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:liruonan@mail.nwpu.edu.cn †SupportedbyJSPSKAKENHI(No.15K04979);E-mail:fujita@yokohama-cu.ac.jp ‡SupportedbyNSFC(No.11471193,11631014);E-mail:ghwang@sdu.edu.cn 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 andv2(a−1),wehave G Pr(Ai) 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

Decomposing edge-colored graphs under color degree constraints

Similar Decomposing edge-colored graphs under color degree constraints