Decomposing edge-colored graphs under color degree constraints

0.15 MB·

by  Ruonan Li

Decomposing edge-colored graphs under color degree constraints
Ruonan Li∗1,2,ShinyaFujita†3 andGuanghuiWang‡4
1DepartmentofAppliedMathematics,NorthwesternPolytechnicalUniversity,Xi’an,710072,P.R.China
2FacultyofEEMCS,UniversityofTwente,P.O.Box217,7500AEEnschede,TheNetherlands
3InternationalCollegeofArtsandSciences,YokohamaCityUniversity,22-2,Seto,Kanazawa-ku,Yokohama,236-0027
Japan
4SchoolofMathematics,ShandongUniversity,Jinan,250100,P.R.China
7
1
0 January12, 2017
2

n
a
J
1 Abstract
1
Foranedge-coloredgraphG,theminimumcolordegreeofGmeanstheminimumnumberofcolorsonedges

] whichareadjacenttoeachvertexofG.WeprovethatifGisanedge-coloredgraphwithminimumcolordegree
O atleast5thenV(G)canbepartitionedintotwopartssuchthateachpartinducesasubgraphwithminimumcolor
C degreeatleast2.Weshowthistheorembyprovingamuchstrongerform. Moreover,wepointoutanimportant
relationshipbetweenourtheoremandBermond-Thomassen’sconjectureindigraphs.
.
h
t
a
m Keywords:Bermond-Thomassen’sconjecture;edge-coloredgraph;vertexpartition
[

1
v 1 Introduction
7
0
0 Whenwetrytosolveaproblemindensegraphs,decomposingagraphintotwodensepartssometimesplaysan
3 importantroleintheproofargument.Thisisbecauseonecanapplyaninductionhypothesistooneofthepartsso
0
astoobtainapartialconfiguration,andthenusetheotherparttoobtainadesiredconfiguration.Motivatedbythis
.
1 naturalstrategy,manyworkhasbeendonealongthisline,andnowwe havea varietyofresultsinthispartition
0
problem.Tonameafew,Stiebitz[8]showedanicetheorem,whichstatesthateverygraphwithminimumdegree
7
atleasta+b+1canbedecomposedintotwopartsAandBsuchthatAhasminimumdegreeatleastaandBhas
1
: minimumdegreeatleastb. Weseethatthebounda+b+1isbestpossiblebyconsideringthecompletegraphof
v
ordera+b+1.Bythesameexample,Thomassen[12,13]conjecturedthatevery(a+b+1)-connectedgraphcan
i
X bedecomposedintotwoparts Aand Binsuchawaythat A isa-connectedand Bisb-connected. Itwasshown
r byThomassenhimself[10] thatif b ≤ 2, thenthe conjectureis true. However,rathersurprisingly,evenforthe
a caseb= 3thisconjectureiswidelyopenuntilnow. Likewise,therearesomeotherpartitionproblemstofindthe
partition V(G) = A∪ B so that both A and B have a certain property,respectively. The digraphversion of this
problemwasproposedatthePragueMidsummerCombinatorialWorkshopin1995:ForadigraphD,letδ+(D)be
theminimumoutdegreeofD. Forintegerssandt,doesthereexistsasmallestvalue f(s,t)suchthateachdigraph
D with δ+(D) ≥ f(s,t) admitsa vertexpartition(D ,D ) satisfyingδ+(D ) ≥ s andδ+(D ) ≥ t? In[1, 2] Alon
1 2 1 2
posedtheproblem: Isthereaconstantcsuchthat f(1,2)≤ c?Weonlyknowthat f(1,1)= 3holdsbyaresultof
Thomassen[11]. Nomuchprogresshasbeenmadeforthisproblem. RecentlyStiebitz[9]proposethisproblem
againwhenhedealswiththecoloringnumberofgraphs.Asobservedfromtheaboveknownresults,itseemsthat
thesepartitionproblemsareverydifficultevenifwerestrictourconsiderationtoaveryspecificcase.
∗SupportedbyCSC(No.201506290097);E-mail:[email protected]
†SupportedbyJSPSKAKENHI(No.15K04979);E-mail:[email protected]
‡SupportedbyNSFC(No.11471193,11631014);E-mail:[email protected]
1
In this paper, we would like to consider a similar problemin edge-coloredgraphs. To state our results, we
introduce some notation and definitions. Throughoutthis paper, all graphs are finite and simple. Let G be an
edge-coloredgraph.Foranedgee∈E(G),weusecol (e)todenotethecolorofe.Foravertexv∈V(G),letdc(v)
G G
bethecolordegreeofvinG,thatis,thenumberofcolorsonedgeswhichareadjacenttov. Theminimumcolor
degreeofGisdenotedbyδc(G)(:=min{dc(v): v∈V(G)}). ForasubgraphHofGwithE(H),∅,letcol (H)be
G G
thesetofcolorsappearedinE(H).Also,forapairofvertex-disjointsubgraphsM,NinG,letcol (M,N)betheset
G
ofcolorsonedgesbetweenMandN inG. ForavertexvofG,letNc(v)=col (v,N (v)).Bydefinition,notethat
G G G
dc(v)= |Nc(v)|. Whenthereisnoambiguity,weoftenwritecol(e)forcol (e),col(H)forcol (H),col(M,N)for
G G G G
col (M,N)anddc(v)fordc(v). Agraphiscalledaproperlycoloredgraph(briefly,PCgraph)ifnotwoadjacent
G G
edgeshavethesamecolor. Letaandbbeintegerswitha ≥ b ≥ 1. Apair(A,B)iscalled(a,b)-feasibleifAand
Baredisjoint,non-emptysubsetsofV(G)suchthatδc(G[A])≥aandδc(G[B])≥b;inparticular,ifGcontainsan
(a,b)-feasiblepair(A,B)withV(G)= A∪BthenwesaythatGhasan(a,b)-feasiblepartition.
Again, motivated by the same complete graph having mutually distinct colored edges (that is, the rainbow
K ),weproposethefollowingconjecture.
a+b+1
Conjecture1.1. Let a,bbeintegerswith a ≥ b ≥ 2, andG be anedge-coloredgraphwith δc(G) ≥ a+b+1.
ThenGhasan(a,b)-feasiblepartition.
Themainpurposeofourpaperistogivethesolutionofthisconjectureforthecasea=b=2.
Theorem1.1. Conjecture1.1istruefora=b=2.
Toconsiderourproblem,utilizingthestructureofminimalsubgraphsHwithδc(H)≥2willbeveryimportant.
An edge-colored graph G is 2-colored if δc(G) ≥ 2. Specifically, we say a graph G is minimally 2-colored if
δc(G)≥ 2holdsbutanypropersubgraphH ofGhasminimumcolordegreelessthan2inH. Bydefinition,note
that,everyPCcycleisaminimally2-coloredgraph. Anedge-coloredgraphobtainedfromtwodisjointcyclesby
joiningapathisageneralizedbowtie(morebriefly,callitg-bowtie).Weallowthecasewherethepathjoiningtwo
cyclesisempty. Inthatcase,theg-bowtiebecomesagraphobtainedfromtwodisjointcyclesbyidentifyingone
vertexineachcycle. NotealsothatK +2K (thatis,agraphobtainedfromtwodisjointtrianglesbyidentifying
1 2
onevertexofeachtriangle)isag-bowtiewithminimumorder.
Wehavethefollowingcharacterizationofminimally2-coloredgraphs,whichwillbeusedtoproveourmain
result.
Theorem 1.2. If an edge-colored graph G is minimally 2-colored, then G is either a PC cycle or a 2-colored
g-bowtiewithoutcontainingPCcycles.
In fact Theorem 1.1 will be given by proving a much stronger result. We generalize the concept of (a,b)-
feasiblepartitionsasfollows.Fork≥2ifV(G)canbepartitionedintokpartsA ,A ,...,A suchthatδc(G[A])≥
1 2 k i
a holds for each 1 ≤ i ≤ k then we say thatG hasan (a ,a ,...,a )-feasible partition. In this paper, we will
i 1 2 k
mainlyfocusonthecase where(a ,a ,...,a ) = (2,2,...,2). Forsimplicity,letuscall2k-feasiblepartitionin
1 2 k
this special case (thus, (2,2)-feasiblepartitions are equivalentto 22-feasible partitions). To state our result, we
shallintroducethefollowingtheorem,whichisontheexistenceofvertex-disjointdirectedcyclesindigraphs.
Theorem1.3(Thomassen[11]). Foreachnaturalnumberkthereexistsa(smallest)number f(k)suchthatevery
digraphDwithδ+(D)≥ f(k)containskvertex-disjointdirectedcycles.
BermondandThomassen[3]conjecturedthat f(k)=2k−1andAlon[1]showedthat f(k)≤64k.
Asabove,fork ≥1let f(k)beafunctionsuchthateverydirectedgraphDsatisfyingδ+(D)≥ f(k)containsk
disjointdirectedcycles. Defineafunctiong(k)asfollows.
2, k =1;
g(k)=
max{f(k)+1,g(k−1)+3}, k ≥2.


Ourmainresultisfollowing.
2
Theorem1.4. LetGbeanedge-coloredgraphwithδc(G)≥g(k).ThenGhasa2k-feasiblepartition.
Wethenfocusonthecaseb=2inConjecture1.1. Weobtainedthefollowingpartialresult.
Theorem 1.5. Let a be an integer with a ≥ 2, and let K be an edge-colored complete graph of order n with
n
δc(K )≥a+3. ThenK hasan(a,2)-feasiblepartition.
n n
Also,in[4],itisshownthatanyedge-coloredcompletebipartitegraphK withδc(K ) ≥ 3containsaPC
m,n m,n
C . Thisyieldsthefollowing.
4
Theorem1.6. Ifan edge-coloredcompletebipartitegraph K satisfiesδc(K ) ≥ a+2, then K admitsan
m,n m,n m,n
(a,2)-feasiblepartition.
RegardingConjecture1.1inthegeneralcase,byusingtheprobabilisticmethod,wegetthefollowingresult.
Theorem1.7. Leta,bbeintegerswith a ≥ b ≥ 1. IfG isanedge-coloredgraphwith |V(G)| = n andδc(G) ≥
2lnn+4(a−1),thenGhasan(a,b)-feasiblepartition.
Althoughourresultsmightlookabitmodest,provingConjecture1.1evenforthecaseb=2seemsquitehard.
ThisisbecausewecouldgiveabigimprovementontheAlon’sbound“64k”ifitistrue.
Theorem1.8. IfConjecture1.1istrueforb=2,then f(k)≤3k−1.
InviewofTheorem1.8,ittellsusthatsolvingConjecture1.1completelyseemsaverydifficultproblem.
Thispaperisorganizedasfollows. InSections2, 3and4, wegivetheproofsofTheorems1.2, 1.4and1.7,
respectively. In Section 5, we prove Theorems 1.5 and 1.8. In particular, Theorem 1.8 is obtained by a much
strongerresult(seeProposition4inSection5).
2 Proof of Theorem 1.2
Inordertoprovethistheorem,wefirstintroduceastructuraltheoremcharacterizingedge-coloredgraphswithout
containingPCcycles.
Theorem2.1(GrossmanandHa¨ggkist[6],Yeo[14]). LetGbeanedge-coloredgraphcontainingnoPCcycles.
Thenthereisavertexz∈V(G)suchthatnocomponentofG−zisjointtozwithedgesofmorethanonecolor.
ProofofTheorem1.2.
Let G be a minimally 2-colored graph. If G contains a subgraph H which is a PC cycle or a 2-colored g-
bowtie without containing PC cycles, thenG = H (otherwise, by deleting vertices in V(G)\V(H) or edges in
E(G)\E(H),weobtainasmaller2-coloredgraph).Hence,itissufficienttoprovethatifGcontainsnoPCcycle,
thenGcontainsa2-coloredg-bowtie. ApplyTheorem2.1toG. SinceGisminimally2-colored,wemayassume
thatGisconnectedandthereisavertexz∈V(G)suchthatG−zconsistsoftwocomponentsH andH withall
1 2
theedgesbetweenzandH hascolorifori=1,2.
i
Let zx x ···x and zy y ···y , respectively, be longest PC paths in G\H and G\H starting from z. Set
1 2 p 1 2 q 2 1
x = z and y = z. Since dc (x) ≥ 2 and dc (y) ≥ 2 for arbitrary vertices x ∈ V(H ) and y ∈ V(H ),
w0ehave p,q ≥0 2andthereexGis\tH2vertices x andyG\Hfo1rsomei, jwith0 ≤ i ≤ p−2and0 ≤ 1j ≤ q−2suchth2at
i j
col(x x),col(x x )andcol(y y ),col(y y ).SinceGcontainsnoPCcycle,wehavecol(x x)=col(xx )
p i p−1 p q j q−1 q p i i i+1
andcol(y y )=col(y y ). Together,thepathxx ···x zy y ···y andcyclesx x ···x x andy y ···y y
q j j j+1 i i−1 1 1 2 j i i+1 p i j j+1 q j
forma2-coloredg-bowtie.
Theproofiscomplete. (cid:3)
3
3 Proof of Theorem 1.4
Firstweprovethefollowingproposition.
Proposition1. LetGbeanedge-coloredgraphwithδc(G)≥a+b−1. IfGcontainsan(a,b)-feasiblepair,then
thereexistsan(a,b)-feasiblepartitionofG.
Proof. Let(A,B)bean(a,b)-feasiblepairsuchthatA∪Bismaximal.If(A,B)isnotan(a,b)-feasiblepartition,
thenA∪B= V(G)\S withS , ∅. Since(A,B)ismaximal,(A,B∪S)isnotafeasiblepair. Hencethereexistsa
vertex xinS suchthatdc (x) ≤ b−1. Recallthatdc(x) ≥ a+b−1. Sodc (x) ≥ a. Thus(A∪x,B)is
G[B∪S] G G[A∪x]
afeasiblepair,whichisacontradictionwiththemaximalityof(A,B). Thisprovesthat(A,B)isan(a,b)-feasible
partitionofG. (cid:3)
Itiseasytocheckthatthefollowingpropositionisalsotrue.
Proposition2. LetGbeanedge-coloredgraphwithδc(G)≥ k (a −1)+1. IfGcontainskdisjointsubgraphs
i=1 i
H1,H2,...,Hk suchthatδc(Hi)≥aifori=1,2,...,k,thenGPadmitsan(a1,a2,...,ak)-feasiblepartition.
Inwhatfollows,wewillkeeptheabovepropositionsinmindandusethesefactsasamatterofcourse.
ProofofTheorem1.4.
We provethe theoremby contradiction. LetG be a counterexamplesuch thatG is chosen accordingto the
followingorderofpreferences.
(i)kisminimum;(ii)|G|isminimum;(iii)|E(G)|isminimum;(iv)|col(G)|ismaximum.
By the choice ofG, we knowthat δc(G) = g(k), k ≥ 2 andG containsno rainbowtriangles. Let S = {u :
v
dc (u)=dc(u)−1}. Thenthefollowingtwoclaimsobviouslyhold:
G−v G
Claim1. S ,∅forallv∈V(G).
v
Claim2. Foreachedgeuv∈ E(G),eitheru∈S orv∈S .
v u
Nowweprovethefollowingclaims.
Claim3. Foreachcolori∈col(G),thesubgraphG inducedbyedgescoloredbyiisastar.
i
Proof. BythechoiceofG,weknowthatG containsnomonochromatictrianglesormonochromaticP ’s. Thus
3
foreverycolori ∈ col(G), eachcomponentofG is a star. IfG containsmorethan onecomponent,thencolor
i i
oneofthecomponentswithacolornotincol(G).Thus,wegetacounterexamplewithmorecolorsthanG,which
contradictstothechoiceofG. (cid:3)
Claim4. Foru,v∈V(G),ifu∈S andv<S ,thenS ∩N (v),∅.
v u u G
Proof. Supposetothecontrarythatthereexistverticesu,v∈V(G)satisfyingu∈S ,v<S andS ∩N (v)=∅.
v u u G
Then col(vu) appears only once at u and more than once at v. By Claim 3, the color col(vu) can only appear
at {v}∪S , particularly, not at S . Now we construct a colored graph G′ by deleting the vertex u and adding
v u
edges {vx : x ∈ S } to G with all of them colored by col(vu) (since S ∩ N (v) = ∅, this is possible without
u u G
resultingmulti-edges). Foreachvertex x ∈ V(G′)\S ,wehavedc (x) = dc(x). Foreachvertexy ∈ S ,wehave
u G′ G u
Nc (y) ⊆ (Nc(y)\col(uy))∪col(vu). Since thecolorcol(vu)doesnotappearatS , we havedc (y) = |Nc (y)| =
G′ G u G′ G′
|Nc(y)| = dc(y). Thisimpliesthatδc(G′) ≥ δc(G) = g(k). Notethat|G′| = |G|−1. BytheassumptionofG,we
G G
knowthatG′mustadmita2k-feasiblepartition.ByTheorem1.2,G′containskdisjointsubgraphsH ,H ,...,H
1 2 k
suchthatH iseitheraPCcycleoraminimally2-coloredg-bowtiewithoutcontainingPCcyclesfori=1,2,...,k.
i
If k E(H)⊆ E(G),thenwecanfinda2k-partitionofGasdesired,acontradiction.If k E(H)* E(G),then
i=1 i i=1 i
allStheedgesinT = ( ki=1E(Hi))\E(G)formamonochromaticstarwiththevertexvasSacenter. Thus,without
lossofgenerality,assuSmethatT ⊆ E(H1).
4
x x
v u v
x v
x v
u y y
(a) |T|=1 (b) |T|=2
Fig.1:Casesof|T|
SinceH iseitheraPCcycleoraminimally2-coloredg-bowtiewithoutcontainingPCcycles,foreachvertex
1
a∈H andeachcolor j∈col(H ),thecolor jappearsatmost2timesatainH . Thuswehave1≤|T|≤2.
1 1 1
If|T|=1,thenletxvbetheuniqueedgeinT. ReplacexvinH withthepathxuv(seeFigure3(a)).Weobtain
1
acoloredgraphH′ inGwithδc(H′)≥2.ThusH′,H ,...,H impliesa2k-feasiblepartitionofG,acontradiction.
1 1 1 2 k
If|T| = 2,thenletT = {vx,vy}. Sincecol(vx) = col(vy),weknowthatH isaminimally2-coloredg-bowtie
1
with v being an end vertex of the connecting path in H . Delete the edges vx,vy and add vertex u and edges
1
uv,ux,uyinH (seeFigure3(b)). Weobtainag-bowtieH′ inG withδc(H′) ≥ 2. ThusH′,H ,...,H impliesa
1 1 1 1 2 k
2k-feasiblepartitionofG,acontradiction. (cid:3)
Claim5. Thereexistsanedgexy∈E(G)suchthatx∈S andy∈S .
y x
Proof. Supposenot. ThenbyClaim 2 , we canconstructanorientedgraph D byorientingeachedgee = uv ∈
E(G)fromutovifandonlyifv∈S . Thend+(v)≥2foreachvertexv∈V(D). LetT(v)={u:col(uv)=i}.
u D i
Subclaim1. Foreachvertexv ∈ V(G)andcolorsi, j ∈ col(G)withi , j, if|T(v)| ≥ 2and|T (v)| ≥ 2,thenthe
i j
followingstatementshold:
(a)T(v)∩T (v)=∅andE(T(v),T (v))=∅.
i j i j
(b)G[T(v)]containsatleastoneedge.
i
Proof. (a) By the definition, we know that T(v)∩T (v) = ∅. Since |T(v)| ≥ 2 and |T (v)| ≥ 2, we know that
i j i j
T(v)∪T (v)⊆ S . Letu ∈ T(v)andu ∈T (v). Thencolorsiand jappearsonlyonceatu andu ,respectively.
i j v i i j j i j
Ifuu ∈ E(G),thenvuu visarainbowtriangle,acontradiction.SowehaveE(T(v),T (v))=∅.
i j i j i j
(b)SupposethatG[T(v)]isemptyforsomecoloriwith|T(v)| ≥ 2. Thenchooseu ∈ T(v). Wehaveu ∈ S
i i i v
andv < S . ApplyClaim4touandv,weobtainS ∩N (v) , ∅. Foreachcolori′ ∈ col(G)with|T (v)|≥ 2,by
u u G i′
Subclaim1(a)andtheassumptionthatG[T(v)]isempty,wehaveE(u,T (v))=∅. Notethat
i i′
N+(v)= T (v).
D i′
|Ti′(v)|≥[2,i′∈col(G)
WehaveN (u)∩N+(v)=∅.RecallthatS ∩N (v),∅andS ⊆ N (u).Theremustexistavertexx∈S ∩N−(v).
G D u G u G u D
ItiseasytocheckthatC = xuvxisarainbowtriangleinG,acontradiction. (cid:3)
Subclaim2. Foreachvertexv∈V(G),thereisexactlyonecolori∈col(G)suchthat|T(v)|≥2.
i
Proof. Givenavertexv,byClaim1,wecanfindavertexu ∈ S . BytheassumptionofG,wehavev < S . Let
v u
i = col(uv). Then |T(v)| ≥ 2. This impliesthatfor each vertexv ∈ V(G), there is at least one colori ∈ col(G)
i
such that |T(v)| ≥ 2. Now, suppose to the contrarythat there exists a vertex v ∈ V(G) and colors i, j ∈ col(G)
i
withi, jsatisfying|T(v)|≥2and|T (v)|≥2. BySubclaim1,wecanchooseedgesuw fromG[T(v)]andu w
i j i i i j j
fromG[T (v)]. LetF =G[v,u,w,u ,w ]. Thenδc(F)≥2. Nowwewilldiscussontheminimumcolordegreeof
j i i j j
G−F.
Ifδc(G−F)≥g(k−1),thenbytheassumptionofG,G−Fhasa2k−1-feasiblepartition.TogetherwithG[V(F)],
weobtaina2k-feasiblepartitionofG,acontradiction. Sowehaveδc(G−F) < g(k−1). Let x ∈ V(G−F)bea
vertexsatisfyingdc (x)=δc(G−F). Sinceδc(G)≥g(k)≥g(k−1)+3and|F|=5,wehave
G−F
4≤|col(x,F)|≤5.
5
Forverticesa∈{u,w}andb∈{u ,w },if|col(x,{a,b,v})|≥3,thenitiseasytocheckthateitherxavxorxbvx
i i j j
isarainbowtriangle,acontradiction. Sowehave|col(x,{a,b,v})|≤ 2. Notethat|col(x,F)|≥ 4. Thisforcesthat
vx < E(G)and|col(x,{u,w,u ,w })| = 4. ThusC = xuvu x isa rainbowcycleoflength4. Supposethatthere
i i j j i j
existsavertexy∈V(G−C)suchthatdc (y)<g(k−1). Then|col(y,C)|≥4. Notethatu,u ∈S . Thuseither
G−C i j v
yuvyor yu vy is a rainbowtriangle, a contradiction. Hence we have δc(G−C) ≥ g(k−1). By the assumption
i j
ofG,thegraphG−C hasa2k−1-feasiblepartition. TogetherwithG[V(C)],wegeta2k-feasiblepartitionofG,a
contradiction. (cid:3)
Subclaim2impliesthatthereareatleastg(k)−1colorsappearonlyonceatvforeachvertexv∈V(G). Thus,
we have δ−(D) ≥ g(k)−1 ≥ f(k). So D containsk disjoint directedcycles, which correspondto k disjointPC
cyclesinG,acontradiction. (cid:3)
Claim6. Foreachedgexy∈E(G)satisfyingx∈S andy∈S ,wehave
y x
(a)|Nc(x)∪Nc(y)−col(xy)|≤g(k)−1,and
G G
(b) N (x) −y = N (y)− x = {v : 1 ≤ i ≤ g(k)− 1}, where col(xv) = col(yv) and col(xv) , col(xv ) for
G G i i i i j
i, j∈[1,g(k)−1]withi, j.
Proof. (a)SinceGcontainsnorainbowtrianglesandcol(xy)appearsonlyonceatxandy,respectively.wehave
col(xu) = col(yu)forallu ∈ N (x)∩N (y). NowletG′ =G/xy. ThenG′ iswelldefinedanddc (v) = dc(v)for
G G G′ G
allverticesinV(G)\{x,y}.Letzbethenewvertexresultedbycontractingtheedgexy.
Supposethat|Nc(x)∪Nc(y)−col(xy)|≥g(k),thendc (z)≥g(k).Thuswehaveδc(G′)≥g(k). Bythechoice
G G G′
of G, we know that G′ must admit a 2k-feasible partition. By Theorem 1.2, G′ contains k disjoint subgraphs
H ,H ,...,H such that H (i = 1,2,...,k) is either a PC cycle or a minimally 2-colored g-bowtie without
1 2 k i
containingPCcycles.
Ifz< k V(H),thenH ,H ,...,H arek-disjointsubgraphsofG. Thisimpliesa2k-feasiblepartitionofG,
i=1 i 1 2 k
acontradiSction.Sowecanassumethatz∈V(H1). Apparently,2≤dH1(z)≤4.
Ifd (z) = 2,thenlet N (z) = {u,v}(seeFigure2). Ifu,v ∈ N (x),thenreplacezwith x. Ifu ∈ N (x)and
H1 H1 G G
v < N (x),thenreplacethepathuzvwithuxyv. Inallcases,wecantransformH intoagraphH′ ⊆ G suchthat
G 1 1
δc(H′) ≥ 2 andV(H′)∩V(H) = ∅ fori = 2,3...,k. Thus H′,H ,...,H implythe existenceofa 2k-feasible
1 1 i 1 2 k
partitionofG,acontradiction.
u v
u v u, v ∈ NG(x) x
z u∈NG(x),v6∈N u v
G(x)
x y
Fig.2: d (z)=2
H1
Ifd (z) = 3,then H mustbeaminimally2-coloredg-bowtiewithzbeinganend-vertexoftheconnecting
H1 1
path. LetN (z)={u,v,w}withu,vonasamecycleinH (seeFigure3). If{u,v,w}⊆ N (x),thenreplacezwith
H1 1 G
x. If{u,v} ⊆ N (x)andw < N (x), thenreplacezw with xyw. If{u,w} ⊆ N (x) andv < N (x), thenreplacezv
G G G G
with xyv. Constructionsoftheremainingcasesaresimilar. Finally,inallcases,wecantransformH intoagraph
1
H′ ⊆Gsuchthatδc(H′)≥2andV(H′)∩V(H)=∅fori=2,3...,k. ThusH′,H ,...,H impliesa2k-feasible
1 1 1 i 1 2 k
partitionofG,acontradiction.
If d (z) = 4, then H is a minimally 2-colored g-bowtie with two cycles overlapped on the vertex z. Let
H1 1
N (z) = {u,v,u′,v′} with u,v on one cycle and u′,v′ on the other cycle (see Figure 4). If {u,v,u′,v′} ⊆ N (x),
H1 G
thenreplacezwith x. If{u,v,u′}⊆ Nc(x)andv′ < Nc(x),thenreplacethepathzv′with xyv′. If{u,v}⊆ Nc(x)and
{u′,v′}∩Nc(x)=∅,thensplitzintotheedgexysuchthattheresultinggraphisstillag-bowtie. If{u,u′}⊆ Nc(x)
and{v,v′}∩Nc(x) = ∅,thensplitzintotheedge xy inanorthogonaldirectionsuchthattheresultinggraphisa
cyclewithonechordxy. Constructionsoftheremainingcasesaresimilar. Finally,inallcases,wecantransform
6
u
N G(x ) x w
∈
u, v, w v
u u
z w u,v∈NG(x),w6∈NG(x) x w
y
v v
u,w
∈
NG(x),v6∈N xu w
G(x) y
v
Fig.3: d (z)=3
H1
u u′
z
v v′
u,v, u′,v′ ∈ N G(xu,)v,′u∈′vN6∈G(xN)G(x) u,v′6∈′uN,vG(∈NG(x)v,uv,′u6∈′N∈GN(xG)(x)
u u′ u u′ ux)u′ u u′
x
x x x y
v y
v v′ v yv′ v′ v v′
Fig.4: d (z)=4
H1
H into a graph H′ ⊆ G such that δc(H′) ≥ 2 and V(H′)∩V(H) = ∅ for i = 2,3...,k. Thus H′,H ,...,H
1 1 1 1 i 1 2 k
impliesa2k-feasiblepartitionofG,acontradiction.
(b)ByClaim6(a)andthefactthatdc(x),dc(y)≥g(k),wehaveNc(x)=Nc(y)anddc(x)=dc(y)=g(k).For
G G G G G G
eachcolor j∈ Nc(x)and j,col(xy),sinceG isastarandthecolor jappearsatxandy,weknowthatx,ymust
G j
beleafverticesofG . Letv bethecenterofG . Theproofiscomplete. (cid:3)
j j j
Nowlet{x,y}∪{v : 1 ≤ i ≤ g(k)−1}bethesetofverticesdescribedinClaim6. Withoutlossofgenerality,
i
letcol(xv) = ifori ∈ [1,g(k)−1]. Let H be thesubgraphofG inducedby{x,y}∪{v : 1 ≤ i ≤ g(k)−1}and
i i
R=G−H.
Claim7. For1≤i≤g(k)−1,col(v,S )={i}.
i vi
Proof. Supposetothecontrarythatthereexistsavertexu ∈ S suchthatcol(uv) , i. Ifu = v forsome jwith
vi i j
1 ≤ j ≤ g(k)−1and j , i,thencol(uv) = j(since xvv xisnotarainbowtriangle). Sincethecolor jappearsat
i i j
least2timesatv (=u),weknowthatu<S ,acontradiction.NowthevertexumustbelongtoV(R). Sinceeach
j vi
G (1≤ j≤g(k)−1)isastarandcol(uv),i,wehavecol(uv)<[1,g(k)−1].Ifv ∈S ,thenbyapplyingClaim
j i i i u
6 to the edgeuv, we have N (u)−v = N (v)−u. Since x ∈ N (v), we have x ∈ N (u), namely,u ∈ N (x),
i G i G i G i G G
a contradiction. So we have v < S . ApplyingClaim 4 to uv, we obtain a vertex v ∈ S ∩N (v). Note that
i u i u G i
col(uv) < [1,g(k)−1]andGcontainsnorainbowtriangle,wehavev ∈ R−u. LetF = G[x,y,v,u,v]. Itiseasy
i i
tocheckthatδc(F)≥2.
Wewillshowthatforeachvertexz∈G−F,|col(z,F)|≤3.Forz∈R∩(G−F),theassertionholdssincezhas
noneighborto xory. Thuswemayassumethatz = v forsome jwith1 ≤ j ≤ g(k)−1and j , i. Ifzv < E(G)
j i
orcol(zv) = j, thenwe havethe desiredconclusion. Sowe mayassumethatzisadjacentto v andcol(zv) = i
i i i
(otherwise, zxvz is a rainbowtriangle). Since thereis no rainbowtriangleandG is a star, we can easily check
i i
thatzu< E(G).Sozsatisfiesthedesiredproperty.
7
Now,δc(G−F)≥g(k)−3≥g(k−1). SoG−F admitsa2k−1-feasiblepartition. TogetherwithG[V(F)],we
obtaina2k-feasiblepartitionofG,acontradiction. (cid:3)
Claim8. Thereexistsavertexv with1≤i≤g(k)−1suchthatS ={x,y}.
i vi
Proof. Suppose not. Then there exists a vertex u ∈ S \{x,y} for all i with 1 ≤ i ≤ g(k)− 1. By Claim 7,
i vi
col(uv)=ifor1≤i≤g(k)−1. LetG′ =G−{x,y}. Thenδc(G′)≥δc(G)≥g(k). BythechoiceofG,thegraph
i i
G′mustadmita2k-feasiblepartition,whichimpliesthatGhasa2k-feasiblepartition,acontradiction. (cid:3)
Wearenowinapositiontoprovethetheorem. Letv bethevertexinClaim8. Sincedc(v) ≤ g(k)−1and
i H i
dc(v)≥g(k),thereisavertexu∈R∩N (v). Notethatu<S . ByClaim2,wehavev ∈S . NowapplyClaim
G i G i vi i u
4totheedgeuv,wehaveS ∩N (u),∅. Thisimpliesthateitherx∈N (u)ory∈N (u),acontradiction.
i vi G G G
ThiscompletestheproofofTheorem1.4. (cid:3)
4 Proof of Theorem 1.7
Lemma1. Letk,x ,x ,...,x bepositiveintegersand x anon-negativeintegerwith0 ≤ x ≤ k. Let{vj : 1 ≤
1 2 k 0 0 2 i
i ≤ k,1 ≤ j ≤ x}be asetof k x verticessuchthateachvertex vj is coloredbyi. Dividetheseverticesinto
i i=1 i i
twosetsS andT,randomlyanPdindependently,with Pr(vj ∈ S) = Pr(vj ∈ T) = 1. Let P (x ,x ,...,x )bethe
i i 2 S 0 1 k
probabilityoftheeventthatthereareatmostx (0≤ x ≤ k)differentlycoloredverticesinS. Then
0 0 2
x0 k 1
P (x ,x ,...,x )≤ ( )k. (4.1)
S 0 1 k
j! 2
Xj=0
→−
Proof. For convenience,we say a vector x = (x ,x ,x ,...,x ) is goodif k,x ,x ,...,x are positiveintegers
0 1 2 k 1 2 k
andx isanon-negativeintegerwith0≤ x ≤ k. ProvingLemma1isequivalenttoverifyInequation(4.1)forall
0 →− 0 2 →− →− →−
goodvectors.Forgoodvectors x =(x ,x ,...,x )and y =(y ,y ,··· ,y ),wesay x < y if(a)or(b)holds.
0 1 k 0 1 k′
(a)k<k′.
(b)k=k′andthereexistst∈[1,k]suchthatx <y andx =y foralliwith0≤i<t.
t t i i
→−
NowwewillproveInequation(4.1)foreverygoodvector x =(x ,x ,...,x ).
0 1 k
Byinduction. First,itiseasytocheckthatInequation(4.1)holdsinthefollowingthreecases: (1) x =0;(2)
0
k = 1;(3) x = 1foralliwith1 ≤ i ≤ k. Nowassumethat x ≥ 1,k ≥ 2, x ≥ 2forsomeiwith1 ≤ i ≤ k,and
i 0 i
→− →− →−
eachgoodvector y with y < x satisfiesInequation(4.1). Considerthevertexv1. Wehave
i
→−
P (x)= Pr(v1 ∈T)P (x ,x ,...,x ,x −1,x ,...,x )+Pr(v1 ∈S)P (x −1,x ,...,x ,x ,...,x ).
S i S 0 1 i−1 i i+1 k i S 0 1 i−1 i+1 k
→− →− →−
Let y =(x ,x ,...,x ,x −1,x ,...,x )and z =(x −1,x ,...,x ,x ,...,x ). Itiseasytoseethat y and
0 1 i−1 i i+1 k 0 1 i−1 i+1 k
→− →− →− →−
z aregoodvectorswith y, z < x. Byinductionhypothesis,wehave
→− x0 k 1
P (y)≤ ( )k
S
j! 2
Xj=0
and
P (→−z)≤ x0−1 k−1 (1)k−1.
S
j ! 2
Xj=0
8
Thus,wehave
P (→−x) ≤ 1 x0 k (1)k+ 1 x0−1 k−1 (1)k−1
S
2 j! 2 2 j ! 2
Xj=0 Xj=0
1 x0 k 1 x0 k−1 1
= ( )k+ ( )k
2 j! 2 j−1! 2
Xj=0 Xj=1
1 x0 k 1 x0 j k 1
= ( )k+ ( )k
2 j! 2 k j! 2
Xj=0 Xj=1
1 x0 k 1 x x0 k 1
≤ ( )k+ 0 ( )k
2 j! 2 k j! 2
Xj=0 Xj=1
1 x x0 k 1
< ( + 0) ( )k
2 k j! 2
Xj=0
x0 k 1
≤ ( )k
j! 2
Xj=0
Theproofiscomplete. (cid:3)
ProofofTheorem1.7.
Proof. Assume V(G) = {1,2,··· ,n}. We divide V(G) into two disjoint parts A,B randomlywith Pr(i ∈ A) =
Pr(i ∈ B) = 1 for each vertex i ∈ V(G). For each vertex i ∈ A, the bad event A means that for vertex i,
2 i
{dc (i)≤a−1}. ByLemma1,wehave
G[A]
a−1 dc(i) 1 dGc(i) dc(i) 1
Pr(Ai)≤ Gj !(2)dGc(i) = Gj !(2)dGc(i) = Pr(X ≥dGc(i)−a+1),
Xj=0 j=dcX(i)−a+1
G
whereX ∼ B(dc(i),1).
G 2
RecallthatChernoff’sbound:Pr[X−E(X)≥nǫ]<e−2nǫ2,whereX ∼ B(n,1). Weget
2
Pr(X ≥dGc(i)−a+1)= Pr(X− dGc2(i) ≥ dGc2(i) −a+1)<e−2(dGc2(i)−a+1)2/dGc(i).
Sincedc(i)≥δc(G)>2(a−1),wehave
G
Pr(Ai)<e−2(dGc2(i)−a+1)2/dGc(i) ≤e−2(δc2(G)−a+1)2/δc(G).
Similarly,foreachvertex j∈ B,thebadeventBjmeansthat{dGc[B](j)≤b−1}andPr(Bj)<e−2(δc2(G)−b+1)2/δc(G) ≤
e−2(δc2(G)−a+1)2/δc(G). So
Pr(( Ai)∪( Bj))≤ Pr(Ai)+ Pr(Bj)<ne−2(δc2(G)−a+1)2/δc(G).
[i∈A [j∈B Xi∈A Xj∈B
Ifne−2(δc2(G)−a+1)2/δc(G) ≤ 1,whichmeans1−Pr[( Ai)∪( Bi)] > 0,then δc(2G) −2(a−1)+ 2δ(ac−(G1))2 ≥ lnn. The
i∈A i∈B
lastinequalityholdsbytheconditionthatδc(G) ≥S2lnn+4S(a−1). Thusthereexistsapartitionsuchthatneither
eventA norB happens.Sowehavean(a,b)-feasiblepartition. (cid:3)
i i
9
5 From (a,2)-feasible partitions to Bermond-Thomassen’s conjecture
Firstly,wegivetheproofofThorem1.5.
ProofofTheorem1.5.
Inordertoprovethetheorem,weusethefollowingfact.
Lemma2. [5]Inanyrainbowtriangle-freecoloringofacompletegraph,thereexistsavertexpartition(V ,V ...,V)
1 2 t
ofV(K )witht≥2suchthatbetweentheparts,thereareatotalofatmosttwocolorsand,betweeneverypairof
n
partsV,V withi, j,thereisonlyonecolorontheedges.
i j
IfK containsarainbowtriangleC,thenletA =C andB = K −C. Itfollowsthatδc(A) ≥ 2andδc(B)≥ a.
n n
So(A,B)isan(a,2)-feasiblepartition.NowweassumethatK containsnorainbowtriangle.UtilizingLemma2,
n
wecaneasilyfindan(a,2)-feasiblepartition.ThusTheorem1.5holds. (cid:3)
Inthissection, wewillpointouta relationshipbetween(a,2)-feasiblepartitionsin edge-coloredgraphsand
Bermond-Thomassen’sconjectureindigraphs. Infact,Bermond-Thomassen’sconjecturehasnotevenbeencon-
firmed in multi-partite tournaments. Recently, Li et al. [7] revealed a relationship between PC cycles in edge-
coloredcompletegraphsandBermond-Thomassen’sconjectureonmulti-partitetournaments.
Weprovethefollowingproposition.
Proposition3. Fork≥1letd ,...,d bepositiveintegers,andlet f(d ,d ,...,d ),g(d ,d ,...,d )andh(d ,d ,...,d )
1 k 1 2 k 1 2 k 1 2 k
betheminimumvalueswhichmakethefollowingthreestatementstrue:
(1)EveryorientedgraphDwithδ+(D)≥ f(d ,d ,...,d )hasavertex-partition(V ,V ,...,V )withδ+(D[V])≥
1 2 k 1 2 k i
d fori=1,2,...,k.
i
(2)Everyedge-coloredgraphGwithδc(G)≥g(d ,d ,...,d )hasa(d ,d ,...,d )-feasiblepartition.
1 2 k 1 2 k
(3)Everyedge-coloredcompletegraphKwithδc(K)≥h(d ,d ,...,d )hasa(d ,d ,...,d )-feasiblepartition.
1 2 k 1 2 k
Thenwehave
f(d −1,d −1,...,d −1)≤g(d ,d ,...,d )≤h(d +1,d +1,...,d +1).
1 2 k 1 2 k 1 2 k
Proof. Given an oriented graph D, we construct an edge-colored graph G with V(G) = V(D), E(G) = {uv :
uv ∈ A(D)orvu ∈ A(D)} and col (uv) = v if and only if uv ∈ A(D). If δ+(D) ≥ g(d ,d ,··· ,d ), then by
G 1 2 k
the construction, we know that δc(G) ≥ g(d ,d ,··· ,d ). Thus, G admits a partition V ,V ,...,V such that
1 2 k 1 2 k
δc(G[V])≥d fori=1,2,...,k. Inturn,bytheconstruction,wehaveδ+(D[V])≥d −1fori=1,2,...,k. Recall
i i i i
thedefinitionoffunction f. Weknowthat
f(d −1,d −1,...,d −1)≤g(d ,d ,...,d ).
1 2 k 1 2 k
Givenanedge-coloredgraphG,weconstructanedge-coloredcompletegraphKwithV(K)=V(G),col (e)=
K
col (e)foralle∈E(G),col (e)=c foralle∈ E(K)\E(G)andc <col(G).Ifδc(G)≥h(d +1,d +1,...,d +1),
G K 0 0 1 2 k
thenδc(K)≥h(d +1,d +1,...,d +1).Bythedefinitionofh,weknowthatthereexistsapartitionV ,V ,...,V
1 2 k 1 2 k
of K such that δc(K[V]) ≥ d + 1 for i = 1,2,...,k. By the construction of K, we have δc(G[V]) ≥ d for
i i i i
i=1,2,...,k. Recallthedefinitionofg. Weknowthat
g(d ,d ,...,d )≤h(d +1,d +1,...,d +1).
1 2 k 1 2 k
(cid:3)
Remark 1. The existence of f(d ,d ,...,d ) for d ≥ 2 (i = 1,2,...,k) and k ≥ 2 is still unknownaccording
1 2 k i
to [1]. Proposition 3 implies that we could show the existence of f(d ,d ,...,d ) by proving the existence of
1 2 k
g(d +1,d +1,...,d +1)orh(d +2,d +2,...,d +2).
1 2 k 1 2 k
10