WDM and Directed Star Arboricity

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Omid Amini, Frédéric Havet, Florian Huc, Stéphan Thomassé

To cite this version:

Omid Amini, Frédéric Havet, Florian Huc, Stéphan Thomassé. WDM and Directed Star Arboricity.

[Research Report] RR-6179, INRIA. 2007, pp.20. �inria-00132396v3�

inria-00132396, version 3 - 3 May 2007

a p p o r t

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Thème COM

WDM and Directed Star Arboricity

Omid Amini — Frédéric Havet — Florian Huc

— Stéphan Thomassé

N° 6179

Janvier 2007

Unité de recherche INRIA Sophia Antipolis

OmidAmini

∗†

, Frédéri Havet

†

, FlorianHu

‡ †

, Stéphan Thomassé

§

ThèmeCOMSystèmes ommuni ants

ProjetMASCOTTE

Rapportdere her he n° 6179Janvier200720pages

Abstra t: A digraph is

m

-labelled ifeveryar sis labelledby an integerin

{1, . . . , m}

. Motivated by wavelength assignment for multi asts in opti al star networks, we study

n

-ber olourings of labelled digraph whi h are olouringsof the ar sof

D

su h that at ea hvertex

v

, forea h olour

λ

,

in(v, λ) +

out(v, λ) ≤ n

with

in(v, λ)

thenumberofar s oloured

λ

entering

v

and

out(v, λ)

thenumberoflabels

l

su h that there existsan ar leaving

v

of label

l

oloured

λ

. One likesto nd theminimum number of olours

λ

n

(D)

su h that an

m

-labbelleddigraph

D

has an

n

-ber olouring. Inthe parti ular ase, when

D

is

1

-labelled then

λ

n

(D)

is the dire ted star arbori ty of

D

, denoted

dst(D)

. We rst show that

dst(D) ≤ 2∆

−

_{(D) + 1}

and onje turethat if

∆

−

_{(D) ≥ 2}

then

dst(D) ≤ 2∆

−

_(D)

. Wealso prove

that if

D

is sub ubi then

dst(D) ≤ 3

and that if

∆

+

_{(D), ∆}

−

_{(D) ≤ 2}

then

dst(D) ≤ 4

. Finally, we study

λ

n

(m, k) = max{λ

n

(D) | D

is

m

-labelledand

∆

−

_{(D) ≤ k}}

. We show that if

m ≥ n

then

 m

n

 k

n



+

k

n



≤ λ

n

(m, k) ≤

 m

n

 k

n



+

k

n



+ C

m

2

_{log k}

n

forsome onstant

C

.

Key-words: WDM opti alnetworks,multi asting,graph olouring, omplexity

Thisproje thasbeensupportedbyeuropeanproje tsISTFETAEOLUSandCOST293.

∗

É olePolyte hnique

†

Projet Mas otte, CNRS/INRIA/UNSA, INRIA Sophia-Antipolis, 2004 route des Lu ioles BP 93, 06902 Sophia-AntipolisCedex,Fran eoamini,fhavet,fhu sophia .inr ia.f r

‡

ThisauthorispartiallysupportedbyRégionProven eAlpesCted'Azur.

§

Résumé:

Undigrapheestappelé

m

-étiquetési haquear possèdeunlabeldansl'ensemble

{1, . . . , m}

. Motivé par l'allo ation de fréquen es dans les réseaux optiques WDM, nous étudions les

n−

bres olorations d'undigraphe

m

-étiqueté

D

. Celles- isontles olorationsdesar sde

D

tellesquepour haquesommet

v

de

D

et haque ouleur

λ

,

in(v, λ) + out(v, λ) ≤ n

où

in(v, λ)

est le nombre d'ar sentrants

v

olorés ave

λ

,et

out(v, λ)

estlenombredelabels

l

telsqu'ilexisteunar sortantde

v

étiqueté

l

et oloré

λ

. Le but est de trouverle nombre minimumde ouleurs

λ

n,m

(D)

tel quetout digraphe

m

-étiquetéadmette une

n

-bres olorationave enombrede ouleurs. Dansle asparti ulierd'uneseulebreet lorsque

D

est

1−

étiqueté, e irevientàtrouverl'arbori itéétoilede

D

,notée

dst(D)

.

Nous démontrons que pour tout digraphe

D

, on a

dst(D) ≤ 2∆

−

_{(D) + 1}

. Nous étudions ensuite

l'arbori iteéétoiledesdigraphesdedegréborné. Nousprouvonsquepourlesorientationsdesgraphesde

degrémaximumtrois,onatoujours

dst ≤ 3

. Pourlesorientationsrégulièresdesgraphesdedegré

4

, on démontreque

dst ≤ 4

. Cal ulerlavaleurexa tede

dst

est

N P

-durmêmepourla lassedegraphesorientés dedegréauplus

4

. Finalement,nousétudions

λ

n

(m, k) = max{λ

n

(D) | D

est

m

-étiqutéand

∆

−

_{(D) ≤}

k}

. Nous prouvonsque si

m ≥ n

alors

 m

n

 k

n



+

k

n



≤ λ

n

(m, k) ≤

 m

n

 k

n



+

k

n



+ C

m2 log k

n

pour une onstante

C

.

Mots- lés : réseaux optiques WDM, multi asting, allo ation de fréquen es, oloration de graphes,

1 Introdu tion

The originof this paperis the study of wavelength assignment for multi asts in star network. Partial

results are already obtained by Brandt and Gonzalez [4℄. We are given a star network in whi h a

enter node is onne ted by an opti al ber to a set of nodes

V

. Ea h node

v

of

V

sends a set of multi asts

M

1

(v), . . . , M

s(v)

(v)

tothesetsofnodes

S

1

(v), . . . , S

s(v)

(v)

. UsingWDM(wavelength-division multiplexing), dierent signals may be sent at the same time through the same ber but on dierent

wavelengths. The entralnodeisanall-opti altransmitter: hen e,itmayredire tasignalarrivingfrom

a node on aparti ular wavelength to someof the other nodes on the samewavelength. Thereforefor

ea hmulti ast

M

i

(v)

,

v

should sendthemessagetothe entralnodeonasetofwavelengthssothat the entralnoderedire tittoea hnodeof

S

i

(v)

usingoneofthesewavelengths. Theaimistominimizethe totalnumberofusedwavelengths.

Werststudy theveryfundamental asewhentheberisuniqueandea hvertex

v

sendsaunique multi ast

M (v)

to the set

S(v)

of nodes. Let

D

be the digraph with vertex set

V

su h that the out-neighbourhood of a vertex

v

is

S(v)

. Note that this is a digraph and nota multidigraph (there is no multiplear s)as

S(v)

isaset. Thentheproblemistondthesmallest

k

su hthatthereexistsamapping

φ : V (D) → {1, . . . , k}

satisfyingthetwo onditions:

(i)

φ(uv) 6= φ(vw)

; (ii)

φ(uv) 6= φ(u

′

_v)

.

Su hamappingis alleddire tedstar

k

- olouring. Thedire tedstar arbori ity ofadigraph

D

,denoted by

dst(D)

, istheminimuminteger

k

su hthat thereexists adire ted star

k

- olouring. Thisnotionhas beenintrodu edin[6℄andisananalogofthestararbori itydenedin[1℄. Anarbores en eisa onne ted

digraphin whi h everyvertexhas indegree

1

ex ept one, alled root,whi h hasindegree

0

. A forestis thedisjointunionofarbores en es. A starisanarbores en ein whi h theroot dominatesalltheother

verti es. A galaxyisaforestofstars. Clearly,every olour lassofadire tedstar olouringisagalaxy.

Hen e,thedire tedstar arbori ityof adigraph

D

istheminimumnumberofgalaxiesinto whi h

A(D)

maybepartitioned.

For avertex

v

, its indegree

d

−

_(v)

orresponds to the number of multi asts it re eives. A sensible

assumption is that anode re eives abounded numberof multi asts. Hen e, Brandt and Gonzalez [4℄

studied the dire ted star arbori ity of a digraph

D

with maximum indegree

∆

−

. They showed that

dst(D) ≤ ⌈5∆

−

_/2⌉

. This upper bound is tight if

∆

−

_{= 1}

be ause odd ir uits have dire ted star

arbori ity

3

. Howeverit anbeimprovedforlargervalueof

∆

−

_{= 1}

. We onje turethatif

∆

−

_{≥ 2}

,then

dst(D) ≤ 2∆

−

.

Conje ture 1 Everydigraph

D

withmaximumindegree

k ≥ 2

satises

dst(D) ≤ 2k

.

This onje turewouldbetightasBrandt[3℄showedthatforevery

k

,thereisana y li digraph

D

k

su h that

∆

−

_(D

k

) = k

and

dst(D

k

) = 2k

. Note that to provethis onje ture, it is su ient to prove it for

k = 2

and

k = 3

. Indeeda digraphwith maximumindegree

k ≥ 2

has anar -partitioninto

k/2

digraphswith maximum indegree

2

if

k

is evenand into

(k − 1)/2

digraphs with maximumindegree

2

andonewithmaximumindegree

3

. Inse tion2,weshowthat

dst(D) ≤ 2∆

−

_{+ 1}

andsettleConje ture1

fora y li digraphs.

Remark2 Notethatwerestri tourselvestodigraphs,i.e. ir uitsoflengthtwoarepermitted,butnot

multiplear s. Whenmultiplear sareallowed,alltheboundsabovedonothold. Indeedthemultidigraph

T

k

withthreeverti es

u

,

v

and

w

and

k

parallelar s

uv

,

vw

and

wu

satises

dst(T

k

) = 3k

. Moreover,this exampleisextremalsin eeverymultidigraphsatises

dst(D) ≤ 3∆

−

. Indeedletusshowitbyindu tion:

pi kavertex

v

withoutdegree at most

∆

−

in aterminal strong omponent. A strong omponent

C

of adigraph is terminal if there is no ar leaving

C

, i.e. with tailin

C

and head outsideof

C

. If

v

has

noinneighbour,it isisolatedand weremoveit. Otherwise, we onsider any ar

uv

. Its olourmustbe dierentfromthe oloursof the

d

−

_(u)

ar sentering

u

,the

d

+

_(v)

ar sleaving

v

andthe

d

−

_{(v) − 1}

other

ar sentering

v

,soatmost

3∆

−

_{− 1}

ar sintotal. Hen e,removethear

uv

,applyindu tion,andextend the olouringto

uv

. Therefore,formultidigraphs,thebound

dst(D) ≤ 3∆

−

issharp.

Wethen studythedire ted stararbori ityofadigraphbounded withmaximumdegree. Thedegree

of a vertex

v

is

d(v) = d

−

_{(v) + d}

+

_(v)

. It orresponds to the degree of the vertex in the underlying

multigraph. (Wehaveedgeswithmultipli ity

2

ea htimethereisa ir uitoflengthtwointhedigraph.) The maximum degree of adigraph

D

, denoted

∆(D)

, or simply

∆

when

D

is learly understood from the ontext,is

max{d(v), v ∈ V (D)}

. Letus denoteby

µ(G)

, themaximummultipli ityofanedgein a multigraph. ByVizing'stheorem,one an olourtheedgesofamultigraphwith

∆(G) + µ(G)

oloursso thattwoedgeshavedierent oloursiftheyarein ident. Sin ethemultigraphunderlyingadigraphhas

maximummultipli ityatmosttwo,foranydigraph

D

,

dst(D) ≤ ∆ + 2

. We onje turethefollowing: Conje ture 3 Let

D

be adigraphwith maximumdegree

∆ ≥ 3

. Then

dst(D) ≤ ∆

.

This onje turewouldbetightsin eeverydigraphwith

∆ = ∆

−

hasdire tedstararbori ityatleast

∆

. Inse tion3,weproveConje ture3holdswhen

∆ = 3

.

Pinlou and Sopena [9℄ studied a strongerform of dire ted star arbori ity, alled a ir uiti dire ted

stararbori ity. Theyaddtheextra onditionthatany ir uithastohaveat leastthree distin t olours.

Note that su h anotion applies only to oriented graphs that are digraphs without ir uit of length 2.

Indeedsu ha ir uitmaynotre eive

3

olours. Theyshowedthatthea ir uiti dire tedstararbori ity ofasub ubi (i.e. ea hvertexhasdegreeatmost

3

)orientedgraphisat mostfour. Wegiveanewand veryshortproofofthisresult.

ArststeptowardsConje tures1and3would beto provethefollowingstatement whi h isweaker

thanthesetwo onje tures.

Conje ture 4 Let

k ≥ 2

and

D

be adigraph. If

max(∆

−

_{, ∆}

+

_{) ≤ k}

then

dst(D) ≤ 2k

.

This onje tureholds and is farfrom beingtightfor largevaluesof

k

. IndeedGuiduli [6℄ showedthat if

max(∆

−

_{, ∆}

+

_{) ≤ k}

then

dst(D) ≤ k + 20 log k + 84

. Guiduli's proof is basedon thefa t that,when bothoutandindegree arebounded, the olourof anar dependsonthe olouroffew otherar s. This

boundeddependen yallowstheuseoftheLovászLo al Lemma. Thisideawasrstused byAlgorand

Alon [1℄, for the star arbori ityof undire ted graphs. Note also that Guiduli's resultis (almost) tight

sin ethere are digraphs

D

with

max(∆

−

_{, ∆}

+

_{) ≤ p}

and

dst(D) ≥ p + Ω(log p)

(see [6℄). Notealso that similarly asfor Conje ture1,it issu ientto proveConje ture4for

k = 2

andk=

3

. InSe tion 4,we provethat Conje ture4holdsfor

k = 2

. Bytheaboveremark,itimpliesthatConje ture4holdsforall even

k

.

InSe tion5, weinvestigatethe omplexity ofnding thedire ted stararbori ityof adigraph.

Un-surprisingly, this is an NP-hard problem. More pre isely, weshow that determining the dire ted star

arbori ityofadigraphwithout-and indegreeat most

2

isNP- omplete.

Next,westudythemoregeneral(andmorerealisti )probleminwhi hthe enteris onne tedtothe

onodesof

V

with

n

opti albers. Moroverea hnodemaysentseveralmulti asts. Wemodelitasalabelled digraphproblem: We onsideradigraph

D

onvertexset

V

. Forea hmulti ast

(v, S

i

(v))

weaddtheset of ar s

A

i

(v) = {vw, w ∈ S

i

(v)}

withlabel

i

. Thelabelof anar

a

is denoted by

l(a)

. Thusfor every ouple

(u, v)

ofverti esandlabel

i

thereisatmostonear

uv

labelledby

i

. Ifea hvertexsendsatmost

m

multi asts,thereareatmost

m

labelsonthear s. Su hadigraphissaidtobe

m

-labelled. Onewants tondan

n

-berwavelength assignmentof

D

,that isamapping

Φ : A(D) → Λ × {1, . . . , n} × {1, . . . n}

inwhi heveryar

uv

isasso iatedatriple

(λ(uv), f

+

_{(uv), f}

−

_(uv))

(i)

(λ(uv), f

−

_{(uv)) 6= (λ(vw), f}

+

_(vw))

; (ii)

(λ(uv), f

−

_{(uv)) 6= (λ(u}

′

_{v), f}

−

_(u

′

_w))

; (iii) if

l(vw) 6= l(vw

′

₎

then

(λ(vw), f

+

_{(vw)) 6= (λ(vw}

′

_{), f}

+

_(vw

′

₎₎

.

λ(uv)

orrespondstothewavelengthof

uv

,and

f

+

_(uv)

and

f

−

_(uv)

theberusedin

u

and

v

respe tively. Hen e the ondition(i) orrespondsto thefa t that anar entering

v

and anar leaving

v

haveeither dierentwavelengthsordierentbers;the ondition(ii) orrespondsto thefa tthattwoar sentering

v

haveeitherdierentwavelengthsordierentbers;the ondition(iii) orrespondstothefa tthattwo ar s leaving

v

withdierentlabels haveeither dierentwavelengths ordierentbers. Theproblem is thentondtheminimum ardinality

λ

n

(D)

of

Λ

su hthatthereexistsan

n

-berwavelengthassignment of

D

.

The ru ialthingin an

n

-berwavelengthassignmentisthefun tion

λ

whi hassigns olours (wave-lengths)to thear s. Itmustbean

n

-ber olouring, thatisafun tion

φ : A(D) → Λ

,su hthatat ea h vertex

v

, for ea h olour

λ ∈ Λ

,

in(v, λ) + out(v, λ) ≤ n

with

in(v, λ)

the number of ar s oloured

λ

entering

v

and

out(v, λ)

thenumberoflabels

l

su hthatthere existsanar leaving

v

oloured

λ

. On e we havean

n

-ber olouring, one an easily nd asuitablewavelength assignment. Foreveryvertex

v

and every olour

λ

, this isdone byassigning adierentber toea h ar of olour

λ

entering

v

and to ea h set of ar s of olour

λ

leaving

v

and of the samelabel. Hen e

λ

n

(D)

is theminimum numberof olourssu hthat thereexistsan

n

-ber olouring.

Weare parti ularly intested in

λ

n

(m, k) = max{λ

n

(D) | D

is

m

-labelledand

∆

−

_{(D) ≤ k}}

that is

themaximumnumberofwavelengthsthatmaybene essaryifthereare

n

-bersandea hnodesendsat most

m

andre eivesatmost

k

multi asts. Inparti ular,

λ

1

(1, k) = max{dst(D) | ∆

−

_{(D) ≤ k}}

. Soour

abovementionnedresultsshowthat

2k ≤ λ

1

(1, k) ≤ 2k + 1

. BrandtandGonzalezshowedthatfor

n ≥ 2

we have

λ

n

(1, k) ≤

l

k

n−1

m

. In Se tion 6, we study the ase when

n ≥ 2

and

m ≥ 2

. Weshowthat if

m ≥ n

then

 m

n

 k

n



+

k

n



≤ λ

n

(m, k) ≤

 m

n

 k

n



+

k

n



+ C

m

2

_{log k}

n

forsome onstant

C

.

Wealsoshowthatif

m < n

then

 m

n

 k

n



+

k

n



≤ λ

n

(m, k) ≤



k

n − m



.

The lower bound generalizes Brandt and Gonzalez [4℄ results whi h established this inequality in the

parti ular aseswhen

k ≤ 2

,

m ≤ 2

and

k = m

. Thedigraphs used to showthis lower bound are all a y li . Weshowthatif

m ≥ n

thenthis lowerbound istightfora y li digraphs. Moreovertheabove mentionned digraphshave largeoutdegree. Generalizingthe resultof Guiduli [6℄, we show that for an

m

-labelleddigraph

D

withbothin- andoutdegreebounded by

k

thenfew oloursareneeded:

λ

n

(D) ≤

k

n

+ C

′

m

2

log k

n

forsome onstant

C

′

.

2 Dire ted star arbori ity of digraphs with bounded indegree

Our goal in this se tion is to approa h Conje ture 1. It is easy to see that aforest has dire ted star

arbori ity

2

. Hen e, an ideato prove Conje ture 1 would be to showthat everydigraph has an ar -partition into

∆

−

forests. However this statement is false. Indeed A. Frank [5℄ (see also [10℄, p.908)

hara terized digraphs having an ar -partition into

k

forests. Let

D = (V, A)

. For any

U ⊂ V

, the digraphindu edbytheverti esof

U

isdenoted

D[U ]

.

Theorem5(A. Frank) Adigraph

D = (V, A)

hasanar -partitioninto

k

forestsifandonlyif

∆

−

_{(D) ≤}

k

andforevery

U ⊂ V

,thedigraph

D[U ]

,has atmost

k(|U | − 1)

ar s.

However,Theorem5impliesthateverydigraph

D

hasanar -partitioninto

∆

−

₊₁

forests. Indeedforany

U ⊂ V

,

∆

−

_{(D[U ]) ≤ min{∆}

−

_{, |U | − 1}}

,so

D[U ]

hasatmost

min{∆

−

_{, |U | − 1} × |U | ≤ (∆}

−

_{+ 1)(|U | − 1)}

ar s. Hen e,everydigraphhasdire ted stararbori ityatmost

2∆

−

_{+ 2}

.

Corollary 6 Everydigraph

D

satises

dst(D) ≤ 2∆

−

_{+ 2}

.

Wenowlessenthisupperbound byone.

Theorem7 Every digraph

D

satises

dst(D) ≤ 2∆

−

_{+ 1}

.

Theideato proveTheorem 7isto showthat everydigraphhas anar -partitioninto

∆

−

forestsand a

galaxy

G

. Todoso,weproveastrongerresult(Lemma8)byindu tion.

A sink isavertexwithoutdegree

0

. A sour eis avertex withindegree

0

. Amultidigraphis

k

-ni e if

∆

−

_{≤ k}

and if thetailsof parallel ar s, ifany, aresour es. A

k

-de omposition ofa digraph

D

is an ar -partitioninto

k

forestsandagalaxy

G

su hthateverysour eof

D

isisolatedin

G

. Let

u

beavertex of

D

. A

k

-de ompositionof

D

is

u

-suitableifnoar of

G

hashead

u

.

Lemma8 Let

u

beavertexof a

k

-ni e multidigraph

D

. Then

D

has a

u

-suitable

k

-de omposition. Proof. Wepro eedbyindu tion on

n + k

. Wenowdis ussthe onne tivityof

D

:

ˆ If

D

isnot onne ted,weapplyindu tiononevery omponent.

ˆ If

D

isstrongly onne ted,everyvertexhasindegreeat leastone. Rememberalsothatthereis no parallel ar s. Let

v

be anoutneighbourof

u

. There existsaspanning arbores en e

T

withroot

v

whi h ontainsallthear swithtail

v

. Let

D

′

bethedigraphobtainedfrom

D

byremovingthear s of

T

and

v

. Observethat

D

′

is

(k − 1)

-ni e. Byindu tion,ithasa

u

-suitable

(k − 1)

-de omposition

(F

1

, . . . , F

k−1

, G)

. Note that

F

i

,

T

and

G

ontain all the ar s of

D

ex ept those with head

v

. By onstru tion,

G

′

_{= G ∪ uv}

is a galaxysin eno ar of

G

hashead

u

. Let

u

1

, . . . , u

l−1

be the inneighboursof

v

distin t from

u

, where

l ≤ k

. Let

F

′

i

= F

i

∪ u

i

v

,for all

1 ≤ i ≤ l − 1

. Thenea h

F

′

i

isaforest,so

(F

1

, . . . , F

k−1

, T, G

′

₎

isa

u

-suitable

k

-de ompositionof

D

.

ˆ If

D

is onne tedbutnotstrongly onne ted,we onsiderastrongly onne tedterminal omponent

D

1

. Set

D

2

= D \ D

1

. Let

u

1

and

u

2

betwoverti esof

D

1

and

D

2

,respe tively,su hthat

u

isone ofthem.

If

D

2

hasaunique vertex

v

(thus

u

2

= v

), sin e

D

is onne ted and

D

1

is strong, there exists a spanning arbores en e

T

of

D

withroot

v

. Now

D

′

_{= D \ A(T )}

isa

(k − 1)

-ni emultidigraph, so byindu tionithasa

u

-suitable

(k − 1)

-de omposition. Adding

T

tothisde omposition,weobtain a

u

-suitable

k

-de omposition.

If

D

2

hasmorethanonevertex,byindu tion,itadmitsa

u

2

-suitable

k

-de omposition

(F

2

1

, . . . , F

k

2

, G

2

)

. Moreoverthe digraph

D

′

1

obtainedby ontra ting

D

2

to a singlevertex

v

is

k

-ni e and so has a

u

1

-suitable

k

-de omposition

(F

1

1

, . . . , F

k

1

, G

1

)

. Moreover, sin e

v

is asour e, it is isolated in

G

1

.

Hen e

G = G

1

_{∪ G}

2

isagalaxy. Wenowlet

F

i

betheunionof

F

1

i

and

F

2

i

by repla ingthear s of

F

1

i

with tail

v

by the orrespondingar sin

D

. Then

(F

1

, . . . , F

k

, G)

is a

k

-de ompositionof

D

whi hissuitableforboth

u

1

and

u

2

.

2.1 A y li digraphs

It is nothard to show that

dst(D) ≤ 2∆

−

when

D

is a y li . But we will provethis resultin amore onstrained way. A y li

n

-intervalof

{1, 2, . . . , p}

is a set of

n

onse utive numbersmodulo

p

. Now forthedire tedstar olouring,wewillinsistthatforeveryvertex

v

,the(distin t) oloursusedto olour the ar s with head

v

are hosen in a y li

k

-interval of

{1, 2, . . . , 2k}

. Thus, the number of possible setsof oloursused to olourtheenteringar sof avertexdrasti allyfalls from

2k

k

when everyset isa

prioripossible,to

2k

. Notethathaving onse utives oloursonthear senteringavertex orrespondsto having onse utiveswavelengthsonthelinkbetweenthe orrespondingnodeandthe entralone. Thisis

veryimportantforgroomingissues. Formoredetailsaboutgrooming,werefertothetwo omprehensive

surveys[7,8℄.

Weneedforthisthefollowingresultonset ofdistin trepresentatives.

Lemma9 Let

I

1

, . . . , I

k

be

k

nonne essary distin t y li

k

-intervalsof

{1, 2, . . . , 2k}

. Then

I

1

, . . . , I

k

admitaset ofdistin t representatives forming a y li

k

-interval.

Proof. We onsider

I

1

, . . . , I

k

asasetof

p

distin t y li

k

-intervals

I

1

, . . . , I

p

withrespe tivemultipli ity

m

1

, . . . , m

p

su hthat

P

p

i=1

m

i

= k

. Su hasystemwill bedenotedby

((I

1

, m

1

), . . . , (I

p

, m

p

))

. Weshall provethe existen eof a y li

k

-interval

J

, su h that

J

anbepartitionedinto

p

subsets

J

i

,

1 ≤ i ≤ p

, su hthat

|J

i

| = m

i

and

J

i

⊂ I

i

. This provesthe lemma(byasso iatingdistin telementsof

J

i

to ea h opyof

I

i

).

Wepro eedbyindu tion on

p

. Theresultholdstriviallyfor

p = 1

. Wehavetwo ases: ˆ There exists

i

and

j

su hthat

|I

j

\ I

i

| = |I

i

\ I

j

| ≤ max(m

i

, m

j

)

.

Supposewithoutlossofgeneralitythat

i < j

and

m

i

≥ m

j

. Weapplytheindu tionhypothesisto

((I

1

, m

1

), · · · , (I

i

, m

i

+ m

j

), · · · , (I

j−1

, m

j−1

), (I

j+1

, m

j+1

), · · · , (I

p

, m

p

))

, in order to nda y li interval

J

′

, su h that

J

′

admits apartition intosubsets

J

′

r

, su h that for any

r 6= i, j

,

J

′

r

⊂ I

r

is asubset ofsize

m

r

, and

J

′

i

⊂ I

i

isof size

m

i

+ m

j

. Wenowpartition

J

′

i

into twosets

J

i

and

J

j

withrespe tivesize

m

i

and

m

j

,insu h awaythat

(I

i

\ I

j

) ∩ J

′

i

⊆ J

i

. Remarkthatthisispossible exa tlybe auseofourassumption

|I

j

\ I

i

| = |I

i

\ I

j

| ≤ m

i

. Sin e

J

i

⊂ I

i

and

J

j

⊂ I

j

,thisrened partition of

J

′

isthedesiredone.

ˆ Forany

i, j

wehave

|I

j

\ I

i

| = |I

i

\ I

j

| ≥ max(m

i

, m

j

) + 1

.

Ea h

I

i

interse ts exa tly

2m

i

− 1

other y li

k

-intervalson lessthan

m

i

elements. Sin e there are

2k

y li

k

-intervalsintotaland

P

p

i=1

(2m

i

− 1) = 2k − p < 2k

,we on ludetheexisten eofa y li

k

-interval

J

whi hinterse tsea h

I

i

inanintervalofsizeatleast

m

i

.

Let usprovethat one an partition

J

in thedesired way. ByHall'smat hing theorem, itsu es to provethatforeverysubset

I

of

{1, . . . , p}

,

|

S

i∈I

I

i

∩ J| ≥

P

i∈I

m

i

.

Suppose for a ontradi tion that asubset

I

of

{1, . . . , p}

violatesthis inequality. Su h a subset will be alled ontra ting. Without lossof generality, we assumethat

I

is a ontra tingset with minimum ardinality and that

I = {1, . . . , q}

. Remark that by the hoi e of

J

, we have

q ≥ 2

.

The set

K :=

S

i∈I

I

i

∩ J

onsists of one or two intervals of

J

, ea h ontaining one extremity of

J

. By the minimality of

I

,

K

must be a single interval (if not, one would take

I

∞

(resp.

I

∈

), all the elements of

I

whi h ontains the rst (resp. these ond) extremity of

J

. Then one of

I

1

or

I

2

would be ontra ting). Thus, one of the two extremities of

J

is in every

I

i

,

i ∈ I

. Without loss of generality, we may assume that

(I

1

∩ J) ⊂ (I

2

∩ J) ⊂ · · · ⊂ (I

q

∩ J)

. Now, for every

2 ≤ i ≤ q

,

|I

i

\ I

i−1

| = |(I

i

∩ J) \ (I

i−1

∩ J)| ≥ max(m

i

, m

i−1

) + 1 ≥ m

i

+ 1

. But

|

S

i∈I

I

i

∩ J| = |(I

1

∩ J)| +

P

q

i=2

|(I

i

∩ J) \ (I

i−1

∩ J)|

. So

|

S

i∈I

I

i

∩ J| ≥

P

q

i=1

m

i

+ q − 1

,whi h isa ontradi tion.



Theorem10 Let

D

be an a y li digraph with maximum indegree

k

.

D

admits a dire ted star

2k

- olouring su h that for every vertex, the olours assigned to its entering ar s are in luded in a y li

k

-intervalof

{1, 2, . . . , 2k}

.

Proof. Byindu tiononthenumberofverti es,theresultbeingtrivialif

D

hasonevertex. Supposenow that

D

hasatleasttwoverti es. Then

D

hasasink

x

. Bytheindu tionhypothesis,

D \ x

hasadire ted star

2k

- olouring

c

su hthat foreveryvertex,the oloursassignedtoitsenteringar sarein ludedin a y li

k

-interval. Let

v

1

, v

2

, . . . , v

l

betheinneighboursof

x

in

D

,where

l ≤ k

be ause

∆

−

_{(D) ≤ k}

. For

ea h

1 ≤ i ≤ l

,let

I

′

i

bea y li

k

-intervalwhi h ontainsallthe oloursofthear swithhead

v

i

. Weset

I

i

= {1, . . . , 2k} \ I

i

′

. Clearly,

I

i

isa y li

k

-intervaland thear

v

i

x

anbe olouredbyanyelementof

I

i

. ByLemma9,

I

1

, . . . , I

l

haveaset ofdistin trepresentativesin ludedin a y li

n

-interval

J

. Hen e assigning

J

to

x

,and olouringthear

v

i

x

bytherepresentativeof

I

i

givesadire ted star

2k

- olouring

of

D

.



Theorem 10 is tight : Brandt [3℄ showed that for every

k

, there is an a y li digraph su h that

∆

−

_(D

k

) = k

and

dst(D

k

) = 2k

. His onstru tion is the spe ial ase of the onstru tion given in Proposition21for

n = m = 1

.

3 Sub ubi digraphs

Re allthat asub ubi digraphis agraphwith degreeatmostthree. Inthisse tion, werstshowthat

thedire tedstararbori ityofasub ubi digraphisatmost

3

,soprovingConje ture3when

∆ = 3

. We then giveaveryshort proofof aresultofPinlou and Sopena assertingthat thea ir uiti dire ted star

arbori ityofasub ubi digraphisatmost

4

.

3.1 Dire ted star arbori ity of sub ubi digraphs

Theaimofthissubse tionistoprovethefollowingtheorem:

Theorem11 Everysub ubi digraphhas dire tedstararbori ityatmost

3

.

Todo,weneedtoestablishsomelemmastoenableustoextendsomepartialdire tedstar olouringinto

dire tedstar olouringofthewholedigraph. Theselemmasneedthefollowingdenition. Let

D = (V, A)

be adigraph and

S

a subset of

V ∪ A

. Suppose that ea h element

x

of

S

is assigned a list

L(x)

. A olouring

c

of

S

isan

L

- olouringif

c(x) ∈ L(x)

forevery

x ∈ S

.

Lemma12 Let

C

bea ir uitinwhi h everyvertex

v

re eives alist

L(v)

of two olours among

{1, 2, 3}

andea h ar

a

re eives the list

L(a) = {1, 2, 3}

. Then thereisno

L

- olouring

c

of the ar sand verti es su hthat

c(x) 6= c(xy)

,

c(y) 6= c(xy)

, and

c(xy) 6= c(yz)

, for all ar s

xy

and

yz

if andonly if

C

isodd andallthe verti eshave thesame list.

Proof. Assume rst that every vertex is assigned thesame list, say

{1, 2}

. If

C

is odd, it is simple mattertoseethatwe annotndthederired olouring. Indeed,ifallverti eshavethesame olour,then

weshouldhaveanar olouringof

C

withtwo olourswhi hisimpossible. Iftwoadja enteverti es,say

x

and

y

, havedifefrent olours

1

and

2

, then

xy

shouldhave olour

3

,

yz

(

z

theotherneighbhourof

y

) should have olour

1

andso

z

should have olourIf

C

is even, we olourtheverti esby

1

andthear s alternatelyby

2

and

3

.

Nowassume that

C = x

1

x

2

. . . x

k

x

1

and

x

1

and

x

2

are assigned dierentlists. Say

L(x

1

) = {1, 2}

and

L(x

2

) = {2, 3}

. We olour thear

x

1

x

2

by

3

, thevertex

x

2

by

2

and thear

x

2

x

3

by

1

. Thenwe olour

x

3

,

x

3

x

4

,...,

x

k

. It remainsto olour

x

k

x

1

and

x

1

. Two asesmayhappen: Ifwe an olour

x

k

x

1

by1or2,wedoitand olour

x

1

by

2

or

1

respe tively. Otherwisethesetof oloursassignedto

x

k

and

x

k−1

x

k

is

{1, 2}

. Hen e,we olour

x

k

x

1

with

3

,

x

1

by

1

,andre olour

x

1

x

2

by

2

and

x

2

by

3

.



Lemma13 Let

D

beasub ubi digraphwith novertexofoutdegreetwoandindegreeone. Assumethat every ar

a

has alist of olours

L(a) ⊂ {1, 2, 3}

su hthat:

ˆ If the headof

a

isasink

s

(

a

is alledaleaving ar ),

|L(a)| ≥ d

−

_(s)

.

ˆ If

a

isnot aleavingar andthe tailof

a

isasour e(

a

is alledanentering ar ),

|L(a)| ≥ 2

. ˆ In other ases

|L(a)| = 3

.

ˆ If a vertexis the head of at least two entering ar s the union of their lists of olours ontains at

leastthree olours.

ˆ If allthe verti es ofan odd ir uitarethe tailsofentering ar s, the unionofthe listsof olours of

these entering ar s ontainsatleastthree olours.

Then

D

hasadire tedstar

L

- olouring.

Proof. We olourthegraphindu tively. Consideraterminalstrong omponent

C

of

D

. Sin e

D

has novertexwithindegreeoneandoutdegree two,

C

indu eseitherasingletonora ir uit.

1) Assumethat

C

isasingleton

v

whi histheheadofauniquear

a = uv

. If

u

hasindegree0, olour

a

with a olourof its list. If

u

hasindegree 1,and thus totaldegree2, olour

a

bythe olourof its list and removethis olour from the list of the ar with head

u

. If

u

is the headof

e

and

f

, observethat

L(e)

and

L(f )

haveatleasttwo oloursandtheirunionhaveatleastthree olours. To on lude, olour

a

witha olourinitslist,removethis olourfrom

L(e)

and

L(f )

,remove

a

,split

u

intotwoverti es,onewith head

e

, and theother with head

f

. Now, hoosein their respe tive listsdierent oloursforthear s

e

and

f

toform thenewlist

L(e)

and

L(f )

.

2) Assumethat

C

isasingleton

v

whi histheheadofseveralar s,in luding

a = uv

. Inthis ase,we redu e

L(a)

toasingle olour,removethis olourfromtheotherar swithhead

v

andsplit

v

into

v

1

whi hbe omestheheadof

a

,and

v

2

whi h be omestheheadof theotherar s.

3) Assume that

C

isa ir uit. Everyar entering

C

hasalistofat leasttwo olours. We anapply Lemma 12to on lude.



Proof of Theorem 11. Assume for ontradi tion that the digraph

D

has star arbori ity more than threeandisminimumwithrespe ttothenumberofar sforthisproperty. Observethat

D

hasnosour e, otherwisewesimplydeleteitwithitsin identar s,applyindu tionandextendthe olouringsin ear s

leavingfromasour e anbe olouredarbitrarily. Let

D

1

bethesubdigraphof

D

indu edbytheverti es ofindegreeat most

1

. Wedenote by

D

2

thedigraphindu edbytheotherverti es,andby

[D

i

, D

j

]

the setofar swithtailin

D

i

andheadin

D

j

. We laimthat

D

1

ontainsnoeven ir uit. Ifnot,wesimply removethe ar s of this even ir uit, apply indu tion and extend the olouring to thear s of the even

A riti al set of verti esof

D

2

is either avertexof

D

2

with indegreeat least twoin

D

1

, oran odd ir uitof

D

2

havingallitsinneighboursin

D

1

. Observethat riti alsetsaredisjoint. Forevery riti al set

S

,wesele ttwoar sentering

S

from

D

1

, alledsele tedar sof

S

.

Let

D

′

bedigraphindu edbythear set

A

′

_{= A(D}

1

) ∪ [D

2

, D

1

]

. Nowwedenea oni t graph on thear sof

D

′

inthefollowingway:

ˆ Twoar s

xy

,

yv

of

D

′

arein oni t, allednormal oni t at

y

. ˆ Twoar s

xy

,

uv

of

D

′

are also in oni t ifthere exists twosele tedar s ofthesame set

S

with tails

y

and

v

. These oni tsare alledsele ted oni tsat

y

and

v

.

Letusanalysethe stru tureofthe oni tgraph. Observerstthat anar isin oni t withthree

ar s: onenormal oni tatitstailandatmosttwo(normalorsele ted)atitshead.

We laim thatthere isno

K

4

inthe oni t graph. Suppose thereis one,thenthereis

4

ar s whi h are pairwise in oni t. Sin e ea h ar has degree

3

, it has a normal oni t at its tail, the digraphs indu ed by these four ar s ontains a ir uit. It annot be a ir uit of even length (

2

or

4

) so it has length

3

. Itfollowsthat thefour ar s

a, b, c, d

areasin Figure1below. Let

D

∗

bethedigraphobtained

from

D

by removingthe ar s

a, b, c, d

and theirfour in ident verti es. Byminimality of

D

,

D

∗

admits

adire ted star

3

- olouringwhi h anbe extended to

D

asdepi ted belowdepending ifthetwoleaving ar sare olouredthesameordierently. Thisprovesthe laim.

2

1

3

3

1

1

2

2

3

1

2

1

3

3

1

2

1

2

3

2

a

b

c

d

a

b

c

d

Figure1: A

K

4

inthe oni t graphandthetwowaysofextendingthe olouring.

Brook'sTheoremassertsthateverysub ubi graphwithout

K

4

is3- olourable.Sothe oni tgraph admits a

3

- olouring

c

. This givesa olouring of thear s of

D

′

. Let

D

′′

bethe digraph and

L

be the list-assignmentonthear sof

D

′′

obtainedasfollow:

ˆ Removethear sof

D

1

from

D

,

ˆ Assigntoea har of

[D

2

, D

1

]

thesingletonlist ontainingthe olourithasin

D

′

,

ˆ Forea har

uv

of

[D

1

, D

2

]

,thereisauniquear

tu

in

A(D

′

₎

, soassignthelist

L(uv) = {1, 2, 3} \

c(tu)

.

ˆ Assignthelist

{1, 2, 3}

totheotherar s.

ˆ Ifthereareverti eswithindegreeoneandoutdegreetwo(theywerein

D

1

),splitea hoftheminto onesour eofdegreetwoandasinkofdegreeone.

Note that there is atrivial one-to-one orresponden e between

A(D

′′

₎

and

A(D) \ A(D

′

₎

. By the

denitionof oni tgraphand

D

′′

,one aneasily he kthat

D

′′

and

L

satisesthe onditionofLemma13.

Hen e

D

′′

admits adire ted star

L

- olouring whi h union with

c

isadire ted star

3

- olouringof

D

, a

3.2 A ir uiti dire ted star arbori ity

A dire ted star olouring is a ir uiti ifthere is no bi oloured ir uits, i.e. ir uits for whi h only two

oloursappearsonitsar s. Thea ir uiti dire tedstararbori ityofadigraph

D

adigraphistheminimum number

k

of olourssu hthatthereexistsana ir uiti dire tedstar

k

- olouringof

D

. Inthissubse tion, wegiveashortalternativeproofofthefollowingtheoremdue toPinlouandSopena.

Theorem14 (Pinlouand Sopena[9℄) Every sub ubi orientedgraph has a ir uiti dire ted star

ar-bori ity atmost

4

.

Inordertoprovethistheorem,weneedthefollowinglemma.

Lemma15 Let

D

beana y li sub ubi digraph. Let

L

bealist-assignmentonthe ar sof

D

su hthat for everyar

uv

,

|L(uv)| ≥ d(v)

. Then

D

admits adire tedstar

L

- olouring.

Proof. Weprovetheresultbyindu tion onthenumberof ar sof

D

,the resultholdingtriviallyif

D

hasnoar s.

Sin e

D

is a y li ,it hasanar

xy

with

y

asink. Let

a

bea olourin

L(xy)

. Foranyar

e

distin t from

xy

, set

L

′

_{(e) = L(e) \ {a}}

if

e

in ident to

xy

(andthus hashead in

{x, y}

sin e

y

is asink), and

L

′

_{(e) = L(e)}

otherwise. Thenin

D

′

_{= D − xy}

,wehave

|L

′

_{(uv)| ≥ d(v)}

. Hen e,byindu tion hypothesis,

D

′

admits a dire ted star

L

′

- olouring that an be extended in a dire ted star

L

- olouring of

D

by

olouring

xy

with

a

.



Proof ofTheorem14.

Let

V

1

betheset ofverti esof outdegree at most

1

and

V

2

= V \ V

1

. Then everyvertex of

V

2

has outdegreeatleast

2

andsoindegreeatmost

1

.

Let

M

bethesetof ar swithtailin

V

1

and headin

V

2

. Colourallthear sof

M

with

4

. Moreover for every ir uit

C

in

D[V

1

]

and

D[V

2

]

hoosean ar

e(C)

and olourit by

4

. Note that, by denition of

V

1

and

V

2

,thear

e(C)

isnotin identto anyar of

M

and

C

istheunique i r uit ontaining

e(C)

. Letusdenote

M

4

thesetofar s oloured

4

. Then

M

4

isamat hingand

D − M

4

isa y li .

Weshallnowndadire tedstar olouringof

D −M

4

with

{1, 2, 3}

that reatesanybi oloured ir uit. Ifsu ha ir uitexists,

4

would beoneof its olourbe ause

D − M

4

is a y li andallitsar s oloured

4

wouldbein

M

be ausethear sof

M

4

\ M

is in aunique ir uitwhi hhas auniquear oloured

4

. Hen e wejust haveto be areful when olouring ar sin thedigraph indu edbythe endverti esof the

ar sof

M

.

Letus denotethear sof

M

by

x

i

y

i

,

1 ≤ i ≤ p

and set

X = {x

i

, 1 ≤ i ≤ p}

and

Y = {y

i

, 1 ≤ i ≤ p}

. Then

x

i

∈ V

1

and

y

i

∈ V

2

. Let

E

′

bethesetof ar swithtailin

Y

andheadin

X

. Let

H

bethegraph withvertexset

E

′

su hthat anar

y

i

x

j

isadja entto anar

y

k

x

l

if (a) either

k = l

,

(b) or

j = k

and

i > j

and

l > j

.

Sin eavertexof

X

hasindegreeatmost

2

andavertexof

Y

hasoutdegreeat most

2

,

H

hasmaximum degree

3

. Moreover

H

ontainsno

K

4

be ausetwoar sof

E

′

with sametail

y

k

arenotadja ent in

H

. Hen e,by Brooks Theorem,

H

has avertex- olouringin

{1, 2, 3}

whi h is orrespondsto a olouring

c

ofthear sof

E

′

. Sin e(a) issatised

c

isadire ted star olouring. Moreoverthis olouring reates no bi oloured ir uits: indeeda ir uit ontainsasubpath

y

i

x

j

y

j

x

l

with

i > j

and

k > j

,whosethreear s are oloureddierentlyby(b).

Let

D

′

_{= D − (M}

4

∪ E

′

)

. Forany ar

uv

in

D

′

, let

L(uv) = {1, 2, 3} \ {c(wv) | wv ∈ E

′

_}

. The

set

L(uv)

isthesetof oloursin

{1, 2, 3}

thatmaybeassignedto

uv

without reatingany oni twith thealready olouredar s.

D

′

isa y li and

|L(uv)| ≥ d(v)

, so by Lemma15, it admitsadire ted star

Remark16 Note that in the a ir uiti dire ted star

4

- olouring provided in the proof of Theorem14 thear s oloured

4

formamat hing.

4 Dire ted star arbori ity of digraphs with maximum in and

out-degree two

The goal of this se tion is to prove that every digraph with outdegree and indegree at most two has

dire tedstararbori ityatmostfour.

Theorem17 Let

D

beadigraph withmaximumin andoutdegreeatmosttwo. Then

dst(D) ≤ 4

. Thus, onje ture4holdsfor

k = 2

andhen eforalleven

k

. However,the lassofdigraphswith inand outdegree at mosttwois ertainly notaneasy lasswith respe t to dire ted stararbori ity,as wewill

showinSe tion 5.

Inorder to proveTheorem 17, it su es to show that

D

ontains agalaxy

G

whi h spans all the verti es of degree four. Then

D

′

_{= D − A(G)}

has maximum degree at most

3

. So, by Theorem 11,

dst(D

′

_{) ≤ 3}

,so

dst(D) ≤ 4

. Hen e Theorem17isdire tlyimplied bythefollowinglemma:

Lemma18 Let

D

be adigraph with maximum indegreeand outdegree two. Then

D

ontains agalaxy whi h spansthe set ofverti eswith degreefour.

Inordertoprovethislemma,weneedsomepreliminaries:

Let

V

beaset. Anordered digraphon

V

isapair

(≤, D)

where: ˆ

≤

isapartial orderon

V

.

ˆ

D

isadigraphwith vertexset

V

.

ˆ

D

ontainstheHassediagramof

≤

(i.e. when

x ≤ y ≤ z

implies

x = y

or

y = z

,then

xz

is anar of

D

).

ˆ If

xy

isanar of

D

,theverti es

x, y

are

≤

- omparable.

Thear s

xy

of

D

thusbelongtotwodierenttypes: theforward ar swhen

x ≤ y

,andtheba kward ar swhen

y ≤ x

.

Lemma19 Let

(≤, D)

be anordereddigraphon

V

. Assumethateveryvertex isthetail ofatmostone ba kwardar andatmosttwoforwardar sandthatthe indegreeof everyvertexof

D

isatleast2,ex ept possibly one vertex

x

with indegree 1. Then

D

ontains twoar s

ca

and

bd

su h that

a ≤ b ≤ c

,

b ≤ d

and

c 6≤ d

,allfour verti esbeing distin tex ept possibly

a = b

.

Proof. Letus onsider a ounterexamplewithminimum

|V |

.

An interval is asubset

I

of

V

whi h has a minimum

m

and a maximum

M

su h that

I = {z :

m ≤ z ≤ M }

. An interval

I

is goodifeveryar withtailin

I

andheadoutside

I

hastail

M

andevery ba kwardar in

I

hastail

M

.

Let

I

bean intervalof

D

. Thedigraph

D/I

obtainedfrom

D

by ontra ting

I

isthe digraphwith vertexset

(V \ I) ∪ {v

I

}

su hthat

xy

isanar ifandonlyeither

v

I

∈ {x, y}

/

and

xy ∈ A(D)

, or

x = v

I

andthere exists

x

I

∈ I

su h that

x

I

y ∈ A(D)

,or

y = v

I

andthere exists

y

I

∈ I

su hthat

xy

I

∈ A(D)

. Similarly, thepartial order

≤

/I

obtainedfrom

≤

by ontra ting

I

is thepartial orderon

(V \ I) ∪ {v

I

}

su h that

x ≤

/I

y

ifand only if either

v

I

∈ {x, y}

/

and

x ≤ y

, or

x = v

I

and there exists

x

I

∈ I

su h

that

x

I

≤ y

, or

y = v

I

and there exists

y

I

∈ I

su h that

x ≤ y

I

. It follows from the denitions that

(≤

/I

, D/I)

isanordereddigraph. Notethat if

x ≤

/I

v

I

then

x ≤ M

with

M

themaximumof

I

.

The ru ial point is that if

I

a good intervalof

D

for whi h the on lusion ofLemma 19holds for

(≤

/I

, D/I)

, thenitholds for

(≤, D)

. Indeed, supposethere exists twoar s

ca

and

bd

of

D/I

su h that

a ≤

/I

b ≤

/I

c

,

b ≤

/I

d

and

c 6≤

/I

d

. Notethatsin e

I

isgood

v

I

6= c

. Let

M

bethemaximumof

I

. If

v

I

∈ {a, b, c, d}

/

,then

ca

and

bd

givesthe on lusionfor

D

.

If

v

I

= a

then

cM

isanar . Letusshowthat

M ≤ b

. Indeedlet

x

beamaximalvertexin

I

su hthat

x ≤ b

and

y

aminimalvertexsu hthat

x ≤ y ≤ b

. Sin etheHassediagram of

≤

isin ludedin

D

then

xy

isanar so

x = M

sin e

I

isgood. Thus

cM

and

bd

arethedesiredar s.

If

v

I

= b

then

M d

isanar and

a ≤ M

,so

ca

and

M d

arethedesiredar s. If

v

I

= d

thenthereexists

d

I

∈ I

su h that

bd

I

,so

ca

and

bd

I

arethedesiredar s.

Hen e to get a ontradi tion, it issu ient to nd agoodinterval

I

su h that

(≤

/I

, D/I)

satises thehypothesesofLemma19.

Observethat thereareat leasttwoba kwardar s. Indeed,iftherearetwominimalelementsfor

≤

, thereareatleastthreeba kwardar sheadingtothesepoints(sin eoneofthem anbe

x

). Andifthere isauniqueminimum

m

,byletting

m

′

minimal in

V \ m

,at leasttwoar sareheadingto

m, m

′

.

Let

M

beavertexwhi h isthetailofaba kwardar andwhi his minimalfor

≤

forthisproperty. Sin etwoar s annothavethesametail,

M

isnotthemaximumof

≤

(ifany). Let

M m

betheba kward ar withtail

M

.

We laimthattheinterval

J

withminimum

m

andmaximum

M

isgood. Indeed,bythedenitionof

M

, noba kwardar hasitstailin

J \ {M }

. Moreover,anyforwardar

bd

withitstailin

J \ {M }

and itsheadoutside

J

wouldgiveour on lusion(with

a = m

and

c = M

),a ontradi tion.

Now onsider agood interval

I

with maximum

M

whi h is maximalwith respe t to in lusion. We laim that if

x ∈ I

, then there is at least onear entering

I

, and if

x /

∈ I

, there are at least twoar s entering

I

withdierenttails.

Call

m

1

theminimumof

I

and

m

2

anyminimalelementof

I \ m

1

. Firstassumethat

x

isin

I

. There areatleastthreear swithheads

m

1

or

m

2

. Oneofthemis

m

1

m

2

,oneofthem anbewithtail

M

,but thereisstill oneleftwithtailnotin

I

. Nowassumethat

x

isnotin

I

. There areatleasttwoar swith heads

m

1

or

m

2

andtailsnotin

I

. Ifthetailsaredierent,wearedone. Ifthetailsarethesame,say

v

, observethat

vm

1

and

vm

2

are bothba kwardof bothforward(otherwise

v

wouldbein

I

). Sin eboth annotbeba kward

vm

1

and

vm

2

areforward. Hen e theintervalwithminimum

v

andmaximum

M

isagoodinterval, ontradi tingthemaximalityof

I

. Thisprovesthe laim.

This laimimpliesthat

(≤

/I

, D/I)

satisesthehypothesesofLemma19,yieldinga ontradi tion.



Proofof Theorem18. Let

G

beagalaxyof

D

whi hspansamaximumnumberofverti esofdegree four. Suppose for ontradi tionthat somevertex

x

withdegreefourisnotspanned.

Analternating pathisanorientedpathendingat

x

,startingbyanar of

G

,andalternatingwithar s of

G

andar sof

A(D) \ A(G)

. Wedenoteby

A

thesetofar sof

G

whi hbelongtoanalternatingpath. Claim1 Every ar of

A

isa omponent of

G

.

Proof. Indeed, if

uv

belongs to

A

, it startssome alternatingpath

P

. Thus, if

u

hasoutdegree more thanonein

G

, thedigraphwithset ofar s

A(G)△A(P )

isagalaxyand spans

V (G) ∪ x

.



Claim2 Thereisno ir uitsalternatingar sof

A

andar sof

A(D) \ A

.

Proof. Assume that there is su h a ir uit

C

. Considera shortestalternating path

P

startingwith somear of

A

in

C

. Nowthedigraphwithar s

A(G)△(A(P ) ∪ A(C))

isagalaxywhi hspans

V (G) ∪ x

,

Wenowendow

A ∪ x

withapartialorderstru turebyletting

a ≤ b

ifthereexistsanalternatingpath startingat

a

and endingat

b

. The fa t that this relationis apartial order relies onClaim 2. Observe that

x

isthemaximumofthis order.

Wealso onstru tadigraph

D

onvertexset

A ∪ x

andallar s

uv → st

su hthat

us

or

vs

is anar of

D

(and

uv → x

su hthat

ux

or

vx

isanar of

D

).

Claim3 Thepair

(D, ≤)

isanordereddigraph. Moreoveranar of

A

isthetailofatmostoneba kward ar andtwoforwardar sand

x

isthe tailof atmosttwoba kwardar s.

Proof. The fa t that the Hasse diagram of

≤

is ontainedin

D

follows from thefa t that if

uv ≤ st

belongsto theHasse diagramof

≤

, there isanalternatingpath startingby

uvst

, in parti ular,thear

vs

belongsto

D

, andthus

uv → st

in

D

.

Suppose that

uv → st

and then

vs

or

us

is an ar of

D

. If

vs

is an ar , then be ause there is no alternating ir uit,

st

follows

uv

onsomealternatingpathso

uv ≤ st

. Inthis ase,

uv → st

isforward. If

us

is an ar of

D

, we laim that

st ≤ uv

. Indeed, if an alternating path

P

starting at

st

does not ontain

uv

, thegalaxywith ar s

(A(G)△A(P )) ∪ {us}

spans

V (G) ∪ x

ontradi tingthemaximalityof

G

. Inthis ase,

uv → st

isba kward.

Itfollowsthatanar

uv

of

A

isthetailofatmostoneba kwardar sin ethisar and

uv

arethetwo ar sleaving

u

in

D

andthetailofatmosttwoforwardar ssin e

v

hasoutdegreeatmost

2

. Furthermore, sin e

x

hasoutdegreeatmosttwo,itfollowsthat

x

isthetailofat mosttwoba kwardar s.



Claim4 The indegreeof every vertexof

D

istwo.

Proof. Let

uv

be avertex of

D

whi h startsan alternatingpath

P

. If

u

hasindegree lessthan two, and thus does notbelong to the setof verti esof degreefour, thegalaxywith ar s

A(G)△A(P )

spans moreverti esofdegreefour than

G

,a ontradi tion. Let

s

and

t

bethetwoinneighboursof

u

in

D

. An element of

A ∪ x

ontains

s

otherwise thegalaxywith ar s

(A(G)△A(P )) ∪ {su}

spans

V (G) ∪ x

and ontradi tsthemaximalityof

G

. Similarlyanelementof

A ∪ x

ontains

t

.

Observethatthesameelementof

A ∪ x

annot ontainboth

s

and

t

(eitherthear

st

orthear

ts

),

otherwisethear s

su

and

tu

wouldbebothba kwardorforward,whi hisimpossible.



At this stage, in order to apply Lemma 19, we just need to insure that theba kwardoutdegree of

everyvertexisat mostone. Sin etheonlyelementof

D

whi histhetailoftwoba kwardar sis

x

,we simplydeleteanyofthesetwoba kwardar s. Theindegreeofavertexof

D

de reasesbyonebutweare stillfullling thehypothesisofLemma19.

Hen ea ordingto thislemma,

D

ontainstwoar s

ca

and

bd

su h that

a ≤ b ≤ c

,

b ≤ d

and

c 6≤ d

. Keepinmindthat

a, b, c, d

areelementsof

A ∪ x

. Inparti ular,thereisanalternatingpath

P

ontaining

a, b, d

(in thisorder)whi h doesnot ontain

c

. Setting

a = a

1

a

2

and

c = c

1

c

2

, notethat theba kward ar

ca

orrespondsto thear

c

1

a

1

in

D

. Werea ha ontradi tion by onsidering thegalaxywithar s

(A(G)△A(P )) ∪ {c

1

a

1

}

whi hspans

V (D

′

_{) ∪ x}

.



5 Complexity

Thedigraphswithdire tedstararbori ity

1

arethegalaxies. Soone anpolynomiallyde ideif

dst(D) =

1

. De iding whether

dst(D) = 2

or notis alsoeasy sin ewejust haveto he k that the oni t graph (withvertexsetthear sof

D

,twodistin tar s

xy, uv

beingin oni twhen

y = u

or

y = v

)isbipartite. Howeverforlargervalue,asexpe ted,itisNP- ompletetode ideifadigraphhasdire tedstararbori ity

Theorem20 Thefollowing problemis NP- omplete:

INSTANCE:Adigraph

D

with

∆

+

_{(D) ≤ 2}

and

∆

−

_{(D) ≤ 2}

.

QUESTION:Is

dst(D)

atmost

3

?

Proof. The proof is a redu tion to

3

-edge- olouring of

3

-regular graphs. To see this, onsider a

3

-regulargraph

G

. Itadmitsanorientation

D

su hthateveryvertexhasin andoutdegreeatleast1. Let

D

′

be the digraph obtainedfrom

D

by repla ingeveryvertexwith indegree

1

and outdegree

2

by the subgraph

H

depi tedinFigure2whi hhasalsooneenteringar (namely

a

)andtwoleavingar s(

b

and

c

). It iseasyto he kthat inanydire ted star

3

- olouringof

H

, thethreear s

a

,

b

and

c

getdierent

a

1

b

c

2

1

3

3

2

2

3

1

2

3

1

Figure2: Thegraph

H

andoneofitsdire tedstar

3

- olouring

olours. Moreoverifthesethreear sarepre olouredwiththreedierent olours,we anextendthistoa

dire tedstar

3

- olouringof

H

. Su ha olouringwith

a

oloured

1

,

b

oloured

2

and

c

oloured

3

isgiven in Figure 2. Furthermore,a vertex with indegree

2

and outdegree

1

must haveits three in identar s oloureddierentlyinadire tedstar

3

- olouring. So

dst(D

′

_{) = 3}

ifandonlyif

G

is

3

-edge olourable.



6 Multiple bers

In this se tion we onsider the problem with

n ≥ 2

bers. More pre isely, we give some bounds on

λ

n

(m, k)

. Werstgivealowerboundon

λ

n

(m, k)

.

Proposition 21

λ

n

(m, k) ≥

 m

n

 k

n



+

k

n



Proof. Considerthefollowing

m

-labelleddigraph

G

n,m,k

withvertexset

X ∪ Y ∪ Z

su hthat :

ˆ

|X| = k

,

|Y | = 2

(m+1)k

2

and

|Z| = m

|Y |

k

.

ˆ Forany

x ∈ X

and

y ∈ Y

thereisanar

xy

(ofwhateverlabel).

ˆ Foreveryset

S

of

k

verti esof

Y

andinteger

1 ≤ i ≤ m

,thereisavertex

z

i

S

in

Z

whi hisdominated byalltheverti esof

S

viaar slabelled

i

.

Suppose there exists an

n

-ber olouring of

G

n,m,k

with

c <



m

n



k

n

 +

k

n



olours. For

y ∈ Y

and

1 ≤ i ≤ m

,let

C

i

(y)

bethesetof oloursassignedtothear slabelled

i

leaving

y

. For

0 ≤ j ≤ n

,let

P

j

the setof oloursusedon

j

ar sentering

y

(andne essarilywithtwodierentsbers). Then

P

n

as

k

ar senter

y

. Moreover

(P

0

, P

1

, . . . , P

n

)

isa partitionof the set of olours so

P

n

j=0

|P

j

| = c

. Now ea h olourof

P

j

mayappearin atmost

n − j

ofthe

C

i

(y)

,so

m

X

i=1

|C

i

(y)| ≤

n

X

j=0

(n − j)|P

j

| = n

n

X

j=0

|P

j

| −

n

X

j=0

j|P

j

| = cn − k.

Be ause

|Y | = 2

(m+1)k

2

,thereisaset

S

of

k

verti es

y

of

Y

havingthesame

m

-uple

(C

1

(y), . . . , C

m

(y)) =

(C

1

, . . . , C

m

)

. Without loss of generality, we may assume

|C

1

| = min{|C

i

| | 1 ≤ i ≤ m}

. Hen e

|C

1

| ≤

cn−k

_m

. But the vertex

z

1

S

has indegree

k

so

|C

1

| ≥ k/n

. Sin e

|C

1

|

is an integer, we have



cn−k

m



≥ |C

1

| ≥ ⌈k/n⌉

. So

c ≥

m

n



k

n

 +

k

n

. Sin e

c

is an integer, we get

c ≥



m

n



k

n

 +

k

n



, a ontradi tion.



Notethat thegraph

G

n,m,k

isa y li . Thefollowinglemmashowsthat,if

m ≥ n

,one annotexpe t better lowerbounds by onsidering a y li digraphs. Indeed

G

n,m,k

is the

m

-labelled a y li digraph withindegreeatmost

k

forwhi h an

n

-ber olouringrequiresthemore olours.

Lemma22 Let

D

beana y li

m

-labelleddigraphwith

∆

−

_{≤ k}

. If

m ≥ n

then

λ

n

(D) ≤



m

n



k

n

 +

k

n



.

Proof. Sin e

D

is a y li ,itsvertexset admitsanordering

(v

1

, v

2

, . . . , v

p

)

su hthat if

v

j

v

j

′

isanar then

j < j

′

.

Byindu tion on

q

, we shall nd an

n

-ber olouring of

D[{v

1

, . . . , v

q

}]

together with sets

C

i

(v

r

)

,

1 ≤ i ≤ m

and

1 ≤ r ≤ q

,of

⌈k/n⌉

olourssu hthat,inthefuture,assigninga olourin

C

i

(v

r

)

toanar labelled

i

leaving

v

r

willfullllthe onditionof

n

-ber olouringat

v

r

.

Startingthepro essiseasy. We maytakeas

C

i

(v

1

)

any

⌈k/n⌉

-setssu hthat a olourappearsin at most

n

ofthem.

Suppose now that we have an

n

-ber olouring of

D[{v

1

, . . . , v

q−1

}]

and that, for

1 ≤ i ≤ m

and

1 ≤ r ≤ q − 1

,theset

C

i

(v

r

)

is determined. Letus olourthear s entering

v

q

. Ea h ofthese ar s

v

r

v

q

maybeassignedoneofthe

⌈k/n⌉

oloursof

C

l(v

r

v

q

)

(v

r

)

. Sin ea olourmaybeassignedto

n

ar s(using dierentbers)entering

v

q

, one an assigna olourandbertoea h su h ar . Itremainsto determine the

C

i

(v

q

)

,

1 ≤ i ≤ m

.

For

0 ≤ j ≤ n

, let

P

j

bethesetof oloursassignedto

j

ar sentering

v

q

. Let

N =

P

n

i=0

(n − j)|P

j

|

and

(c

1

, c

2

, . . . , c

N

)

beasequen eof olourssu hthatea h olourof

P

j

appearsexa tly

n − j

timesand onse utively. For

1 ≤ i ≤ m

, set

C

i

(v

q

) = {c

a

| a ≡ i mod m}

. As

n ≤ m

, a olour appearsat most on einea h

C

i

(v

q

)

. Moreover,

N = n



m

n



k

n

 +

k

n

 − k ≥ m 

k

n



. Sofor

1 ≤ i ≤ m

,

|C

i

(v

q

)| ≥



k

n



.



Lemma22givesatightupperbound on

λ

n

(D)

fora y li digraphs. Weshallproveanupperbound for general digraphs. To do so, we srt give an upper bound on

λ

n

(D)

for

m

-labelled digraphs with boundedin-andoutdegreInthis ase,oone anderivefromthefollowingtheoremofGuidulithatfew

oloursareneeded. Notethat thegraphs

G

n,m,k

requireslots of oloursbut haveverylargeoutdegree.

Theorem23 (Guiduli [6℄) If

∆

−

_{, ∆}

+

_{≤ k}

then

dst(D) ≤ k + 20 log k + 84

. Moreover

D

admits a dire ted star olouring with

k + 20 log k + 84

olours su h that for ea h vertex

v

there are at most

10 log k + 42

olours assignedtoitsleavingar s.

The proof of Guiduli's Theorem an be modied to obtain the following statement for

m

-labelled digraphs.

Theorem24 Let

f (n, m, k) =

 k + (10m

2

_{+ 5) log k + 80m}

2

_{+ m + 21}

n



and

D

bean

m

-labelleddigraph with

∆

−

_{, ∆}

+

_{≤ k}

. Then

λ

n

(D) ≤ f (n, m, k)

. Moreover

D

admitsa

n

-ber olouringwith

f (n, m, k)

su h thatfor ea h vertex

v

andea hlabel

l

,thereare atmost

g(m, k) = ⌈(10m + 5) log k + 40m + 21⌉

olours assignedtothe ar slabelled

l

leaving

v

.