AUSTRALASIAN JOURNAL OF COMBINATORICSVolume 54 (2012), Pages 107114Some results on incidencecoloring, star arboricityand dominationnumberPakKiuSunWaiCheeShiuDepartmentofMathematicsHongKongBaptistUniversity,KowloonTongHongKonglionel@hkbu.edu.hk wcshiu@hkbu.edu.hkAbstractTwoinequalitiesareestablishedconnectingthegraphinvariantsofinci-dencechromaticnumber,stararboricityanddominationnumber. Usingthese, upperandlowerboundsarededucedfortheincidencechromaticnumberofagraphandfurtherreductionsaremadetotheupperboundforaplanargraph. Itisshownthatcubicgraphswithordersnotdivis-iblebyfourarenot4-incidencecolorable. SharpupperboundsontheincidencechromaticnumbersaredeterminedforCartesianproducts ofgraphs,andforjoinsandunionsofgraphs.1 IntroductionAn incidence coloringseparatesthewholegraphintodisjoint independentincidencesets. SinceincidencecoloringwasintroducedbyBrualdi andMassey[4], mostre-searchhas concentratedondeterminingtheminimumnumber of independent in-cidencesets, alsoknownastheincidencechromaticnumber, whichcancoverthegraph. Theupperboundontheincidencechromaticnumberof planargraphs[7],cubic graphs [13] and a lot of other classes of graphs were determined [6, 7, 15, 17, 21].However, for general graphs, only an asymptotic upper bound is obtained [9]. There-fore, to nd an alternative upper bound and lower bound on the incidence chromaticnumberforallgraphsisthemainobjectiveofthispaper.InSection2,wewillestablishaglobalupperboundfortheincidencechromaticnumber interms of chromaticindexandstar arboricity. This result reduces theupper boundonthe incidencechromaticnumber of the planar graphs. AgloballowerboundwhichinvolvesthedominationnumberwillbeintroducedinSection3.This research was partially supported by the Pentecostal Holiness Church Incorporation (HongKong)108 PAKKIUSUNANDWAICHEESHIUFinally,theincidencechromaticnumberofgraphsconstructedfromsmallergraphswillbedeterminedinSection4.All graphs in this paper are connected. Let V (G) and E(G) (or Vand E) be thevertex-setandedge-setofagraphG,respectively. LetthesetofallneighborsofavertexubeNG(u)(orsimplyN(u)). Moreover,thedegreedG(u)(orsimplyd(u))ofuisequalto |NG(u)|andthemaximumdegreeofGisdenotedby(G)(orsimply). AnedgecoloringofGisamapping: E(G) C,whereCisacolor-set, suchthatadjacentedgesofGareassigneddistinctcolors. Thechromaticindex

(G)ofagraphGistheminimumnumberof colorsrequiredtolabel all edgesof Gsuchthatadjacentedgesreceiveddistinctcolors. Allnotationsnotdenedinthispapercanbefoundinthebooks[3]and[22].Let D(G) be a digraph induced from G by splitting each edge uv E(G) into twooppositearcs uvand vu. Accordingto[15],incidencecoloringofGisequivalenttothe coloring of arcs of D(G), where two distinct arcsuvandxyare adjacent providedoneofthefollowingholds:(1)u = x;(2)v= xory= u.Let A(G) be the set of all arcs of D(G). An incidence coloring of G is a mapping: A(G) C,whereCisacolor-set,suchthatadjacentarcsofD(G)areassigneddistinctcolors. Theincidencechromaticnumber, denotedbyi, istheminimumcardinalityofCforwhich:A(G) Cisanincidencecoloring. AnindependentsetofarcsisasubsetofA(G)whichconsistsofnon-adjacentarcs.2 IncidencechromaticnumberandStararboricityA star forest is a forest whose connected components are stars. The star arboricity ofagraph G (introducedby Akiyama and Kano[1]),denoted by st(G),is thesmallestnumberofstarforestswhoseunioncoversalledgesofG.We now establish a connection among the chromatic index, the star arborcity andtheincidencechromaticnumberofagraph. Thisrelation,togetherwiththeresultsby Hakimi et al. [10], provided a new upper bound of the incidence chromatic numberofplanargraphs,k-degenerategraphsandbipartitegraphs.Theorem2.1LetGbeagraph. Theni(G)

(G) + st(G), where

(G)isthechromaticindexofG.Proof: We color all the arcs goingintothe center of astar bythe same color.Thus,half ofthearcsof astar forest can becolored by one color. Sincest(G) is thesmallestnumberof starforestswhoseunioncoversall edgesof G, half of thearcsofGcanbecoloredbyst(G)colors. TheuncoloredarcsnowformadigraphwhichisanorientationofG. WecolorthesearcsaccordingtotheedgecoloringofGandthisisaproperincidencecoloringbecauseedgecoloringismorerestrictive. Hence

(G) +st(G)colorsaresucienttocoloralltheincidencesofG. INCIDENCECOLORING, ARBORICITYANDDOMINATION 109Remarks: Theorem2.1wasindependentlyobtainedbyYang[23] previously. Forfurtherinformationpleaseconsultthewebpage[18].We now obtain the following new upper boundson the incidence chromatic num-bersof planargraphs,a class of k-degenerate graphsand aclass of bipartitegraphs.Corollary2.2Let Gbe aplanar graph. Theni(G) +5for =6andi(G) 12for = 6.Proof: Theboundis truefor 5, sinceBrualdi andMassey[4] provedthati(G) 2. LetGbeaplanargraphwith 7, wehave

(G)=[14, 20].Also, Hakimi et al. [10] provedthat st(G) 5. Therefore, i(G) +5byTheorem2.1. Whilewereducetheupperboundontheincidencechromaticnumberofplanargraphsfrom+7 [7]to+5,Hosseini Dolamaand Sopena[6] reduced theboundto + 2undertheadditionalassumptionsthat 5andgirthg 6.AgraphGis k-degenerate[12] if everysubgraphof Ghas avertexof degreeatmostk. Coloringproblemsonk-degenerategraphs[5, 7, 11] arewidelystudiedbecause of its relative simple structure. Inparticular, Hosseini Dolamaet al.[7]provedthat i(G) +2k 1, where Gis ak-degenerate graph. Moreover,certain importantclasses of graph such as outerplanargraphs and planar graphsare2-degenerateand5-degenerategraphs, respectively[12]. Arestrictedk-degenerategraph is a k-degenerate graph with the subgraph induced by N(vi){v1, v2, . . . , vi1}isacompletegraphforeveryi. Weloweredtheboundforrestrictedk-degenerategraphasfollow.Corollary2.3LetGbearestrictedk-degenerategraph. Theni(G) +k + 2.Proof: ByVizingstheorem,wehave

(G) + 1. Also,thestararboricityofarestrictedk-degenerategraphGislessthanorequal tok + 1[10]. Hencewehavei(G) +k + 2byTheorem2.1. Corollary2.4Let Bbeabipartitegraphwithat most onecycle. Theni(B) + 2.Proof: WhenBisabipartitegraphwithatmostonecycle,st(B) 2[10]. Also,it is well known that

(B) = . These results together with Theorem 2.1 prove thecorollary. 3 IncidencechromaticnumberanddominationnumberAdominatingset SV (G) of agraphGis aset where everyvertexnot inShasaneighborinS. Thedominationnumber, denotedby(G), istheminimumcardinalityofadominationsetinG.110 PAKKIUSUNANDWAICHEESHIUAmaximal star forest is astar forest withmaximumnumber of edges. LetG=(V, E)beagraph, thenumberofedgesofamaximalstarforestofGisequalto |V | (G) [8]. We now use the domination number to form a lower bound on theincidencechromaticnumberofagraph. Thefollowingpropositionreformulatestheideasin[2]and[13].Proposition3.1LetG = (V, E)beagraph. Theni(G) 2|E||V | (G).Proof: Eachedgeof Gisdividedintotwoarcsinoppositedirections. Thetotalnumberof arcsof D(G)isthereforeequal to2|E|. Accordingtothedenitionoftheadjacencyofarcs, anindependentsetofarcsisastarforest. Thus, amaximalindependentsetof arcsisamaximal starforest. Asaresult, thenumberof colorclassrequiredisatleast2|E||V | (G). Corollary3.2LetG = (V, E)beanr-regulargraph. Theni(G) r1 (G)|V |.Proof: ByHandshakinglemma,wehave2|E| =

vVd(v) = r|V |,theresultfollowsfromProposition3.1. Example3.1Corollary 3.2 provides an alternative method to show that a completebipartitegraphKr,r, wherer 2isnotr + 1-incidencecolorable. AsKr,risar-regulargraphwith(Kr,r) = 2,wehavei(Kr,r) r1 (G)|V |=r1 1r> r + 1.

Example3.1canbeextendedtocompletek-partitegraphwithpartitesetsofsamesize.Example3.2Let G be a complete k-partite graph where every partite set is of sizej. Thus,theorderofGiskj= nandGisa(n nk)-regulargraphwith(G)= 2.AccordingtoCorollary3.2,wehavei(G) n nk1 (G)|V |=n nk1 2n.Simplecalculationshowsthatn nk1 2n> n nk+ 1fork 3. Therefore,Gisnot((G) + 1)-incidencecolorable. INCIDENCECOLORING, ARBORICITYANDDOMINATION 111Corollary3.3LetG = (V, E)beanr-regulargraph. Twonecessaryconditionsfori(G) = r + 1(alsofor(G2) = r + 1[17])are:1. ThenumberofverticesofGisdivisiblebyr + 1.2. Ifrisodd,thenthechromaticindexofGisequaltor.Proof: Weprove 1only,2wasproved in [16]. By Corollary 3.2, if G isan r-regulargraphandi(G) =r+ 1, thenr+ 1=i(G) r|V ||V | (G)|V |r + 1(G).Sincetheglobal lowerboundofdominationnumberis

|V | + 1

, weconcludethatthenumberofverticesofGmustbedivisiblebyr + 1. 4 GraphsConstructedfromSmallerGraphsIn this section, the upper bounds on the incidence chromatic number are determinedfor Cartesian productof graphsand for join and union of graphs,respectively. Also,theseboundscanbeattainedbysomeclasses ofgraphs[19].We start byprovingthe followingtheoremabout unionof graphs, where thegraphsmaynotdisjoint.Theorem4.1LetG1andG2begraphs. Theni(G1 G2) i(G1) +i(G2).Proof: If some edge e E(G1)E(G2), then we delete it from either one of the edgeset. Thisprocesswillnotaect I(G1 G2),hence,weassumeE(G1) E(G2) = .Let be a i(G1)-incidence coloring of G1 and be a i(G2)-incidence coloring of G2usingdierentcolorset. Thenisaproper(i(G1) +i(G2))-incidencecoloringofG1G2 with (uv) = (uv) when uv E(G1) and (uv) = (uv) when uv E(G2).

ThefollowingexamplerevealedthattheupperboundgiveninTheorem4.1issharp.Example4.1Letnbeanevenintegerandnotdivisibleby3. LetG1beagraphwithV (G1)= {u1, u2, . . . , un}andE(G1)= {u2i1u2i |1 i n2}. Moreover, letG2beanothergraphwithV (G2)=V (G1)andE(G2)= {u2iu2i+1 | 1 i n2}.Then, itisobviousthati(G1)=i(G2)=2andG1 G2=Cnwherenisnotdivisibleby3. Therefore,i(Cn) = 4 = i(G1) +i(G2). TheCartesianproductof graphsG1andG2, denotedbyG1G2, isthegraphwithvertexsetV (G1) V (G2)speciedbyputting(u, v)adjacentto(u

, v

)ifandonlyif(1)u = u

andvv

E(G2)or(2)v = v

anduu

E(G1).Theorem4.2LetG1andG2begraphs. Theni(G1G2) i(G1) + i(G2).112 PAKKIUSUNANDWAICHEESHIUProof: Let |V (G1)| =mand |V (G2)| =n. G1G2isagraphwithmnverticesandtwotypesofedges: fromconditions(1)and(2)respectively. Theedgesoftype(1)formagraphconsistingofndisjointcopiesofG1,henceitsincidencechromaticnumber equal to i(G1). Likewise, the edges of type (2) form a graph with incidencechromaticnumberi(G2). Consequently,thegraphG1G2isequaltotheunionofthe graphs from (1) and (2). By Theorem 4.1, we have i(G1G2) i(G1)+i(G2).

Wedemonstrate theupper bound given in Theorem 4.2 is sharp by the followingexample.Example4.2LetG1=G2=C3, itfollowsthatG1G2isa4-regulargraph. IfG1G2is5-incidencecolorable,thenthechromaticnumberofitssquareisequalto5[17]. However,allvertices inG1G2areofdistanceatmost 2. Therefore,G1G2isnot5-incidencecolorableandtheboundderivedinTheorem4.2isattained. Finally,weconsidertheincidencechromaticnumberofthejoinofgraphs.Theorem4.3LetG1andG2begraphswith |V (G1)| = mand |V (G2)| = n,wherem n 2. Theni(G1 G2) min{m +n, max{i(G1), i(G2)} +m + 2}.Proof: Ontheonehand, wehavei(G1 G2) m + n. Ontheotherhand, thedisjoint graphs G1 and G2 can be colored by max{i(G1), i(G2)} colors, and all otherarcs in between can be colored by m+2 new colors. Therefore, max{i(G1), i(G2)}+m + 2isalsoanupperboundfori(G1 G2). WeutilizethefollowingexampletoshowthattheupperboundinTheorem4.3issharp.Example4.3Let G1 = Kmand G2 = Kn. Then the upper bound m+n is attainedsinceG1 G2 =Km+n. Ontheotherhand, letG1bethecomplementof KmandG2be the complement of Knwithmn2. Thenthe other upper boundmax{i(G1), i(G2)} + m + 2isattainedbecauseG1 G2 =Km,nandi(G1) =i(G2) = 0. References[1] Jin Akiyama and Mikio Kano, Path factors of a graph, Graphs and applications:Proc.FirstColoradoSymposiumonGraphTheory(FrankHararyandJohnS.Maybee,eds.),Wiley,1985,pp. 121.[2] I.AlgorandN.Alon,Thestararboricityofgraphs,DiscreteMath.75(1989),1122.[3] J.A. BondyandU.S.R. Murty, Graphtheorywithapplications, 1st ed., NewYork: Macmillan Ltd.Press,1976.INCIDENCECOLORING, ARBORICITYANDDOMINATION 113[4] R.A. Brualdiand J.J.Q. Massey, Incidence and strongedge colorings of graphs,DiscreteMath.122(1993),5158.[5] Leizhen Cai and Xuding Zhu, Game chromatic index of k-degenerate graphs, J.GraphTheory36(2001),144155.[6] M.HosseiniDolamaandE.Sopena, Onthemaximumaveragedegreeandtheincidence chromatic number of a graph, Discrete Math. and Theoret. Comp. Sci.7(2005),203216.[7] M. Hosseini Dolama, E. Sopena and X. Zhu, Incidence coloring of k-degeneratedgraphs,DiscreteMath.283(2004),121128.[8] S. Ferneyhough, R. Haas, D.HansonandG. MacGillivray, Starforests, domi-natingsetsandRamsey-typeproblems,DiscreteMath.245(2002),255262.[9] B.Guiduli,Onincidence coloringandstar arboricityofgraphs,DiscreteMath.163(1997),275278.[10] S.L. Hakimi, J. Mitchem andE. Schmeichel, Stararboricity ofgraphs,DiscreteMath.149(1996),9398.[11] A.V. Kostochka, K. NakprasitandS.V. Pemmaraju, Onequitablecoloringofd-degenerategraphs,SIAMJ.DiscreteMath.19(1)(2005),8395.[12] D.R. LickandA.T. White, k-degenerategraphs, Canad. J. Math. 22(1970),10821096.[13] M. Maydanskiy, The incidence coloring conjecture for graphs of maximum degree3,DiscreteMath.292(2005),131141.[14] D.P. Sandersand Y. Zhao, Planar graphs of maximum degree seven are class 1,J.Combin.TheorySer.B83(2001),201212.[15] W.C. Shiu, P.C.B. LamandD.L. Chen, Noteonincidencecoloringforsomecubicgraphs,DiscreteMath.252(2002),259266.[16] W.C. Shiu, P.C.B. LamandP.K. Sun, Cubicgraphswithdierentincidencechromaticnumbers,Congr.Numer.182(2006),3340.[17] W.C. ShiuandP.K. Sun, Invalidproofsonincidencecoloring, DiscreteMath.308(2008),65756580.[18]EricSopena,TheIncidenceColoringPage,athttp://www.labri.fr/perso/sopena/pmwiki/index.php.[19] Pak Kiu Sun, Incidence coloring: Origins, developments andrelation with othercolorings, Ph.D.thesis,HongKongBaptistUniversity,2007.114 PAKKIUSUNANDWAICHEESHIU[20] V.G. Vizing, Someunsolvedproblems ingraphtheory(inRussian), UspekhiMat.Nauk.23(1968),117134.[21] S.D.Wang,D.L.ChenandS.C.Pang,TheincidencecoloringnumberofHalingraphsandouterplanargraphs,DiscreteMath.256(2002),397405.[22] D.B.West,Introductiontographtheory,2nded.,Prentice-Hall, Inc.,2001.[23] D.Yang,Fractionalincidencecoloringandstararboricityofgraphs,ArsCom-bin.(toappear).(Received26 May 2011; revised25 May 2012, 24 July 2012)