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The acircuitic directed star arboricity of subcubic
graphs is at most four
Alexandre Pinlou, Eric Sopena
To cite this version:
of sub ubi graphs is at most four
Alexandre Pinlou,
Eri Sopena
LaBRI, UniversiteBordeaux I,
351, ours de laLiberation
33405 Talen e Cedex, Fran e
E-mail: fAlexandre.Pinlou,Eri .Sopenaglabri.fr
O tober8, 2004
Abstra t
Adire ted starforestisaforest allofwhose omponentsarestars with ar s emanating from the enter to the leaves. The a ir uiti dire tedstararbori ityofanorientedgraphG(thatisadigraphwith nooppositear s)istheminimumnumberofedge-disjointdire tedstar forests whoseunion oversalledges of Gand su h that theunion of anytwosu hforests isa ir uiti . Weshowthateverysub ubi graph hasa ir uiti dire tedstararbori ityatmostfour.
1 Introdu tion
We onsider nitesimpleorientedgraphs,thatisdigraphswithnoopposite
ar s. ForanorientedgraphG,wedenotebyV(G) itssetofverti esandby A(G)its setof ar s.
In[1 ℄,AlgorandAlonintrodu edthenotionofthedire tedstararbori ity
of a digraph G, de ned as the minimum number of edge-disjoint dire ted star forests needed to over A(G). (A dire ted star forest is a forest all of whose omponentsaredire tedstars, thatisstarswithar semanatingfrom
Guiduliprovedthateveryorientedgraphwithindegreeandoutdegree both
lessthanDhas dire tedstar arbori ityat most D+20logD+84 olors. In thispaper, we prove thefollowing
Theorem 1 Every graph with maximum degree at most 3 has a ir uiti
dire ted star arbori ity at most4.
The notion of a ir uiti dire ted star arbori ity arises from the study of ar - oloring of oriented graphs. In [4℄, Cour elle introdu ed the notion of vertex- oloring of oriented graphs as follows: a k-vertex- oloring of an
oriented graph Gis a mappingf from V(G) to a set of k olors su h that (i) f(u)6=f(v) whenever
!
uv isan ar inG, and (ii) f(u)6=f(x)whenever !
uv and !
wx aretwo ar s inG withf(v)=f(w). Vertex- oloringof oriented
graphs have been studied by several authors in the last past years (see e.g. [2 ,6 ℄ or[9℄foran overview).
Re all that an a y li oloring of an undire ted graph U is a proper oloringof U su hthatevery y leinU uses atleastthree olors. Raspaud andSopenaprovedin[8℄thateveryorientationofanundire tedgraphthat
admitsan a y li k- oloring admitsan oriented(k2 k 1
)- oloring.
One ande near - oloringsoforientedgraphsinanaturalwaybysaying that, as in the undire ted ase, an ar - oloring of an oriented graph G is a vertex- oloring of the line digraph of G. (Re all that the line digraph
L(G)of G is given by V(L(G))=A(G) and ( ! uv; ! vw) 2A(L(G)) whenever ! uv 2 A(G) and !
vw 2 A(G).) It is not diÆ ult to see that every oriented graph havinga k-vertex- oloring admits a k-ar - oloring (froma
k-vertex- oloringf,weobtaina k-ar - oloring gbysetting g( !
uv)=f(u)).
By adapting theproof of the above-mentionned result of Raspaud and Sopena,itis notdiÆ ultto prove thateveryorientedgraphwitha ir uiti
dire tedstar arbori ityat most k admits a(k2 k 1
)-ar - oloring.
Thispaperisorganisedasfollows: weintrodu ethemainde nitionsand notationinthenext se tion and prove ourmainresultin Se tion3.
2 De nitions and notation
Intherestof thepaper,orientedgraphswillbesimply alledgraphs. Fora
vertexv,wedenote byd (v) theindegree of v, by d +
(v) itsoutdegree and byd(v) its degree,thatisd(v)=d
+
if(G)=Æ(G)=3 and sub ubi if(G)3.
Wedenoteby !
uvthear fromutovorsimplyuvwheneveritsorientation is notrelevant (therefore uv =
! uv or uv = ! vu). If a= ! uv is an ar , thenu isthetail andv is theheadof a.
For a graph G and a vertex v of V(G), we denote by Gnv the graph obtained from G by removing v together with the set of its in ident ar s; similarly,foran ar aof A(G),GnadenotesthegraphobtainedfromGby
removinga. Thesetwonotionsareextendedtosetsinastandardway: fora setofverti esV
0 ,GnV
0
denotesthegraphobtainedfromGbysu essively removingall verti esof V
0
and theirin ident ar s,and fora set of ar s A 0
,
GnA 0
denotesthegraph obtainedfrom Gbyremovingallar s of A 0
. Thenotionsofarbori itydis ussedinthepreviousse tionmaybede ned in terms of ar - olorings or partitions of the set of ar s. More pre isely, a
k-dire ted-star- oloring (orsimplyk-dst- oloring)ofagraphGisapartition of A(G)into k dire ted star forests fF
1 ;F 2 ;:::;F k g. Equivalently,a k-dst- oloring of G is a k- oloring f of A(G) su h that (i)
! uv; ! vw 2 A(G) ) f( ! uv) 6= f( ! vw) , and (ii) ! uv; ! tv 2 A(G) ) f( ! uv) 6= f( !
tv). The dire ted star arbori ity of G, denoted bydst(G), isthen thesmallestk forwhi h G
admitsa k-dst- oloring.
AgraphGisa ir uiti ifitdoesnot ontainany ir uit. A k-a ir uiti -dire ted-star- oloring (orsimplyk-adst- oloring)ofagraphGisapartition
of A(G)into k dire ted star forests fF 1
;F 2
;:::;F k
g su h that for all i;j 2 [1;k℄; F
i [F
j
is a ir uiti . Equivalently,a k-adst- oloring of G is a k-dst- oloringofGsu hthatno ir uitinGisbi hromati . Thea ir uiti dire ted
stararbori ity ofG,denotedbyadst(G),isthesmallestkforwhi hGadmits ak-adst- oloring.
Note that fromthe above de nitionswe get that every edge- oloring of
an undire tedgraphH is adst- oloring of any orientationof H. Similarly, every a y li edge- oloringof H isanadst- oloringofanyorientationofH. The followingnotationwillbeextensivelyusedintherest of thepaper.
Considera graph G and let A 0 =fa 1 ;a 2 ;:::;a n g be a subsetof A(G). We denotebyC G (a 1 ;a 2 ;:::;a n ),orsimplyC G (A 0
),thesetof ir uitsofGthat ontainall thear s a
1 ;a 2 ;:::;a n .
Drawing onventions. In all the gures, we shall use the following on-vention: avertexwhoseneighborsaretotallyspe i edwillbebla k,whereas avertexwhoseneighborsarepartiallyspe i edwillbewhite. Moreover,an
Suppose that Theorem 1 is false and onsider a minimal ounter-example G. Weprovea seriesoflemmas. Inea h ofthem,weredu e Gto a smaller graphG
0
(thatisjA(G)j>jA(G 0
)j)whi hadmitsa4-adst- oloring f 0
whi h
is also a partial adst- oloring of G (that is an adst- oloring only de ned on some subset A
0
of A(G)). We extend su h a partial adst- oloring f 0
to an adst- oloring f of G. In this ase, it shouldbe understoodthat we set
f(a) = f 0
(a) for every ar a 2 A(G 0
). We then explain how to set f(a) for every un olored a 2 A(G). The existen e of f proves that G does not ontain some spe i on gurations. This set of on gurations will nally
leadto a ontradi tion.
Consider a ir uit C and let u;v 2 V(C). We denote by P C
(u;v) the dire tedpath fromu to v inC.
The followingobservation willbe extensivelyusedinthesequel:
Observation 2 Let C be a ir uit, f an adst- oloring of C, and C 0
the
ir uitobtained fromC byrepla ingP C
(u;v)by adire tedpathP C 0 (u;v). If f 0 is a dst- oloring of C 0 su h that f 0
(a)=f(a) for every a2= P C 0 (u;v) and ff(a); a2 P C (u;v)g ff 0 (a 0 ); a 0 2 P C 0 (u;v)g then f 0 is an adst- oloring of C 0 .
Thisdire tly follows fromthe fa tthatjf 0
(C 0
)jjf(C)j3.
We rst show that a minimal ounter-example to Theorem 1 is ne es-sarilya ubi graph.
Lemma3 IfGisaminimal ounter-exampletoTheorem1,thenÆ(G)3.
Proof. Letv2V(G) withd(v)2. We onsidertwo ases:
Case 1 : d G
(v)=1.
Considerthedanglingar uv inGandlet f 0 beany4-adst- oloring of thegraph G 0 =Gnfvg. We extend f 0 to a4-adst- oloring f of Gby setting f(uv) = a for some olor a distin t from the olors of the at mosttwo ar sin ident to uv.
Case 2 : d G
(v)=2.
Considerthetwoar suvandwvinGandletf 0
beany4-adst- oloring
ofthegraphG 0
obtainedfromGby ontra tinguvinasinglevertexx. We extend f
0
to a 4-adst- oloring f of G by setting f(wv) =f 0
(wx) and f(uv) = a for any a distin t from the olors of the three ar s
Lemma4 If G is a minimal ounter-example to Theorem 1, then G does not ontain any sour e vertex.
Proof. Letv2V(G) bea sour evertex, u 1
,u 2
and u 3
bethethree neigh-borsofvandf
0
beany4-adst- oloringofthegraphG 0
=Gnv. ByLemma3, weknowthatd
+
(v)=3. Ea hofthear s ! vu 1 , ! vu 2 and ! vu 3
hasatleasttwo
available olors. Sin e they an get the same olor, we an extend f 0
to a 4-adst- oloring f of G,a ontradi tion.
We now prove that a minimal ounter-example to Theorem 1 ontains notriangle.
Lemma5 If G is a minimal ounter-example to Theorem 1, then G is
triangle-free.
Proof. If G ontainsthree pairwiseadja ent triangles,then Gis an orien-tationofthe ompletegraphK
4
. ByLemma4weonlyhaveto onsiderthe
twoorientations ofK 4
depi tedonFigures1(a)and1(b)thatbothadmita 4-adst- oloring.
If G ontainstwo adja ent triangles,then G ontainsthe on guration
of Figure 1( ). Consider thegraph G 0 =Gnfw;xg and let f 0 bea 4-adst- oloringofG 0 su hthatf 0 (uv)6=f 0
(yz)(this an bedonesin ewehavetwo possible hoi es for oloring ea h of uv and uz). Suppose without loss of
generalitythat f 0
(uv) =1 and f 0
(yz) =2. In this ase, we an produ ean a y li 4-edge- oloringasdepi tedonFigure1( ). Indeed,this oloringis a properedge- oloringandno pathlinkinguandz isbi hromati . Hen e,for
all possibleorientations of thear s of the on guration, this oloringgives a4-adst- oloring f ofG.
Suppose nally that G ontains the on guration of Figure 1(d), and
let f 0
be any 4-adst- oloring of the graph G 0
obtained from G by on-tra ting the triangle v
1 v
2 v
3
in a single vertex v. Therefore, every ir uit C2C G ( ! u i v i ; ! v j u j ) orrespondsto a ir uitC 0 2C G 0( ! u i v; ! vu j ).
We now extend thepartial adst- oloring f 0
to a 4-adst- oloring f of G asfollows. We distinguishtwo ases:
Case 1 : f 0 (vu 1 )6=f 0 (vu 2 )6=f 0 (vu 3 )6=f 0 (vu 1 ). Withoutlossofgenerality,supposethatf
2 1 4 3 1 (a) 3 1 1 2 3 (b) 3 2 3 1 4 2 1 w y v z u x ( ) u 2 v 1 u 1 v 2 v 3 u 3 (d)
Case 2 : 9 i;j 2f1;2;3g, i6=j,f (v i u i )=f (v j u j )=a. In this ase we ne essarily have
! v i u i ; ! v j u j 2 A(G). Let k 2 f1;2;3g, k6=i;j. We thenset f(v i v k ),f(v j v k ) and f(v i v j )asfollows: 1. f(v i v k
)=bforanyb62fa;f 0 (u k v k )g, 2. f(v j v k
)= forany 62fa;b;f 0 (u k v k )g, 3. f(v i v j
)=d foranyd62fa;b; g.
This an bedonesin ewehave fouravailable olors.
Inboth ases,thanks toObservation 2,we obtaina4-adst- oloring f ofG, a ontradi tion.
Let G be a graph and C a ir uit in G. An ar having exa tly one of its endpointsin C is said to be in ident to C. Moreover, two su h in ident ar sare neighboring iftheirendpointsinC arelinkedbyan ar of C.
The four next lemmas will allow us to prove that a minimal ounter-exampleGto Theorem1 isne essarily a ir uiti .
Lemma6 If G is a minimal ounter-example to Theorem 1, then G does
not ontain a ir uit all of whose verti es have indegree one and outdegree two.
Proof. Supposethat there existsa ir uit C =f ! v 0 v 1 , ! v 1 v 2 , ::: , ! v k 2 v k 1 , ! v k 1 v 0 gin G su h that d + (v i ) =1 and d (v i
) =2 for i2[0;k 1℄and let
f 0
beany4-adst- oloring of thegraph G 0 =GnC. Letf ! v i u i ji2[0;k 1℄g be theset ofar s in identto C.
We extendthepartial oloringf 0
to a 4-adst- oloring f of Gasfollows.
Duetotheorientationof G,C is theonly ir uitofGthatdoesnotbelong to G
0
. Therefore, we onlyneedto olor thear s of C insu ha waythat C isnotbi hromati . Wedistinguishtwo ases dependingon the olorsofthe
ar sin ident to C.
1. Allar s in ident to C are olored with the same olor. In this ase, we olorthear s ofC usingthe threeother remaining olors.
2. Two neighboring ar s in ident to C have distin t olors. Suppose without loss of generality that f
0 ( ! v 0 u 0 ) = 0 and f 0 ( ! v 1 u 1 ) 6= 0 . In
this ase, we set
v 1 v 0 v 2 u 0 1 u2
(a)ThegraphG
u 0 1 u2 v (b)ThegraphG 0
Figure2: The on gurationof Case1of Lemma 7and its redu tion.
(b) 8i2[1;k 2℄; f( ! v i v i+1 )= i forany i 62f i 1 ;f( ! v i+1 u i+1 )g, ( ) f( ! v k 1 v 0 )= k 1 forany k 1 62f 0 ; 1 ; k 2 g.
The ir uitC is learlynotbi hromati sin e 0 6= 1 6= k 1 6= 0 .
Inboth ases,we obtaina 4-adst- oloring f of G, a ontradi tion.
Lemma7 If G is a minimal ounter-example to Theorem 1, then G does
not ontain a ir uit all of whose verti es have indegree two and outdegree one.
Proof. Supposethat there existsa ir uit C =f ! v 0 v 1 , ! v 1 v 2 , ::: , ! v k 2 v k 1 , ! v k 1 v 0
ginG(see Figures2(a)or3(a))su hthatd + (v i )=2and d (v i )=1 fori2[0;k 1℄. Let f ! u i v i
ji2[0;k 1℄g bethe set of ar s in ident to C. ByLemma5,thetailsoftwo neighboringar sin identto C arene essarily
distin t.
We onsidertwo ases dependingon whether theverti esu 0
and u 2
are distin tornot. We rst showthatinboth asesthere existsaredu tionG
0
ofG(seeFigures2(b)and3(b))whi hadmitsa4-adst- oloringf 0 su hthat f 0 ( ! u 0 v 0 )6=f 0 ( ! u 1 v 1 )6=f 0 ( ! u 2 v 2 )6=f 0 ( ! u 0 v 0 ). Case 1 : u 0 6=u 2
(see Figure 2(a)).
Letf 0
beany4-adst- oloring of thegraphG 0
v 1 v 0 v 2 u 1
(a)ThegraphG
v 1 v 0 v 2 u 1 (b)ThegraphG 0
Figure3: The on gurationof Case2of Lemma 7and its redu tion.
identifying v 0 , v 1 and v 2
in a single vertex v (see Figure 2(b)). We
learlyhave f( ! u 0 v)6=f( ! u 1 v)6=f( ! u 2 v)6=f( ! u 0 v). Case 2 : u 0 =u 2
=u(see Figure 3(a)). NotethatbyLemma4 we have u
1 6=u. Letf 0 beany4-adst- oloring ofthegraphG 0
=GnC(seeFigure3(b)). Sin ewehaveatleastthree available olors for the ar s
! uv 0 and ! uv 2 , we an hoose f 0 in su h a waythatf 0 ( ! uv 0 )6=f 0 ( ! u 1 v 1 )6=f 0 ( ! uv 2 )6=f 0 ( ! uv 0 ).
Assume now that f( ! u 0 v 0 ) = 1 , f( ! u 1 v 1 ) 6= 1 and f( ! u 2 v 2 ) 6= 1 . As in theprevious lemma,C is theonly ir uitof G thatdoes notbelong to G
0 . Therefore, we onlyneed to olor thear s of C insu ha way that C is not
bi hromati . We thenset f asfollows:
ounter-example to Theorem 1, there exist two neighboring ar s in ident
withChavingoppositedire tions(withrespe ttoC). Thenexttwolemmas willshowthat thissituationisalso notpossible.
Lemma8 If G is a minimal ounter-example to Theorem 1, then G does
not ontain the on guration depi ted on Figure 4(a).
Proof. SupposethatthegraphG ontainsthe on gurationofFigure4(a), withthear su
0 1 u,u 0 2 u,y 0 1 y,y 0 2 y,v 0 1 v,v 0 2 v,z 0 1 zandz 0 2
zbeingpairwisedistin t, and let f
0
be any4-adst- oloring of the graph G 0
obtained from Gnfw;xg
by adding the ar s ! uy and
!
vz (see Figure 4(b)). Supposethat f 0 ( ! uy) = a andf 0 ( ! vz)=b.
We extend the partial4-adst- oloring f 0 to a 4-adst- oloring f of G as follows. Let E = C G ( ! uw; ! wx; ! xy)[C G ( ! vw; ! wx ; ! xz) and F = C G ( ! uw; ! wx ; ! xz)[ C G ( ! vw; ! wx ; ! xy). We rst set f( ! xy) =a and f( !
xz)=b. Clearly,all ir uits in G not belonging to E[F also belong to G
0
, and thus are already not bi hromati . Moreover, by Observation 2, the ir uits in E will not be
bi hromati . Therefore,we onlyhave to payattentionto the ir uitsinF. We onsidertwo ases dependingonthe olorsaand b:
Case 1 : a6=b. We setf( ! uw)=a, f( ! vw)=band f( !
wx)= forany 2=fa;bg. Sin e
jff( ! uw);f( ! wx);f( ! xz)gj=jfa; ;bgj=3andjff( ! vw);f( ! wx );f( ! xy)gj= jfb; ;agj=3,no ir uitinF is bi hromati .
Case 2 : a=b.
We onsiderthree sub ases.
1. f ! uu 0 1 ; ! uu 0 2 ; ! vv 0 1 ; ! vv 0 2 g T A(G)6=;.
We assume without loss of generality that ! vv 0 1 2 A(G). In this ase, we rst set f( !
uw) = forany 2= fa;f(uu 0 1 );f(uu 0 2 )g and f( !
vw) =d for any d 2= fa; ;f(vv 0 2
)g. Now, we an olorthe ar !
wxwiththefourth olore2=fa; ;dg. Therefore,jff( ! uw),f( ! wx ), f( ! xz)gj =jf ;e;agj=3, jff( ! vw), f( ! wx), f( ! xy)gj=jfd;e;agj = 3,and sono ir uitinF isbi hromati .
y x w u z v 0 1 v z 0 1 u 0 1 u 0 2 y 0 1 y 2 v 0 2 z 0 2 a b
(a)ThegraphG
y u z v 0 1 v z 0 1 u 0 1 u 0 2 y 0 1 y 0 2 v 0 2 z 0 2 a b (b)Theredu tionG 0
Notethatsin ea2= ff(u 0 1 u);f(u 0 2 u);f(v 0 1 v);f(v 0 2 v)g,we ne essar-ilyhave ff( ! u 0 1 u);f( ! u 0 2 u)g\ff( ! v 0 1 v);f( ! v 0 2 v)g 6= ;. Therefore, we
an assume withoutlossof generality thatf( ! u 0 1 u)=f( ! v 0 1 v)= , f( ! u 0 2 u)=dand f( ! v 0 2
v)=e,witha; ;d;ebeingpairwisedistin t. Inthis ase,wesetf(
!
uw)=eandf( !
vw)=d. Now,we an olor thear
!
wxwith the olor . Therefore, jff( ! uw), f( ! wx), f( ! xz)gj =jfe; ;agj =3, jff( ! vw), f( ! wx), f( !
xy)gj = jfd; ;agj = 3, and sono ir uitin F isbi hromati . 3. ! u 0 1 u; ! u 0 2 u; ! v 0 1 v; ! v 0 2 v2A(G) andff( ! u 0 1 u);f( ! u 0 2 u)g=ff( ! v 0 1 v);f( ! v 0 2 v)g.
We assume withoutloss of generality that f( ! u 0 1 u) =f( ! v 0 1 v) = andf( ! u 0 2 u)=f( ! v 0 2 v)=d,a6= 6=d6=a. Wethensetf( ! vw)=a andf( !
uw)=ewith e2=fa; ;dg.
Sin e ! uw and
!
xz are olored with distin t olors, no ir uit in C G ( ! uw; ! wx ; ! xz)is bi hromati .
We still have to set the olor of the ar ! wx. We onsider three sub ases. (a) f ! y 0 1 y; ! y 0 2 yg T A(G)6=;
We assume without loss of generality that ! y 0 1 y 2A(G). So, if f(yy 0 2 ) = (resp. f(yy 0 2 ) = d), we set f( ! wx) = d (resp. f( ! wx) = ), otherwise (f(yy 0 2
) = e), we use either or d.
Therefore, jff( ! wx );f( ! xy);f( ! yy 0 2
)gj =3, and thus no ir uit
inC G ( ! vw; ! wx ; ! xy) isbi hromati . (b) ! yy 0 1 ; ! yy 0 2 2A(G)and ff( ! v 0 1 v);f( ! v 0 2 v)g6=ff( ! yy 0 1 );f( ! yy 0 2 )g.
Weassume withoutlossof generalitythat f( ! yy 0 2 )=e. Now, if f( ! yy 0 1 ) = (resp. f( ! yy 0 1 ) = d) we set f( ! wx ) = d (resp. f( !
wx) = ). This implies that for any i 2 f1;2g, jff( ! wx ), f( ! xy),f( ! yy 0 i
)gj=3,andthusno ir uitinC G ( ! vw; ! wx ; ! xy) is bi hromati . ( ) ! yy 0 1 ; ! yy 0 2 2A(G)and ff( ! v 0 1 v);f( ! v 0 2 v)g=ff( ! yy 0 1 );f( ! yy 0 2 )g.
We an suppose without loss of generality that f( ! yy 0 1 ) = and f( ! yy 0 2 ) = d. We then set f( ! wx) = . If there is no
ar emanating from y 0 1
and olored with a, no ir uit in C G ( ! vw; ! wx ; !
xy) is bi hromati . If there exists an ar ema-natingfrom y
0 1
and oloredwitha, thenthere existsat least
thear yy 0 1
insu hawaythatweforbidbi hromati ir uits
inC G ( ! vw; ! wx ; ! xy).
Inall ases, we obtaina 4-adst- oloring f of G, a ontradi tion.
In the on guration of the previous Lemma, the ar s uu 0 i and vv 0 j on one hand, yy 0 i and zz 0 j
on theother hand, are ne essarily distin tsin e, by Lemma 5, a minimal ounter-example to Theorem 1 ontains no triangle. Thenextlemmadealswiththe asewheretwoar suu
0 i (orvv 0 i )and yy 0 j (or zz 0 j
) arethesame. Without lossofgenerality,we willsupposethatthear s v 0 1 v andz 0 1
z arethe same.
Lemma9 If G is a minimal ounter-example to Theorem 1, then G does
not ontain the on guration depi ted on Figure 5(a).
Proof. SupposethatthegraphG ontainsthe on gurationofFigure 5(a) (in this on guration, two ar s linking a bla k and a white vertex may be the same provided it does not produ e a triangle). We onsidertwo ases
dependingon theorientationof thear vz.
Case 1 : !
vz2A(G).
Considerthegraph G 0 1
(see Figure 5(b))obtained fromGnfw;xg by
addingthear s ! uv and
!
zy (see Figure 5(b)) andlet f 0 1 be any 4-adst- oloringof G 0 1 . Assume that f 0 1 ( ! uv) = a, f 0 1 ( ! vz) = b and f 0 1 ( ! zy) = (seeFigure5(b)). Weextendthepartial4-adst- oloringf
0 toa 4-adst- oloringf ofG asfollows. We rst set f( ! uw) = a, f( ! wx) = f( ! vz) = b and f( ! xy) = (see Figure 5(a)). By Observation 2, no ir uit in G is thus bi hromati . We then olor thear s
! vw and ! xz so that f( ! vw) 2= fa;b;f(v 0 v)g and f( ! xz)2= fb;f(zz 0 )g. Case 2 : ! zv2A(G).
Consider the graph G 0 2
obtained from Gnfw;xg by adding the ar s ! uz and ! vy and let f 0 2 be any 4-adst- oloring of G 0 2 . Assume that f 0 2 ( ! uz) =a, f 0 2 ( ! zv) = b and f 0 2 ( !
vy) = (see Figure 5( )). We extend thepartial4-adst- oloring f
0
to a 4-adst- oloring f of Gasfollows.
As in the previous ase, we set f( ! uw) = a, f( ! wx) = f( ! vz) = b and f( !
xy) = (see Figure 5(a)). By Observation 2, we only have to pay attentionto the ir uitsinC
G ( ! vw; ! wx ; !
y x w u z b a v 0 v z 0 u 0 1 u 0 2 y 0 1 y 2 b
(a)ThegraghG
u a y y 0 2 y 0 1 u 0 2 u 0 1 z 0 z v 0 v b (b)Theredu tionG 0 1 z b z 0 v 0 u y v u 0 1 u 0 2 y 0 2 y 0 1 a ( )Theredu tionG 0 2
2
1
4
3
1
2
v
u
w
x
Figure 6: Theorientation ! K
4
su h thatadst( ! K 4 )=4 and !
xz insu h away that f( ! vw) 2= fa;b;f(v 0 v)g and f( ! xz)=a(this anbedonesin ef 0 (zz 0 )6=a). Sin e a6=b6=f( ! vw)6=a,no ir uitin C G ( ! vw; ! wx ; ! xz) isbi hromati .
Inboth ases we obtaina4-adst- oloring f ofG, a ontradi tion.
Usingthepreviouslemmas, we an nowprove ourmainresult.
ProofofTheorem1. ByLemmas6,7,8and9,aminimal ounter-example Gto Theorem1doesnot ontainany ir uit. Therefore, any4-dst- oloring of G is a 4-adst- oloring of G. Moreover, it follows from the de nitions
thatanyk-edge- oloringoftheunderlyingundire tedgraphofGisa k-dst- oloring of G. Therefore, by Vizing's theorem [11 ℄, the graph G admits a
4-edge- oloringand thusa 4-adst- oloring, a ontradi tion.
The bound given in Theorem 1 is optimal. To see that, onsider the
orientation ! K
4
of the omplete graphK 4
given on Figure 6. If we want to olor this graph with three olors, the only way to olor the ar s
! uv, ! xv, ! xu, ! wx and !
vwis learlytheonedepi tedon Figure6. Butinthis ase, we
needone more olorforthear !
wuand thus, adst( ! K
4 )=4.
4 Dis ussion
In [3 ℄ Burnstein proved that every graph with maximum degree 4 admits
an a y li 5-vertex- oloring. Sin e the linegraph of a sub ubi graph has maximum degree at most 4, we get that every sub ubi graph admits an a y li 5-edge- oloring and thus a 5-adst- oloring. Our result shows that
this bound an be de reased to 4 when onsidering oriented graphs and a ir uiti ar - olorings.
We also provided an oriented ubi graph with a ir uiti dire ted star
Fromourresult,wegetthateveryorientedgraphwithmaximumdegree
threeadmitsa42 4 1
=32-ar - oloring. However, everysu hgraphadmits an11-vertex- oloring [10 ℄ andthusan 11-ar - oloring.
In a ompanion paper [7℄ we show that every K 4
-minor free oriented
graphGhasa ir uiti dire ted stararbori ityatmostminf(G); (G)+ 2g,where (G)standsforthemaximumindegreeofG. This lassofgraphs ontains in parti ular outerplanar graphs. It would thus be interesting to
determinethea ir uiti dire ted star arbori ityof planargraphs.
Referen es
[1℄ I. Algorand N.Alon. Thestararbori ityofgraphs. Dis reteMath., 75:11{22, 1989.
[2℄ O. V. Borodin, A. V. Kosto hka, J. Ne set
ril, A. Raspaud, and E. Sopena. On the maximum average degree and the oriented
hro-mati numberof a graph. Dis reteMath., 206:77{89, 1999.
[3℄ M.I. Burnstein. Every 4-valent graph has an a y li 5- oloring.
Soobs . Akad. Nauk Gruzin. SSR 93, pages21{24, 1979.
[4℄ B.Cour elle. Themonadi se ondorder-logi ofgraphsVI:on
sev-eral representationsof graphsby relationnalstu tures. Dis reteAppl. Math., 54:117{149, 1994.
[5℄ B. Guiduli. Onin iden e oloringand star arbori ityof graphs. Dis- reteMath., 163:275{278, 1997.
[6℄ A. V. Kosto hka, E. Sopena, and X. Zhu. A y li and oriented hromati numbersof graphs. J. Graph Theory,24:331{340, 1997.
[7℄ A. Pinlou and E. Sopena. The a ir uiti dire ted star arbori ity of K
4
minor-free graphs. 2004. Manus ript.
[8℄ A. Raspaud and E. Sopena. Good and semi-strong olorings of ori-ented planar graphs. Information Pro essing Letters, 51(4):171{174,
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fren h). PhDthesis,LaBRI, Universite deBordeaux I, 1997.
