ELSEVIERDiscreteMathematics149(1996)93-98 DISCRETE MATHEMATICS Stararboricityo f graphs S. L. Haki mi a,j . Mi t c h e mb,E. S c h me i c h e l b,, a Departmentof EleetriealandComputerEngineering,UCDavis,Davis,CA95616,USA b Departmentof MathematicsandComputerScience,SanJoseStateUniversity,SanJose, CA95192,USA Received16September1993; revised9August1994 Abstract Wedevelopaconnectionbetweenvertexcoloringingraphsandstararboricitywhichallows ustoprovethateveryplanargraphhasstararboricityatmost 5.Thissettlesanopenprobl em raisedindependentlybyAlgorandAlonandbyRingel.Wealsoshowthatdecidingi f agraph hasstararboricity2isNP-complete, evenfor2-degenerategraphs. 1.I nt roduct i on Ou r t e r mi n o l o g y a ndnot a t i onwi l l be s t andar de x c e p t asi ndi cat ed. Ag o o d r e f e r e n c ef or a n y undef i nedt e r ms is[5]. Ast ari sat r eewi t hat mo s t one v e r t e x wi t hd e g r e e l a r ge r t han1.Ast arJorest( a l s ot e r me d ame adowin[ 10] ) isaf or e s t wh o s e c o mp o n e n t s ar ee a c hst ar s. Ast ar colorinyo f Gi sapa r t i t i ono f E ( G) i nt os t ar f or es t s . Equi va l e nt l y, as t ar c ol or i ngo f G i sac ol or i ngo f E ( G) s ucht hat t her ei snomo n o c h r o ma t i c 3- pat h; t hus s t ar c o l o r i n g is ar at her nat ur al e xt e ns i ono f or di na r ye dge col or i ng. Th e st ararbori ci t yo f G, de not e d s t ( G) , i st hemi n i mu mn u mb e r o f e dge - di s j oi nt s t ar f or e s t s wh o s e uni onc o v e r s allo fE( G) . We cal l Gk-st ar-col orabl ei f st(G)1}l=1;inthat casejiscalledthecolorof thecentervunderc~t. Astarcoloringcg~of Giscalled st ri ct i f eachedge joining twocentersunderoK' hasthecolorof oneof thetwocenters. Apropervertexcoloringc~ofGiscalledacycl i c(uni cycl i c)i f everypairofcolor classesunder~inducesagraphwhosecomponentsareallacyclic(whosecomponents eachcontainatmostonecycle).Clearlyanacycliccoloringisunicyclic.Givena vertexcoloringcgof G,astarcoloringcg~ofGissaidtorespect cgi fthecolorofeachcentervundercg~ matchesthecolorof vunderc~. S.L.Hakimi et al./ Discrete Mathematics 149(1996) 93-9895 Wenowproveatheoremwhichestablishesaninterestingconnectionbetweenuni- cyclicvertex-coloringandstrictstar-coloring.Wewillthenapplythistheoremtoes- tablishatightupperboundforst (G)whenGisplanar,andalsotoobtainasufficient conditionforak-degenerategraphtobe(k+1)-star-colorable. Theorem1.I f Ghasauni~Tclick-vertex-coloring,thenGhasastrictk-star-coloring inwhicheachvertexisgood. BeforeprovingTheorem1,weneedthefollowinglemma. Lemma1.Let Bbeabipartitegraphinwhicheachcomponentcontainsatmostone cycle.Let cgbeany2-vertex-coloringofB.ThenBhasastrict2-star-coloringwhich respectscgandunderwhicheveryvertexisgood. Pr oof . Let1and2denotethecolorsin~. Ineachacycliccomponentof B,select anyvertexastheroot,andcoloreachedgeewiththecolorof thevertexincidentto ewhichisclosertotheroot.Ineachunicycliccomponentof B,colortheedgesofthecyclealternately1and2.Thencoloreachremainingedgeewiththecolorof the vertexincidenttoewhichisnearerthecycle.[] ProofofTheorem1.Letc~=(Vi,V2. . . . , Vk)beaunicyclick-vertexcoloringof G. Foreachintegerpairi , j [1,k],thesubgraphGijinducedbyVi tOVjisabipartite graphinwhicheachcomponentcontainsatmostonecycle.ByLemma1,G~jhasa strict2-star-coloring~' whichrespectsthe2-vertex-coloringof Giginducedbyc~and suchthateveryvertexisgoodunderc~,. I f we2-star-coloreachGijinthisway,it iseasytoseethatweobtainastrictk-star-coloringof Ginwhicheveryvertexis good. In1980,Borodinprovedthefollowingresult[6]. TheoremD.Everyplanargraphisacyclically5-colorable. Sinceanacyclicvertexcoloringistriviallyunicyclic,TheoremDandTheorem1 togetherimmediatelyimplythefollowingresult,whichsettlestheproblemraisedby AlgorandAlonandbyRingel. Theorem2.Everyplanargraphhasastrict5-star-coloringinwhicheveryvertexis good. TheupperboundinTheorem2istightsinceAlgorandAlon[2]constructedexam- plesof planargraphswhichcannotbe4-star-colored. Theorem1alsohasapplicationforstar-coloringk-degenerategraphs[5,p.272].It iseasytoseethati fGisk-degenerate,thena(G)