Alon
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Discrete Mathematics 75 (1989) 11-22 North-Holland 11 THE STAR ARBORICITY OF GRAPHS I. ALGOR and N. ALON* Department of Mathematics, Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel A star forest is a forest whose connected components are stars. The star arboricity St(G) of a graph G is the minimum number of star forests whose union covers all edges of G. We show that for every d-regular graph G, id < St(G) =Z id + O(ds(log d)f), and that there are d-regular graphs G with St(G) > id + Q(log d). We also observe that the star arboricity of any planar graph is at most 6 and that there are planar graphs whose star arboricity is at least 5. 1. Introduction All graphs considered here are finite, undirected and simple unless otherwise specified. A stur forest is a forest, whose connected components are stars. The star arboricity st(G) of a graph G is the minimum number of star forests in G whose union covers all edges of G. This notion was introduced in [4], where the authors show that the star arboricity of the complete graph on II vertices is [n/2] + 1. In [5], th e author determines the star arboricity of every complete multipartite graph with equal color classes G and shows that it does not exceed [d/2] + 2, where d is the degree of regularity of G. Notice that by a trivial edge-counting the star arboricity of every d-regular graph is greater than id, and in view of the results above one may be tempted to suspect that St(G) s $d + O(1) for every d-regular graph G. This would also resemble the linear arboricity conjecture. A linear forest is a forest whose connected components are paths. The linear urboricity la(G) of a graph G is the minimum number of linear forests in G whose union covers all edges of G. The linear arboricity conjecture, raised in [2], asserts that for every d-regular graph G, la(G) = [(d + 1)/2]. This conjecture is proved for d G 6, d = 8 and d = 10 in [2, 3, 9, 16, 17, 11, 121. In [l] it is shown that for every E > 0 and every d-regular graph G, id < la(G) < (4 + E)d, provided d > do(E). Here we observe that the star arboricity St(G) of a d-regular graph G can be bigger than id by more than an additive constant. In fact, we show that there are d-regular graphs G with St(G) 3 id + Q(log d). On the other hand, st(G) cannot be much bigger than id. Our main result is that the star arboricity of any d-regular graph G does not exceed id + O(dg(log d)f). This result is proved in *Research supported in part by Allon Fellowship and by a grant from the United States Israel Binational Science Foundation. 0012-365X/89/$3.50 @ 1989, Elsevier Science Publishers B.V. (North-Holland)
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DiscreteMathematics75(1989)11-22 North-Holland 11 THESTARARBORICITYOFGRAPHS I.ALGORandN.ALON* DepartmentofMathematics,SacklerFacultyofExactSciences,Tel AvivUniversity,TelAviv, I srael Astarforestisaforestwhoseconnectedcomponentsarestars.ThestararboricitySt(G)ofa graphGistheminimumnumberofstarforestswhoseunioncoversalledgesofG.Weshow thatforeveryd-regulargraphG,idid+ Q(logd).Wealsoobservethatthestararboricityofanyplanar graphisatmost6andthatthereareplanargraphswhosestararboricityisatleast5. 1.Introduction Allgraphsconsideredherearefinite,undirectedandsimpleunlessotherwise specified.Asturforestisaforest,whoseconnectedcomponentsarestars.The stararboricityst(G)ofagraphGistheminimumnumberofstarforestsinG whoseunioncoversalledgesofG.Thisnotionwasintroducedin[4],wherethe authorsshowthatthestararboricityofthecompletegraphonIIverticesis [n/2]+1.In[5],th eauthordeterminesthestararboricityofeverycomplete multipartitegraphwithequalcolorclassesGandshowsthatitdoesnotexceed [d/2]+2,wheredisthedegreeofregularityofG.Noticethatbyatrivial edge-countingthestararboricityofeveryd-regulargraphisgreaterthanid,and inviewoftheresultsaboveonemaybetemptedtosuspectthatSt(G)s $d+ O(1)foreveryd-regulargraphG.Thiswouldalsoresemblethelinear arboricityconjecture.Alinearforestisaforestwhoseconnectedcomponentsare paths.Thelinearurboricityla(G)ofagraphGistheminimumnumberoflinear forestsinGwhoseunioncoversalledgesofG.Thelineararboricityconjecture, raisedin[2],assertsthatforeveryd-regulargraphG,la(G)=[(d+ 1)/2].This conjectureisprovedfordG6,d= 8andd= 10in[2,3,9,16,17,11,121.In[l] itisshownthatforeveryE >0andeveryd-regulargraphG,id< la(G)< (4+E)d,providedd> do(E). HereweobservethatthestararboricitySt(G)ofad-regulargraphGcanbe biggerthanidbymorethananadditiveconstant.Infact,weshowthatthereare d-regulargraphsGwithSt(G)3id+ Q(logd).Ontheotherhand,st(G)cannot bemuchbiggerthanid.Ourmainresultisthatthestararboricityofany d-regulargraphGdoesnotexceedid+ O(dg(logd)f).Thisresultisprovedin *ResearchsupportedinpartbyAllonFellowshipandbyagrantfromtheUnitedStatesIsrael BinationalScienceFoundation. 0012-365X/89/$3.50@1989,ElsevierSciencePublishersB.V.(North-Holland) 121. A&or,N.Alon Section2byprobabilisticarguments,inamethodthatresemblestheoneusedin [l]butcontainssomeadditionalideas.InSection3weobservethatthereare d-regulargraphsGwithst(G)2id+Q(logd).InSection4thestararboricityof planargraphsisconsidered.WeobservethatforanyplanargraphG,St(G)c6 andconstructplanargraphsGwithSt(G)25.ThefinalSection5containssome concludingremarksandopenproblems. 2.Anupperboundforthestararboricityofregulargraphs Inthissectionweprovethefollowingtheorem. Theorem2.1.Thereisapositiveconstantbsuchthat foreveryd31thestar arboricityofanyd-regulargraphdoesnotexceedid+4* d$. (logd)f+b. Noticethatanimmediatecorollaryofthistheoremisthefollowing. Corollary2.2.ForeveryE >0thereisad,, =do(e)sothat forevery,d>d,,the stararboricityofandd-regularGsatisfies id p-kQ-k) + ( p+ (4-o(l))logp 1 d;+(:-o(l))logd.0 4.Thestararhoricityofplanargraphs Themainresultofthissectionisthatthestararboricityofanyplanargraphis atmost6andthatthereareplanargraphsGwithSt(G)35. FirstweshowthatifGisplanarthenSt(G)s6. ThearboricityofagraphG,A(G),istheminimumnumberofforestsinG whoseunioncoversE(G). LetGbeagraphandputqn= max{]E(H)]: HisasubgraphofGwithn vertices}.AwellknowntheoremofNashWilliams[15]statesthatA(G)= ma{TqJ (n-1>1>.Lemma4.1.I fGis a forestthenSt(G)=Z2. Proof.ItisclearlysufficienttoassumethatGisatree. FixuEV(G),andletd(u,V)bethedistanceofufromI_JinG.Thenthetwo starforests HI=((~7w)6E(G)14 u,v)=2i,d(w,v)=2i+l,i~O} and H2={(u,w)~E(G))d(u,v)=2i+1,d(w,v)=2i+2,i~0} coveralledgesofG.Cl AsaconsequenceofLemma4.1weconcludethatforeverygraphG: A(G)cSt(G)s2A(G). IfG=(V,E)isplanarthenq,,G3n-6andhence,fromNashWilliamstheorem wegetA(G)c3,whichimpliesthatSt(G)s6. 18I. Algor,N.Alon ItiseasytofindaplanargraphGsuchthatSt(G)=4,butitbecomesmore difficulttofindonewithabiggerstararboricity. WenextshowhowtoconstructaplanargraphGsatisfyingSt(G)35.Let G=(V,E),beagraphonV={v,,...,v,}andletC=(H,,...,H,)beastar decompositionofG.WesaythatvE VisagoodvertexinthedecompositionCif I {i1dH,(v)>1}1 c1.Thismeansthatvistakenasacenterofanon-trivialstar (i.e.astarwithmorethanoneedge)inatmostoneforest.Equivalently:vs columninA,,,hasatmostoneelementbiggerthan1. ThedecompositionCisgoodifalltheverticesaregood. LetGbeaplanargraph.Byaddingedgesandvertices(ifnecessary)toG,G canbeembeddedinaplanargraphG,withminimumdegree5. LetG2bethegraphconsistingof7disjointcopiesofG,.Finally,letHobea graphobtainedfromGzbytriangulatingeachofitsfaces.Wehavethus associatedeveryplanargraphGwitha(non-unique)planartriangulation H=H,. Lemma4.2.LetGbeplanar.If HG hasa decompositioninto4 star foreststhenG hasa gooddecompositioninto4star forests. Proof.H,isplanar,ithasminimumdegree6=5andI E(H,)j=3I V(H,)j-6. LetC=(H,,. . . , H4)beadecompositionofHGintostarforests.Weclaimthat inCthereareatmost6verticeswhicharenotgood(=badvertices).Indeed,in A12I E(=3 I V1 -6andtherearenocolumnswithfourls,since6(H,)= 5.;:;~thenumberoflsinacolumnisatmost3,andasthetotalnumberofls is1~3 I V1 -6thereareatmost6columnswithlessthan31s.Obviously, columnswith3lsaregoodandhencethereareatmost6badvertices.ButHG contains7disjointcopiesofG,(containingG),henceatleastinoneofthe copiesofG,alltheverticesaregood.Thus,ifwerestrictCtothatG,wegeta gooddecompositionofG,(andhenceofG),asclaimed.0 NextweshowthatthereexistsaplanargraphGwithnogooddecomposition into4starforests.ThisimpliesthatH(;isplanarandst(H,;)25. Lemma4.3.LetGbeplanarwith agooddecompositionC=(H,,. . ., H,).ifu andu areverticeswith morethan6commonneighborsinG(seeFig.l),thenthey mustbetakenascentersofnon-trivialstarsindifferentforests. Proof.SinceCisagooddecompositionavertixucanbetakenasanon-trivial centeratmostonce.Intheotherforestsitsdegreeisatmost1.Ifuanduhaver commonneighbors(r27)thenwhenu(v)istakenasanontrivialcenter,the correspondingstarmustcoveratleastr-3ofthecommonneighbors. Sincethereareno2disjointsetsofr-3verticesofthecommonneighbors(as r 27),uandvmustbetakenasnontrivialcentersindifferentforests.0 Stararboricityofgraphs19 v Fig.1. Theorem4.4.Thereexistsa planargraphGsuchthat St(G)25. Proof.ConsiderFig.2.Adashedlinebetween2verticesmeansthattheyhave7 commonneighbors(likeuandvinFig.1).Afulllinebetweentwoverticesxand yis likeadashedlinewiththeadditionaledge(x,y).SupposethegraphinFig.2, G,hasagooddecompositioninto4starforests.InviewofLemma4.3if xandy areconnectedbyaline(dashedorfull),theymustbetakenasnontrivialcenters indifferentforests.The7verticesinthemiddleformanoddcyclesotheyare takenasnontrivialcentersin3forests(ormore).uandvareconnectedtoallof themsotheyarenontrivialcentersinthe4thforest.Butuandvcannotbe nontrivialcentersinthesameforestbecausetheyhave7commonneighbors. Thus,Gdoesnothaveagooddecompositioninto4starforestsandhence St(&)35.0 WeknowthatforeverygraphGA(G)cSt(G)c2A(G).Anaturalquestionto consideristhedeterminationofthemaximumstararboricityofagraphG satisfyingA(G)=k.Weconcludethissectionbyshowingthatfork=2this maximumis4,evenifwerestrictourselvestoplanargraphs. Proposition4.5.Thereis a planargraphGsuchthat A(G)=2andSt(G)=4. i___-___________._-i Fig.2. 20I.Al gor,N.Alon .... Fig.3. Proof.ConsiderFig.3.Clearlyitisaplanargraph(edgeslike(u,V)canbe drawnsurroundingthegraph),anditsarboricityis2(thefulllinesformonetree andthedashedlinestheother).WerefertothisgraphasG=(V,E),I V(=n. Weknowthat2cSt(G)c4,butclearlySt(G)23sinceitcontainsaK,. SupposeSt(G)=3,andletC=(H,,Hz,H3)beadecompositioninto3star forests.Ghas4verticesofdegree3andtherestareofdegree4.Hence [El=2n-2.Ifdeg(v)=4itscolumninAG,Cis(uptoapermutation)oneof thefollowingfourtypes: Inthefirst3typesthenumberoflsislessthanhalfofthesumofthecolumn, andinthefourthtypeitispreciselyhalf.ButinAG,CI24 cb,C;l=, aii,soif therearecolumnsfromthefirstthreetypestheymustbebalancedbycolumnsof 1 verticesofdegree3whosecolumnsare 0 1.Sincethereareonly4verticesof 1 degree3,thereareonlyafewcolumnsfromthefirst3types(nomorethan6). Hencewecanassumethattherearenosuchbadcolumns(sinceifthegraphis takentobelongenough,thereisalongenoughsectionwithnobadverticesof degree4).Thus,wemayassumethatthecolumnsofverticesofdegree4areall 2 oftype 0 1(uptoapermutation).ClearlytherearealsoveryfewKi,,starsin 1 C,becausethenumberoftheKr,istarsinCisexactly1 -[El=1 -(2~-2).If deg(v)=3thenthereareatmost3lsinitscolumnandifdeg(v)=4,thereare exactly2ls,hence 1 -I,?51=1 -(2n-2)0(sayE= 0.01)apolynomialtime (deterministicorrandomized)algorithmforproducingthedesiredstarforests. Finally,itwouldbeinterestingtodetermineifthemaximumstararboricityof aplanargraphis5or6. References [1]N.Alon,Thelineararboricityofgraphs,.IsraelJ.Math.62(1988)311-325. [2]J.Akiyama,G.ExooandF.Harary,CoveringandpackingingraphsIII,cyclicandacyclic invariants,Math.Slovaca30(1980)405-417. [3]J.Akiyama,G.ExooandF.Harary,CoveringandpackingingraphsIV,Lineararboricity, Networks11(1981)69-72. [4]J.AkiyamaandM.Kano,Pathfactorsofagraph,in:GraphTheoryanditsApplications(Wiley andSons,NewYork,1984). [S]Y.Aoki,Thestararboricityofthecompleteregularmultipartitegraphs,preprint. [6]B.Bollobls,RandomGraphs(AcademicPress,London,1985). [7]C.Berge,GraphsandHypergraphs(North-Holland,Amsterdam,1976). [8]B.BollobasandA.Thomason,Graphswhichcontainallsmallgraphs,Europ.J.Combinatorics 2(1981)13-15. [9]H.Enomoto,Thelineararboricityof5-regulargraphs,Technicalreport,Dep.ofInformation Sci.,Univ.ofTokyo,1981. [lo]P.ErdosandL.Lovasz,Problemsandresultson3-chromatichypergraphsandsomerelated questions,inInfiniteandFiniteSets,A.Hajnaletal.,Eds,(North-Holland,Amsterdam,1975) 609628. [ll]H.EnomotoandB.Peroche,Thelineararboricityofsomeregulargraphs,J.GraphTheory8 (1981)309-324. [12]F.Guldan,Thelineararboricityoflo-regulargraphs,Math.Slovaca36(1986)225-228. [13]R.L.GrahamandJ.H.Spencer,Aconstructivesolutiontoatournamentproblem,Canad. Math.Bull.14(1971)45-48. [14]R.L.Graham,B.L.RothschildandJ.H.Spencer,RamseyTheory(Wiley-Interscience,New York,1980)79-80. [15]C.St.J.A.Nash-Williams,Decompositionoffinitegraphsintoforests,J.LondonMath.Sot.39 (1964)12. [16]B.Peroche,Onpartitionofgraphsintolinearforestsanddissections,preprint. [17]P.Tomasta,Noteonlineararboricity,Math.Slovaca32(1982)239-242.
