arXiv:1307.1242v1 [math.CO] 4 Jul 2013Staredgecoloringofsomeclassesofgraphsudmila Bezegov, Borut Luar, Martina Mockoviakov,Roman Sotk, Riste krekovskiJuly 22, 2013AbstractAstaredgecoloringof a graph is a proper edge coloring without bichromatic pathsandcyclesoflengthfour. Inthispaperweestablishtightupperboundsfortreesandsubcubic outerplanar graphs, and derive an upper bound for outerplanar graphs.Keywords: Star edge coloring, star chromatic index1 IntroductionAstaredgecoloringof a graphG is a proper edge coloring where at least three distinctcolors are used on the edges of every path and cycle of length four, i.e., there is neitherbichromaticpathnorcycleof lengthfour. Theminimumnumberof colorsforwhichGadmitsastaredgecoloringiscalledthestarchromaticindex anditisdenotedbyst(G).The star edge coloring was initiated in 2008 by Liu and Deng [8], motivated by thevertex version (see [1, 3, 4, 6, 7, 9]). Recently, Dvok, Mohar and mal [5] determinedupper and lower bounds for complete graphs.Theorem1(Dvok,Mohar,mal). ThestarchromaticindexofthecompletegraphKnsatises2n(1 + o(1)) st(Kn) n222(1+o(1))log n(log n1/4).Inparticular, forevery>0thereexistsaconstant csuchthat st(Kn)cn1+foreveryn 1.They asked what is the true order of magnitude of st(Kn), in particular, if st(Kn) =O(n). From Theorem 1, they also derived the following near-linear upper bound in termsof the maximum degree for general graphs.Institute of Mathematics, Faculty of Science, Pavol Jozef afrik University, Koice, Slovakia. Supportedin part by Science and Technology Assistance Agency under the contract No. APVV-0023-10 (R. Sotk), Slo-vak VEGA Grant No. 1/0652/12 (M. Mockoviakova, R. Sotk), and VVGS UPJ No. 59/12-13 (M. Mockovi-akova).All the authors where supported in part by bilateral project SK-SI-0005-10 between Slovakia and Slove-nia. E-Mails: roman.sotak@upjs.sk,{ludmila.bezegova,martina.mockovciakova}@student.upjs.skFaculty of Information Studies, Novo mesto, Slovenia & Institute of Mathematics, Physicsand Mechanics, Ljubljana, Slovenia. Partially supported by ARRS Program P1-0383. E-Mails:{borut.luzar,skrekovski}@gmail.com1Theorem2(Dvok, Mohar, mal). LetGbeanarbitrarygraphofmaximumdegree. Thenst(G) st(K+1) O_log log log _2,andthereforest(G) 2O(1)log .In addition, Dvoak et al. considered cubic graphs and showed that the star chromaticindex lies between4 and7.Theorem3(Dvok, Mohar, mal).(a) IfGisasubcubicgraph,thenst(G) 7.(b) IfGisasimplecubicgraph,thenst(G) 4,andtheequalityholdsifandonlyifGcoversthegraphofthe3-cube.A graph G coversa graph H if there is a locally bijective graph homomorphism fromG toH. While there exist cubic graphswith the star chromaticindex equal to6,e.g.,K3,3or Heawood graph,no example of a subcubic graph that would require7 colors isknown. Thus, Dvoak et al. proposed the following conjecture.Conjecture1(Dvok, Mohar, mal). IfGisasubcubicgraph,thenst(G) 6.Inthis paper wedetermineatight star chromaticindexof subcubicouterplanargraphs and trees, and an upper bound for star chromatic index of outerplanar graphs.Throughoutthe paper,wemostly use the terminologyused in [2]. In a graphwithcolorededges, wesaythat acolor c appears atvertexv, whenever thereisanedgeincident withvand colored byc. In a rooted tree, the root is considered to be atlevel0, its neighbors atlevel 1 (1-level vertices),etc. An edge is oflevel i (i-level edge) if itstwo endvertices are at leveli andi + 1. A path of four edges is referred to as a4-path.2 TreesA tight upper bound for acyclic graphs is presented in this section. For example, a starchromatic index of every tree with a -vertex whose all neighbors are -vertices achievesthe upper bound.Theorem4. LetTbeatreewithmaximumdegree. Thenst(T) _32_.Moreover,theboundistight.Proof. Firstweprovethetightnessofthebound. LetTbearootedtreewitharootvof degree and let each of its neighborsvihave 1 leafs. Thus, Thas( 1)1-level edges. Inanoptimal coloringwith + kcolorssupposethateachedgevviiscoloredbyi. Note thatfor anydistinctneighborsvi, vjof vwehavethatifthecolorjappears atvithe colori does not appear atvj, and vice versa. Thus, colors1, . . . , may appear together at most( 1)/2 times at1-level edges. Moreover, each of thecolors + 1, . . . , + kmayappearateachvi, sotogethertheyappearktimesat1-level edges. Hence, it must hold( 1)/2 + k ( 1) ,2which implies thatk ( 1)/2 and gives the tightness of our bound.Now, wepresentaconstructionofastaredgecoloringof TusingcolorsfromasetC, where|C| =_32_. First, weselectanarbitraryvertexvof TtobetherootofT. Moreover,to every vertex of degree at least2 inTwe add some edgessuch thatits degree increasesto. In particular,everyvertexinThas degree or1 after thismodication. Foreveryvertexuof Twedenoteitsneighborsbyn0(u), . . . , n1(u);here we always choose its eventual predecessor regarding the root v to be n0(u). By t(u),foru=v,we denote an integeri such thatu=ni(n0(u)),and byC(u) we denote thecurrent set of colors of the edges incident tou.Weobtaininthefollowingway. Firstwecolortheedgesincidenttotherootvsuch that(vni(v))= i + 1, fori=0, 1, . . . , 1. Then,we continueby coloring theedges incidentto the remaining vertices,in particular,we rst color the edges incidentto the vertices of level 1, then the edges incident to the vertices of level 2 etc.Let ubeak-level vertexwhoseincident k-level edges arenot coloredyet. LetC(n0(u)) =C\ C(n0(u)) bethesetof colors that arenot usedfortheedges inci-dentton0(u). Since|C(n0(u))| =_2_,forthe edgesuni(u)fori=1, . . . ,_2_weuse distinct colors from the setC(n0(u)).Next, we color the remaining edges incident tou. There are _2_ 1 of such edges.We color the edges one by one fori=1, . . . ,_2_ 1() with colors fromC(n0(u)) asfollows. Letj= (t(u) + i) mod and set(uni(u)) = (n0(u)nj(n0(u))) . (1)Notice that after the above procedure is performed for each vertexu all the edges ofTare colored.It remains to verify that is indeed a star edge coloring ofT. It is easy to see thattheedgesincidenttothesamevertexreceivedistinctcolors, soweonlyneedtoshowthat there is no bichromatic4-path in T. Suppose, for a contradiction, that there is suchapathP=wxyztwith(xw)=(yz)and(xy)=(tz). Assume, withoutlossofgenerality,thatd(v, w)> d(v, z). Note also thaty= n0(x), w= ni(x), for somei > 0,andz= nj(y), for somej 0. Since(xw) = (yz)(xni(x)) = (n0(x)nj(n0(x))) ,by (1) we have thatj= (t(x) + i) mod . Hence,z= nt(x)+i(y) and by() we have1 i _2_ 1. (2)Now, we consider two cases regarding the value ofj. First, supposej= (t(x) + i) mod = 0 (it means thati + t(x) = ). Since(zt) = (yx)(n0(y)n(n0(y))) = (ynt(x)(y)) ,as above, by (1), we have that = (t(y) + t(x)) mod and by()1 t(x) _2_ 1. (3)3By summing (2) and (3) and usingi + t(x) = , we obtain = i + t(x) 2_2_ 2 < ,a contradiction. Second, letj> 0. Then,y= n0(z) and(yx) = (zt)(n0(z)nt(x)(n0(z))) = (zn(z)) .By (1),t(x) = (t(z) + ) mod and1 _2_ 1 . (4)Moreover, sincej= t(x) + i andt(z) = j, we havet(x) = (t(z) + ) mod = (t(x) + i + ) mod .So,i + = and, similarly as in the previous case, by summing (2) and (4) we obtain = i + 2_2_ < ,a contradiction that establishes the theorem.3 OuterplanargraphsIn this section we consider outerplanar graphs. First, we use the bound from Theorem 4to derive an upper bound for outerplanar graphs.Theorem5. LetGbeanouterplanargraphwithmaximumdegree. Then,st(G) _32_+ 12 .Proof. LetG be an outerplanar graph andv a vertex of G. LetTbe a rooted spanningtree ofG with the rootvsuch that for any vertexu V (G) isdG(v, u)= dT(v, u). ByTheorem 4, there exists a star edge coloring ofTwith at most _32_ colors.Now, we consider the edges of G which are not contained in T. Let uw E(G)\E(T)be an edge such that d(v, u) = k, d(v, w) = , and k . By the denition of T, we havethat either k= or k= 1. In the former case, we refer to uw as a horizontal edge(orak-horizontal edge), and in thelatter case,wecallit adiagonal edge(or ak-diagonaledge) (see Fig. 1 for an example).Since outerplanar graphs do not contain K4-minors, we have that the horizontal edgesinduce a linear forest in G, i.e., a graph in which every component is a path. It is easy tosee that3 colors suce for a star coloring of a path. We use three additional colors forallthek-horizontal edgeswhenkisodd,andanotherthreecolorsfor thek-horizontaledges whenk is even, six additional colors altogether. Obviously, no bichromatic4-pathis introduced inG.It remains to color the diagonal edges. LetLkbe the bipartite graph whose verticesare thek-level and(k + 1)-level vertices ofG, and two vertices are adjacent if they are4Figure1: An outerplanargraphwiththeedgesofthespanningtreede-picted solid, the horizontal edges dotted and the diagonal edges dashed.adjacent in T(rededges), or they are connected by a k-diagonal edge (blueedges). SinceG isK2,3-minor free,Lkis acyclic.Everyk-level vertex is incident to at most two blue edges inLk, and every(k + 1)-level vertex is incident to at most one blue edge inLk. If ak-level vertexwis incidentto twoblueedgeswxandwy,noneofitssuccessorsinTisincidentto ablue edgeinLk, and the predecessors ofx andy inTare incident to at most one blue edge inLk.This properties imply that thek-diagonal edges can be colored with two colors suchthat no bichromatic path is introduced inG. For instance, we rst color the edges inci-dent to eventualk-level vertices that are incident to twok-diagonal edges, and then theremaining edges. We use 2 distinct colors for the k-diagonal edges whenk 0(mod3),another two whenk 1(mod3), and two whenk 2(mod3), so6 altogether.Hence, we obtain a star edge coloring ofG with at most32 +12 colors.Inaspecial casewhenamaximumdegreeof anouterplanargraphis3, weprovethat ve colors suce for a star edge coloring. Moreover,the bound is tight. Considera5-cyclev1v2v3v4v5with one pendent edge at every vertex (see Figure 2). In case thatv1v2v3v4v51 213445Figure2: A subcubic outerplanar graph with the star chromatic index5.onlyfour colorssuce,allfourshouldbe usedontheedgesofthecycle andonlyonecouldbeusedtwice, sothecoloringof thecycleisuniqueuptothepermutationofcolors. Consider the coloring of the cycle in Figure 2. Only the color4 may be used forthependentedgeincidenttov4. But, inthiscase, allthecolorsareforbiddenforthependent edge incident tov1, which means that we need at least5 colors for a star edgecoloring of the graph. In particular, we may color all the pendent edges of the graph bycolor5 and obtain a star5-edge coloring.5Beforeweprovethetheorem, wepresentatechnical lemma. Acactus graphisa(multi)graphin which every block is a cycle or an edge. We say that a cactus graph isalightcactus graph if the following properties hold: all the vertices have degree2 or4; only the vertices of degree4 are cutvertices; oneverycyclethereareatmosttwoverticesofdegree2. Moreover, iftherearetwo vertices of degree2 on a cycle, then they are adjacent.Lemma6. Everyouterplanarembeddingofalightcactusgraphadmitsaproper4-edgecoloringsuchthatnobichromatic4-pathexistsontheboundaryoftheouterface.Proof. The proof is by induction on the numbernc of cutvertices of a light cactus graphL. Thebasecasewithnc=0istrivial,sinceLhasatmosttwoedges. Incasewhennc= 1,L is isomorphic to one of the three graphs depicted in Fig. 3, together with thecolorings of the edges.1234(i)12341(ii)112344(iii)Figure3: The three light cactus graphs with one cutvertex.Now, let L containk cutvertices,k 2. Then, there is a blockMof L that containsat least two cutvertices ofL. Letv1, v2, . . . , v,2 k, be the cutvertices ofL inM.Everyviisincidenttotwoedgest1iandt2iof Lsuchthatt1i, t2i/ E(M), wecallsuchedgestentacles. Note that these two edges belong to a same block.Let Li be the component of L\E(M) containing vi (and so t1iand t2i). Let Hi be thegraph obtained from Li by connecting the vertex vi with a new vertex ui with two edgeseiandfi. By induction hypothesis, everyHiadmits a coloring with at most4 colors.Next, letLMbe a subgraph ofL consisting of the cycleMand all its tentacles. Wecolor the edges ofLMas follows. IfMis an even cycle, we color the edges ofMby thecolors1 and2,and the tentaclesbycolors3 and4 in such a waythat thetentaclesofcolor3 are always on the same side, i.e., contraction of the edges ofMresults in a starwith the edges alternately colored by3 and4 (see Fig. 4).In case whenMis an odd cycle, it contains a pair of adjacent cutverticesu andv ofL. Without loss of generality, we choose u to be incident to an eventual vertex of degree2 of M. Wecolor theedgeuvwithcolor3,andthe remainingedgesofMalternatelybythecolors1and2startingattheedgeincidenttou. Next, wecolorthetentaclesby colors3 and4 similarly as in the previous case such that the tentacles of color3 arealways on the same side, in particular the edge of color4 incident tou is consecutive tothe edge of color 1 incident to u on the boundary of the outerface of LM. In the end, werecolor the tentacles incident to the verticesu andv as follows.In case when M is a cycle of length 3 with a vertex of degree 2, we recolor the tentacleofcolor3incidenttoubycolor2,andthetentaclesofcolor4and3incidenttovbycolors1 and4, respectively (see the left graph in Fig. 5). Otherwise, ifMis not a cycleof length3 with a vertex of degree2, we recolor the tentacle of color3 incident tou bycolor2, and the tentacle of color3 incident to v by color1 (see the right graph in Fig. 5for an example). Observe that everyfour consecutive edgesalongthe boundaryof the6121 1223333344444Figure4: An example of an edge coloring of a graphLM, whenMis aneven cycle.211233444432u v3141 23 4214Figure5: Examples of an edge coloring of a graph LM, when Mis an oddcycle.outer face in LMreceive at least three distinct colors. Moreover, every three consecutiveedgesalongtheboundaryoftheouterfaceof LMstartingatatentaclereceivethreedistinct colors.Finally, wepermutethecolorsineachcomponentLitomatchthecoloringof LM(weidentifytheedgeseiandfiwiththetentaclesof Mincidenttovi). Itiseasytosee that we obtain an edge coloring withoutbichromatic4-path along the boundaryofthe outer face of the graphL. Namely,letPbe a4-pathalong the outer face inL. Ifat least threeedgesofPare inLM,thenPis notbichromatic. On theother hand,ifLMcontains at most two edges ofP, then all the edges ofPbelong to some componentLi. Hence, we have shown that every outerplanarembedding of a light cactus graphLadmits a proper 4-edge coloring such that no bichromatic4-path exists on the boundaryof the outer face, and so establish the lemma.Theorem7. LetGbeasubcubicouterplanargraph. Then,st(G) 5 .Proof. LetG be a minimal counterexample to the theorem, i.e., a subcubic outerplanargraph with the minimum number of edges that does not admit a star edge coloring withatmost5colors. WealsoassumethatGisembeddedintheplane. Werstdiscussa structural property ofG and then show that there exists a star5-edge coloring inG,hence obtaining a contradiction.Suppose thatG contains a bridgeuv, whereu andvhave degree at least2 (we calltheedgeuvanontrivial bridge). Let GuandGvbethetwocomponentsof G uv7containing the vertexu andv, respectively. LetHu= G[V (Gu) {v}], i.e., a subgraphof Ginducedbythe verticesinV (Gu) {v},andsimilarly,letHv=G[V (Gv) {u}].By the minimality of G, there exist star 5-edge colorings of Hu and Hv. We may assume(eventually after a permutation of the colors), that uv is given the color 5 in both graphsHuandHvand allthe edgesincidentto the edgeuvare coloredbydistinct colors. Itiseasytoseethatthecoloringsof HuandHvinduceastar 5-edgecoloringof G, acontradiction.Hence, we may assume that G has no nontrivial bridge, so G is comprised of an outercycleC=v1v2 . . . vk, somechordsof Candsomependentedges. Now, weconstructa star5-edgecoloringof G. Wewillshowthatthereexistsastar edgecoloringoftheedges ofCby the four colors in such a way that every chord ofCis incident to the fourdistinctly colored edges. That means that after we color the chordsand pendent edgesby the fth color a star5-edge coloring ofG is obtained.First,weremovethependentedgesandthenmaximizethenumberofchordsof Cbyjoiningpairsofnonadjacent 2-verticesincidenttocommoninnerfaces. Weobtaina2-connected subcubic outerplanar graphGwhich is either isomorphic toC3or everyinner facefof G is incident to at most two2-vertices. Moreover, if fis incident to two2-vertices, they are adjacent.Now, letGdenote the graph obtained fromGby contracting the chords. Observethat G is an outerplanar multigraph with vertices of degree 2 or 4, where the 4-verticesarethecutvertices obtainedbychordcontractions, andthe 2-vertices of Garethe2-vertices ofG. In fact,Gis a light cactus graph (see Fig. 6 for an example).Figure6: An example of the graphsG,GandG.It remainsto nd a star coloringofG. By Lemma6,we can color the edgesofGwithfourcolorssuchthatnobichromatic4-pathexistsontheboundaryoftheouterface. This coloring induces a partial edge coloring ofG, where only the chords ofCandthependentedgesremainnoncolored. Wecolorthembycolor 5. Sinceall theedgesincident to any edge of color 5 receive distinct colors, we obtain a star 5-edge coloring ofG. This completes the proof.In Theorem 5, we proved that _32_+12 colors suce for a star edge coloring of everyouterplanar graph. Let us mention that the constant 12 can be decreased to 9 using moreinvolved analysis. However,we prefer to present a short proof, since the precise boundisprobablyconsiderablylower. Inparticular, webelievethatthefollowingconjectureholds.Conjecture2. LetGbeanouterplanargraphwithmaximumdegree 3. Then,st(G) _32_+ 1 .For graphswithmaximumdegree=2,i.e. for pathsand cycles,thereexist staredgecoloringswithatmost3colorsexceptforC5whichrequires4colors. Incaseofsubcubic outerplanar graphs the conjecture is conrmed by Theorem 7.8References[1] Albertson, M. O., Chappell, G. G., Kiersted, H. A., Knden, A., andRamamurthi,R. Coloring with no2-coloredP4s. Electron. J.Combin.1(2004),#R26.[2] Bondy,J.A.,andMurty,U.S.R. Graphtheory, vol. 244 ofGraduateTextsinMathematics. Springer, New York, 2008.[3] Bu, Y., Cranston, N. W., Montassier, M., Raspaud, A., andWang, W.Star-coloring of sparse graphs. J.GraphTheory62(2009), 201219.[4] Chen, M., Raspaud, A., andWang, W. 6-star-coloringof subcubic graphs. J.GraphTheory72, 2 (2013), 128145.[5] Dvok,Z., Mohar,B., and mal,R.Star chromatic index. J.GraphTheory72(2013), 313326.[6] Grnbaum, B. Acyclic coloring of planar graphs. Israel J.Math.14(1973),390412.[7] Kierstead, H. A., Kndgen, A., andTimmons, C. Star coloringbipartiteplanar graphs. J.GraphTheory60(2009), 110.[8] Liu,X.-S.,andDeng,K. An upper bound on the star chromatic index of graphswith 7. J.LanzhouUniv.(Nat.Sci.)44(2008), 9495.[9] Neetil, J., anddeMendez, P. O. Coloringsandhomomorphismsofminorclosed classes. AlgorithmsCombin.25(2003), 651664.9