhood. So,the olouring

c

of

H

anbeextendedtoa

k

-frugallist olouringof

G

, ontradi tingthe hoi e

5 Frugal Colouring and Cy li Colouring

Inthisse tion,wedis ussthelink betweenfrugal olouringand y li olouringofplanegraphs. A plane graph

G

is aplanar graph with a pres ribed planar embedding. The size (numberof verti esinitsboundary)ofalargestfa eof

G

isdenotedby

∆

∗

(G)

.

A y li olouring of aplane graph

G

is avertex olouringof

G

su h that anytwoverti es in identto the samefa ehavedistin t olours. This on eptwasintrodu ed byOre and Plum-mer [18℄, whoalso provedthat aplane graphhasa y li olouring usingat most

2 ∆

∗

olours. Borodin[2℄(seealsoJensenandToft[15,page37℄) onje turedthatanyplanegraphhasa y li olouring with



3

2

∆

∗



olours, and proved this onje ture for

∆

∗

_{= 4}

. The best known upper boundin thegeneral aseisdueto Sandersand Zhao[20℄,whoprovedthatanyplanegraphhas a y li olouringwith



5

3

∆

∗



olours.

There appears to be astrong onne tion between bounds on olouring the square of planar graphsand y li olourings of plane graphs. Oneshould only ompare Wegner's onje ture in Se tion 2withBorodin's onje tureabove,and thesu essivebounds obtainedforea h ofthese onne tions. Nevertheless, thesimilar looking bounds for these types of olouringshave always requiredindependentproofs. Noexpli itrelationthatwouldmakeitpossibletotranslatearesult ononeofthetypesof olouringintoaresultfortheothertype,haseverbeenderived.

Inthis se tionwe show thatif there isan even

k

≥ 4

sothat Borodin's onje ture holds for allplanegraphswith

∆

∗

≤ k

,andourConje ture2.2istrueforthesamevalue

k

,thenWegner's onje tureistrueuptoanadditive onstantfa tor.

Theorem5.1

Let

k

≥ 4

beanevenintegersu hthateveryplanegraph

G

with

∆

∗

_{(G) ≤ k}

hasa y li olouring usingatmost

3

2

k

olours. Then, if

G

isaplanar graphsatisfying

χ

k

(G) ≤



∆(G)−1

k

 + 3

,wealso have

χ(G

2

_{) = χ}

1

(G) ≤



3

2

∆(G) +

9

2

k

− 1

.

Proof Let

G

beaplanargraphwithagivenembeddingand let

k

≥ 4

beanevenintegersu h that

t

= χ

k

(G) ≤



∆(G)−1

k

 + 3

. Consider an optimal

k

-frugal olouring

c

of

G

, with olour lasses

C

1

, . . . , C

t

. For

i

= 1, . . . , t

, onstru t the graph

G

i

as follows: Firstly,

G

i

has vertex set

C

i

,whi h weassumetobeembeddedintheplaneinthesamewaytheywerefor

G

. Forea h vertex

v

∈ V (G) \ C

i

with exa tly two neighbours in

C

i

, we add an edge in

G

i

between these twoneighbours. For avertex

v

∈ V (G) \ C

i

with

ℓ

≥ 3

neighboursin

C

i

, let

x

1

, . . . , x

ℓ

bethose neighboursin

C

i

ina y li orderaround

v

(determinedbytheplaneembeddingof

G

). Nowadd edges

x

1

x

2

, x

2

x

3

, . . . , x

ℓ−1

x

ℓ

and

x

ℓ

x

1

to

G

i

. These edgeswill formafa e ofsize

ℓ

in thegraph wehave onstru tedsofar. Callsu hafa e aspe ialfa e. Notethatsin e

C

i

is a olour lassin a

k

-frugal olouring,this fa ehassizeatmost

k

.

Do the above for all verti es

v

∈ V (G) \ C

i

that have at least two neighbours in

C

i

. The resulting graphis aplane graph with some fa es labelled spe ial. Addedges to triangulate all fa esthatarenotspe ial. Theresultinggraphisaplanegraphwithvertexset

G

i

andeveryfa e size at most

k

. Fromthersthypothesisit followsthat we an y li ly olourea h

G

i

with

3

2

k

new olours. Sin eeverytwoverti esin

C

i

thathavea ommonneighbourin

G

areadja entin

G

i

or are in ident to the same (spe ial) fa e, verti es in

C

i

that are adja ent in the square of

G

re eivedierent olours. Hen e, ombiningthese

t

olourings,usingdierent oloursforea h

G

i

, we obtaina olouringof thesquare of

G

, using atmost

3

2

k

·



∆(G)−1

k

 + 3
≤ 

3

2

∆ +

9

2

k

− 1

olours.

2

Sin eBorodin[2℄provedhis y li olouring onje tureinthe ase

∆

∗

= 4

,wehavethefollowing orollary.

Corollary 5.2

If

G

isaplanar graphsothat

χ

4

(G) ≤



∆(G)−1

4

 + 3

,then

χ(G

2

_{) ≤}



3

6 Frugal Edge Colouring

An importantelementin theproof in[9℄of Theorem3.2 mentionedearlieristhederivation ofa relationbetween(list) olouringsquareofplanargraphsandedge(list) olouringsofmultigraphs. Be auseofthis,itseemstobeopportunetohaveashortlookatafrugalvariantofedge olourings ofmultigraphsingeneral.

Ifweneed to properly olour theedges of amultigraph

G

, the minimum number of olours requiredis the hromati index,denoted

χ

′

_(G)

. Thelist hromati index

ch

′

(G)

isdened anal-ogously as the minimum length of list that needs to be given to ea h edge so that we an use oloursfromea hedge'slisttogiveaproper olouring.

A

k

-frugal edge olouring ofa multigraph

G

is a(possibly improper) olouringof the edges of

G

su h that no olour appears more than

k

times on the edges in ident with avertex. The leastnumberof oloursina

k

-frugaledge olouringof

G

,the

k

-frugaledge hromati number (or

k

-frugal hromati index),isdenoted by

χ

′

k

(G)

.

Notethata

k

-frugaledge olouringof

G

isnotthesameasa

k

-frugal olouringoftheverti es ofthelinegraph

L(G)

of

G

. Sin etheneighbourhoodofanyvertexinthelinegraph

L(G)

anbe partitioned into at mosttwo liques, everyproper olouring of

L(G)

isalso a

k

-frugal olouring for

k

≥ 2

. A 1-frugal olouring of

L(G)

(i.e., avertex olouring of the square of

L(G)

) would orrespondto aproperedge olouringof

G

in whi h ea h olour lassindu es amat hing. Su h olouringsareknownasstrongedge olourings, see,e.g.,[7℄.

Thelistversionof

k

-frugaledge olouring analsobedened in thesameway: givenlistsof size

t

forea hedgeof

G

,oneshouldbeabletonda

k

-frugaledge olouringsu hthatthe olour ofea hedgebelongstoitslist. Thesmallest

t

withthispropertyis alledthe

k

-frugaledge hoi e number,denoted

ch

′

k

(G)

.

Frugaledge olouringsanditslistversionwerestudiedunderthenameimproperedge- olourings andimproper L-edge- olourings byHiltonetal [12℄.

It is obvious that the hromati index and the edge hoi e numbers are alwaysat least the maximumdegree

∆

. Thebest possibleupperbounds in terms ofthe maximumdegreeonly are givenbythefollowingresults.

Theorem6.1

(a) For asimplegraph

G

wehave

χ

′

_{(G) ≤ ∆(G) + 1}

. (Vizing [22℄) (b) For amultigraph

G

wehave

χ

′

_{(G) ≤}



3

2

∆(G)



. (Shannon[21℄) ( ) For abipartite multigraph

G

wehave

ch

′

(G) = ∆(G)

. (Galvin [8℄) (d) For amultigraph

G

wehave

ch

′

(G) ≤



3

2

∆(G)



. (Borodinet al [3℄) WewilluseTheorem6.1( )and(d)toprovetworesultsonthe

k

-frugal hromati indexandthe

k

-frugal hoi e number. The rstresultshowsthat foreven

k

, themaximumdegree ompletely determines thevalues of these twonumbers. This result wasearlier proved in [12℄ in aslightly moregeneralsetting,involvingamore ompli atedproof.

Theorem6.2 (Hilton et al [12℄)

Let

G

beamultigraph, andlet

k

be aneveninteger. Then wehave

χ

′

k

(G) = ch

′

k

(G) =



1

k

∆(G)



.

Proof Itisobviousthat

ch

′

k

(G) ≥ χ

′

k

(G) ≥



1

k

∆



,soitsu estoprove

ch

′

k

(G) ≤



1

k

∆



. Let

k

= 2 ℓ

. Withoutlossofgenerality,we anassume

∆

isamultipleof

k

and

G

isa

∆

-regular multigraph. (Otherwise,we anaddsomenewedgesand,ifne essary,somenewverti es. Ifthis larger multigraphis

k

-frugal edge hoosable with lists of size



1

k

∆



, then so is

G

.) As

k

, and hen e

∆

, is even, we annd an Euler tour in ea h omponent of

G

. By given these tours a dire tion,weobtainanorientation

D

oftheedgesof

G

su hthatthein-degreeandtheout-degree ofeveryvertexis

1

2

∆

. Letusdenethebipartitemultigraph

H

= (V

1

∪ V

2

, E)

asfollows:

V

1

, V

2

areboth opiesof

V

(G)

. Foreveryar

(a, b)

in

D

,weaddanedge between

a

∈ V

1

and

b

∈ V

2

.

Sin e

D

is adire ted multigraph with in- andout-degree equalto

1

2

∆

,

H

is a

(

1

2

∆)

-regular bipartitemultigraph. Thatmeanswe ande omposetheedgesof

H

into

1

M

1

, M

2

, . . . , M

∆/2

. Denedisjointsubgraphs

H

1

, H

2

, . . . , H

ℓ

asfollows: for

i

= 0, 1, . . . , ℓ − 1

set

H

i+1

= M

i

k

∆+1

∪ M

i

k

∆+2

· · · ∪ M

i+1

k

∆

. Noti ethat ea h

H

i

is abipartitemultigraphof regular degree

1

k

∆

.

Now,supposethat ea hedge omeswithalistof oloursofsize

1

k

∆

. (Ifwehadtoaddedges tomake

∆

amultipleof

k

orthemultigraph

∆

-regular,thengivearbitraryliststo theseedges.) Ea h subgraph

H

i

hasmaximumdegree

1

k

∆

,sobyGalvin's theoremwe anndaproperedge olouringof ea h

H

i

su h that the olourof ea h edge isinside itslist. We laimthat thesame olouringofedgesin

G

is

k

-frugal. Forthisweneedthefollowingobservation:

Observation Let

M

be a mat hing in

H

. Then the set of orresponding edges in

G

form a subgraphofmaximum degreeatmosttwo.

Toseethis,remarkthatea hvertexhastwo opiesin

H

: onein

V

1

andonein

V

2

. The ontribution oftheedgesof

M

to avertex

v

in theoriginalmultigraphisthenatmosttwo,atmostonefrom ea h opyof

v

.

To on lude,weobservethat ea h olour lassin

H

istheunionofatmost

ℓ

mat hings,one in ea h

H

i

. So at ea h vertex, ea h olour lassappearsat most twotimes thenumberof

H

i

's, i.e.,atmost

2 ℓ = k

times. Thisisexa tlythe

k

-frugality onditionweset outto satisfy.

2

Foroddvaluesof

k

wegiveatightupperboundofthe

k

-frugaledge hromati number.

Theorem6.3

Let

k

beanodd integer. Then wehave



∆(G)

k

 ≤ χ

′

k

(G) ≤ ch

′

k

(G) ≤



3 ∆(G)

3 k−1



.

Proof Again,allwehavetoproveis

ch

′

k

(G) ≤



3 ∆(G)

3 k−1



.

Let

k

= 2 ℓ + 1

. Sin e

3 k − 1

is even and not divisible by three, we an again assume, withoutlossof generality, that

∆

iseven and divisibleby

3 k − 1

, and that

G

is

∆

-regular. Set

∆ = m (3 k − 1) = 6 ℓ m + 2 m

. Usingthesameideaasinthepreviousproof,we ande ompose

G

intotwosubgraphs

G

1

, G

2

,where

G

1

is

(6 ℓ m)

-regularand

G

2

is

(2 m)

-regular. (Alternatively,we anusePetersen'sTheorem [19℄thateveryevenregularmultigraphhasa2-fa tor,tode ompose theedgesetin2-fa tors,and ombinethese2-fa torsappropriately.) Sin e

1

2 ℓ

·6 ℓ m =

3

3 k−1

∆

,by Theorem6.2 weknowthat

G

1

hasa

2 ℓ

-frugaledge olouringusing the oloursfromea hedge's lists. Similarly we have

3

2

· 2 m =

3

3 k−1

∆

, and hen e Theorem 6.1(d) guarantees that we an properly olourthe edgesof

G

2

using olours from thoseedges' lists. The ombinationof these two olouringsisa

(2 ℓ + 1)

-frugallistedge olouring,asrequired.

2

NotethatTheorem6.3isbestpossible: For

m

≥ 1

,let

T

(m)

bethemultigraphwiththreeverti es and

m

paralleledgesbetweenea hpair. If

k

= 2 ℓ + 1

isodd,thenthemaximumnumberofedges with thesame oloura

k

-frugal edge olouring of

T

(m)

anhaveis

3 ℓ + 1

. Hen e the minimum numberof oloursneededfora

k

-frugaledge olouringis

l

3 m

3 ℓ+1

m

=

l

3

3 k−1

∆(T

(m)

₎

m

. 7 Dis ussion

Asthisis oneoftherstpapersonfrugal olouring,manypossibledire tionsforfuture resear h are still open. An intriguing questionis inspired by the resultson frugal edge olouring in the previousse tion. Theseresultsdemonstrateanessentialdieren ebetweenevenandodd

k

asfar as

k

-frugaledge olouringis on erned. Based on what we think are the extremal examplesof planargraphsfor

k

-frugalvertex olouring,alsoourConje ture2.2givesdierentvaluesforeven andodd

k

. Butforfrugalvertex olouringsofplanargraphsingeneralwehavenotbeenableto obtainresults that are dierentfor even and odd

k

. Mostof our resultsfor vertex olouringof planargraphsare onsequen esofProposition3.1andknownresultson

L(k, 1)

-labellingofplanar graphs,forwhi hnofundamentaldieren ebetweenoddandeven

k

haseverbeendemonstrated. Hen e,amajorstepwouldbetoprovethatProposition 3.1isfarfrom tightwhen

k

iseven.

Ase ond lineof future resear h ouldbeto investigatewhi h lassesof graphshave

k

-frugal hromati numberequaltotheminimumpossiblevalue



∆

k

 + 1

. Corollary3.7andTheorems4.1 and 4.2givebounds forplanar graphswith largegirthand outerplanargraphswith large maxi-mum degreethat are very lose to thebest possiblebound. We onje ture that,in fa t, planar graphswithlargeenoughgirthandouterplanargraphsoflargeenoughmaximumdegreedosatisfy

χ

k

(G) =



∆(G)

_k

 + 1

forall

k

≥ 1

.

Referen es

[1℄ J.A. Bondy and U.S.R. Murty, Graph Theory with Appli ations. Ma millan, London and Elsevier,NewYork,1976.

[2℄ O.V. Borodin,Solution of the Ringelproblem on vertex-fa e oloringof planargraphsand oloringof1-planargraphs(in Russian).MetodyDiskret.Analyz.41(1984),1226. [3℄ O.V. Borodin, A.V. Kosto hka, and D.R. Woodall, List edge and list total olourings of

multigraphs.J.Combin.TheorySer.B71(1997),184204.

[4℄ R.Diestel,GraphTheory,3rdedition.Springer-Verlag,BerlinandHeidelberg,2005. [5℄ Z.Dvo°ák,D.Král,P.Nejedlý,andR.’krekovski,Coloringsquaresofplanargraphswithno

short y les.Manus ript,2005.

[6℄ L. Esperet and P. O hem, Oriented olorings of 2-outerplanar graphs. Inform. Pro ess. Lett.101(2007),215219.

[7℄ J.-L.FouquetandJ.-L.Jolivet,Strongedge- oloringsofgraphsandappli ations tomulti-

k

-gons.ArsCombin.16(1983),141150.

[8℄ F.Galvin,Thelist hromati indexofabipartitemultigraph.J.Combin.TheorySer. B63 (1995),153158.

[9℄ F.Havet,J. van denHeuvel, C.M Diarmid, and B. Reed,List olouring squaresof planar graphs.Inpreparation.

[10℄ T.J. Hetherington and D.R. Woodall, List- olouring the square of an outerplanar graph. Manus ript,2006.

[11℄ J. van den Heuvel and S. M Guinness, Coloring the square of a planar graph. J. Graph Theory42(2003),110124.

[12℄ A.J.W.Hilton,T.Slivnik, andD.S.G.Stirlingilton, Aspe ts ofedgelist- olourings.Dis rete Math.231(2001),253264.

[13℄ H. Hind, M. Molloy, and B. Reed, Colouring a graph frugally. Combinatori a 17 (1997), 469482.

[14℄ H. Hind, M. Molloy, and B. Reed, Total oloring with

∆ + poly(log ∆)

olors. SIAM J. Comput.28(1999),816821.

[15℄ T.R.Jensen,B.Toft,GraphColoringProblems.John-Wiley&Sons,NewYork,1995. [16℄ K.-W. Lih and W.-F. Wang Coloring the square of an outerplanar graph. Taiwanese J. of

Math.10(2006),10151023.

[17℄ M. Molloy and M.R. Salavatipour, A bound on the hromati number of the square of a planargraph.J.Combin.TheorySer.B94(2005),189213.

[18℄ O.OreandM.D.Plummer,Cy li olorationofplanegraphs.In: Re entProgressin Combi-natori s,Pro eedingsoftheThird WaterlooConferen eonCombinatori s, A ademi Press, SanDiego(1969)287293.

[19℄ J.Petersen,DieTheoriederregulärenGraphs.A taMath.15 (1891)193220.

[20℄ D.P.SandersandY.Zhao,Anewboundonthe y li hromati number.J.Combin.Theory Ser.B83(2001),102111.

[21℄ C.E.Shannon,Atheoremon olouringlinesof anetwork.J.Math. Physi s28(1949),148 151.

[22℄ V.G.Vizing,Onanestimateofthe hromati lassofa

p

-graph(inRussian).MetodyDiskret. Analiz.3(1964),2530.

[23℄ W.-F. Wang and K.-W. Lih, Labelingplanar graphswith onditionson girthand distan e two.SIAMJ.Dis reteMath. 17(2003),264275.

[24℄ G.Wegner,Graphswithgivendiameteranda oloringproblem.Te hni alReport,University ofDortmund,1977.

Contents

1 Introdu tion 3

1.1 Furthernotationanddenitions. . . 3 2 Frugal Colouring ofPlanar Graphs 3 3 Frugal Colouring and

L(p, q)

-Labelling 5 4 Frugal Colouring ofOuterplanar Graphs 7 5 Frugal Colouring and Cy li Colouring 8

6 Frugal Edge Colouring 9

Unité de recherche INRIA Sophia Antipolis

2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France)

Unité de recherche INRIA Futurs : Parc Club Orsay Université - ZAC des Vignes

4, rue Jacques Monod - 91893 ORSAY Cedex (France)

Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique

615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France)

Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France)

Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier (France)

Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France)

Éditeur

INRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France)

http://www.inria.fr