Circular edge-colorings of cubic graphs with girth six
aa r X i v : . [ m a t h . C O ] M a y Circular edge-colorings of cubic graphs withgirth six
Daniel Kr´al’ ∗ Edita M´aˇcajov´a † J´an Maz´ak ‡ Jean-S´ebastien Sereni § Abstract
We show that the circular chromatic index of a (sub)cubic graphwith odd-girth at least 7 is at most 7 / A classical theorem of Vizing [14] asserts that the chromatic index of everycubic bridgeless graph, i.e., the smallest number of colors needed to properlyedge-color such a graph, is 3 or 4. Cubic cyclically 4-edge-connected graphs ∗ Institute for Theoretical Computer Science (ITI), Faculty of Mathematics andPhysics, Charles University, Malostransk´e n´amˇest´ı 25, 118 00 Prague 1, Czech Repub-lic. This research was partially supported by the grant GACR 201/09/0197. E-mail: [email protected] . Institute for Theoretical Computer Science is supported asproject 1M0545 by Czech Ministry of Education. † Department of Computer Science, Faculty of Mathematics, Physics and Infor-matics, Comenius University, Mlynsk´a dolina, 842 48 Bratislava, Slovakia. Thisauthor’s work was partially supported by the APVV project 0111-07. E-mail: [email protected] . ‡ Department of Computer Science, Faculty of Mathematics, Physics and Informatics,Comenius University, Mlynsk´a dolina, 842 48 Bratislava, Slovakia. This author’s workwas partially supported by the grant UK/384/2009 and by the APVV project 0111-07.E-mail: [email protected] . § CNRS (LIAFA, Universit´e Denis Diderot), Paris, France, and Department of AppliedMathematics (KAM) - Faculty of Mathematics and Physics, Charles University, Prague,Czech Republic. This author’s work was partially supported by the ´Egide eco-net project16305SB. E-mail: [email protected] . snarks and it is known that smallestcounter-examples, if any, to several deep open conjectures in graph theory,such as the Cycle Double Cover Conjecture, must be snarks.Our research is motivated by edge-colorings of cubic bridgeless graphswith no short cycles. The Girth Conjecture of Jaeger and Swart [5] assertedthat there are no snarks with large girth. This conjecture was refuted by Ko-chol [8] who constructed snarks of arbitrary large girth. Hence, it is naturalto ask whether it can be said that snarks of large girth are close to being3-edge-colorable in some sense.One of the relaxations of ordinary colorings are circular colorings, intro-duced by Vince [13]. A ( p, q ) -coloring of a graph G is a coloring of verticeswith colors from the set { , . . . , p } such that any two adjacent vertices receivecolors a and b with q ≤ | a − b | ≤ p − q . Circular colorings naturally appearin different settings, which is witnessed by several equivalent definitions ofthis notion as exposed in the surveys by Zhu [15, 16].The infimum of the ratios p/q such that G has a ( p, q )-coloring is the circular chromatic number of G . It is known that the infimum is the minimumfor all finite graphs and the ceiling of the circular chromatic number of a graphis equal to its chromatic number. Thus, the circular chromatic number is afractional relaxation of the chromatic number. The circular chromatic index of a graph is the circular chromatic number of its line-graph.Zhu [15] asked whether there exist snarks with circular chromatic indexclose or equal to 4, and as there are snarks with arbitrary large girth, it isalso interesting to know whether there exist such snarks of arbitrary largegirth. Afshani et al. [1] showed that the circular chromatic index of everycubic bridgeless graph is at most 11 / ε >
0, there exists g such that every cubic bridgeless graph with girthat least g has circular chromatic index at most 3 + ε . This latter resultwas generalized to graphs with bounded maximum degree [7]. Moreover, thecircular chromatic indices of several well-known classes of snarks have beendetermined [2, 3, 4, 10].The Petersen graph is the only cubic bridgeless graph that is known tohave the circular chromatic index equal to 11 /
3. In fact, it is the only exampleof a cubic bridgeless graph with circular chromatic index greater than 7 / Conjecture 1.
Every cubic bridgeless graph different from the Petersengraph has circular chromatic index strictly less than / , maybe, at most / . In their paper, Kaiser et al. [6], formulated a problem to determine thesmallest girth g such that every cubic bridgeless graph with girth at least g has circular chromatic index at most 7 /
2, and they showed that g ≤ g ≥ /
2. This impliesthat g = 6, i.e., the circular chromatic index of every cubic bridgeless graphwith girth at least 6 is at most 7 /
2. Our result also applies to subcubicgraphs with odd-girth at least 7.
The core of our argument is formed by decomposing the graph obtained bycontracting a 2-factor of a cubic bridgeless graph into trails. We first intro-duce notation related to such decompositions and then prove their existence. An abstract map is a graph with multiple edges, loops and half-edges allowedwith a fixed cyclic ordering of the ends around each vertex. Formally, anabstract map ( V, E, ϕ ) is comprised of a vertex-set V and an edge-set E .Each edge has two ends: one of them is incident with a vertex, and the othermay, but need not, be incident with a vertex. An edge that has exactly oneend incident with a vertex is a half-edge . An edge with both ends incidentwith the same vertex is a loop . The degree d v of a vertex v ∈ V is the numberof ends incident with v (in particular, loops are counted twice). Moreover,for each vertex, there is a cyclic ordering of the ends incident with it. Theseorderings are represented by a surjective mapping ϕ : V × N → E suchthat ϕ ( v, n ) is an edge incident with v and ϕ ( v, n ) = ϕ ( v, n + d v ) for every( v, n ) ∈ V × N . An edge has two pre-images in { v } × { , . . . , d v } if and only ifit is a loop incident with v . An abstract map naturally yields an embeddingof the corresponding multi-graph on a surface. Deleting a vertex v ∈ V from an abstract map G = ( V, E, ϕ ) yields anabstract map G ′ = ( V ′ , E ′ , ϕ ′ ) with V ′ = V \ { v } , E ′ = ϕ ( V ′ × N ) and ϕ ′ being the restriction of ϕ to V ′ × N . In other words, the vertex v and half-3dges and loops incident with v are removed. The other edges incident with v become half-edges.A trail in an abstract map is a sequence of mutually distinct edges e , e , . . . , e k such that • e i and e i +1 have a common end-vertex v i , for i ∈ { , . . . , k − } ; and • v i − = v i unless e i is a loop, for i ∈ { , . . . , k − } .If e is not a half-edge, then the trail starts at a vertex . Similarly, if e k is nota half-edge, the trail ends at a vertex . Furthermore, a trail that consists ofa single half-edge either starts or ends at a vertex. For a given trail, a linearordering on its edges is naturally defined. Let T and T be two edge-disjointtrails and v a vertex. If T ends at v and T starts at v , then we can link thetwo trails by identifying the end of T with the beginning of T : thereby, weobtain a new trail that first follows the edges of T and then those of T . Wealso define the linking of a trail T that starts and ends at the same vertexwith itself: in this case, we obtain the same trail T except that the orderingof the edges becomes cyclic. A trail obtained in this way is closed . Trailsthat are not closed are open . Note that a trail that starts and ends at thesame vertex can be either open or closed.Trails W , . . . , W K form a compatible decomposition of an abstract map G if all of them are open, every edge is contained in exactly one of the trailsand for every vertex v of odd degree of G , there is an index i v such that thefollowing pairs of edges are consecutive (regardless of their order) in some ofthe trails (and thus are not the same edge): • ϕ ( v, i v ) and ϕ ( v, i v + 5); • ϕ ( v, i v + 1) and ϕ ( v, i v + 3); and • ϕ ( v, i v + 2) and ϕ ( v, i v + 4).Note that if an abstract map G has a compatible decomposition, then it hasno vertices of degree 1 or 5. We now prove that every abstract map with no vertices of degree 1, 3 or 5 hasa compatible decomposition. In the next two lemmas, which form the baseof our inductive argument, abstract maps with a single vertex are analyzed.4 e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e Figure 1: Compatible decompositions in the first (the top two pictures) andthe last four (the remaining pictures) main cases in the proof of Lemma 1.The edges incident with the vertex v are drawn with solid lines and the wayin which they are joined to form the trails of the decomposition is indicatedby dashed lines. Lemma 1.
Every abstract map G that has a single vertex and the degree ofthis vertex is has a compatible decomposition.Proof. Let v be the vertex and let e i be the image of ϕ ( v, i ) for each i ∈{ , , . . . , } . Further, let W i be the set of trails obtained by the followingprocess: initially, each edge of G forms a single trail. We then link the twotrails starting or ending with e i and e i +5 , next we link the two trails starting orending with e i +1 and e i +3 , and last the two trails starting or ending with e i +2 and e i +4 . This operation does not always yield a compatible decomposition,since it may create closed trails. For instance, W i contains a closed trail if e i = e i +5 , or if e i +1 = e i +4 and e i +2 = e i +3 (in this last case, after havinglinked e i +1 with e i +3 , thereby obtaining a trail T , the last linking amountsto linking T with itself). On the other hand, if all the obtained trails areopen, then W i is a compatible decomposition, with i being the index i v ofthe definition.Our goal is to show that at least one of the sets W i is composed only ofopen trails. Since the degree of v is odd, the vertex v is incident with at leastone half-edge. By symmetry, we assume that this half-edge is e .If W contains a closed trail, then one of the following four cases applies5recall that e is a half-edge): • e = e , • e = e , • e = e and e = e , or • e = e and e = e .Similarly, if W contains a closed trail, then one of the following four casesapplies: • e = e , • e = e , • e = e and e = e , or • e = e and e = e .Comparing the two sets of four possible cases, we conclude that if none of W and W is compatible, then at least one of the following six cases applies(these cases are referred to as main cases in Figures 1 and 2): • e = e and e = e , • e = e , • e = e , e = e and e = e , • e = e , e = e and e = e , • e = e , e = e and e = e , or • e = e , e = e and e = e .In the first case, the set W contains only open trails. In the last four cases,the sets W , W , W and W , respectively, are composed of open trails (seeFigure 1 for an illustration).We now focus on the second case. If W contains a closed trail (and e = e ), it holds that e = e . However, in this case, the set W contains noclosed trails (see Figure 2). This finishes the proof of the lemma.6 e e e e e e e e e e e e e Figure 2: Compatible decompositions in the second main case in the proofof Lemma 1. e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e Figure 3: Compatible decompositions in the last four cases in the proof ofLemma 2. Additional loops that can also be present are drawn with dottedlines.
Lemma 2.
Every abstract map G that has a single vertex and the degree ofthis vertex is different from , and has a compatible decomposition.Proof. Let v be the only vertex of the graph G . If the degree of v is even,then there is nothing to prove as every decomposition into open trails iscompatible. If the degree of v is 7, then the statement follows from Lemma 1.Hence, we assume that the degree of v is odd and it is at least 9. Let e i bethe image of ϕ ( v, i ) for i ∈ { , , . . . , } .As the degree of v is odd, v is incident with at least one half-edge. Withoutloss of generality, we can assume that e is the half-edge. Let W i be the setof trails defined as in the proof of Lemma 1. Assume that both the sets W and W contain a closed trail; if any of them were composed of open trailsonly, then it would form a compatible decomposition.As in the proof of Lemma 1, we infer from the facts that both W and W contain a closed trail that one of the following six cases applies (replace e with e and W i with W i +1 for i ∈ { , } in the analysis done in the proof7 e e e e e e e e e e e e e e e e e e Figure 4: Compatible decompositions in the case where e = e and e = e in the proof of Lemma 2. Two cases are distinguished based on whether thedegree of v is equal to 9 or not.of Lemma 1): • e = e and e = e , • e = e , • e = e , e = e and e = e , • e = e , e = e and e = e , • e = e , e = e and e = e , or • e = e , e = e and e = e .In the last four cases, the sets W , W , W and W , respectively, contain noclosed trails (see Figure 3 for an illustration). Let us focus on the first twocases, now.Suppose that e = e and e = e . If W contains a closed trail, then e = e . Further, if W contains a closed trail, then e = e . Thus, theset W contains no closed trails (see Figure 4), and hence is a compatibledecomposition.Assume now that e = e . If W contains a closed trail, then e = e .Further, if W contains a closed trail, then e = e . Again, the set W isthen a compatible decomposition (see Figure 5).To summarize, we have shown that at least one of the sets W i , i ∈{ , , , , } , is composed of open trails only, and thus it forms a compatibledecomposition.Using Lemma 2, we show that every abstract map with no vertices ofdegree 1, 3 or 5 has a compatible decomposition.8 e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e Figure 5: Compatible decompositions in the case where e = e in the proofof Lemma 2. Two cases are distinguished based on whether the degree of v is equal to 9 or not. Additional loops that can also be present are drawnwith dotted lines. Lemma 3.
Every abstract map G without vertices of degree , or has acompatible decomposition.Proof. The proof proceeds by induction on the number of vertices of G . If G has one vertex, then the statement follows from Lemma 2. Otherwise,let v be an arbitrary vertex of G and let G ′ be an abstract map obtainedfrom G by removing v . By the induction hypothesis, G ′ has a compatibledecomposition W ′ into trails.Let W be the set of trails obtained from W ′ by adding the set of loopsand half-edges incident with v in G . Observe that W is a set of trails of G in which there is no trail “traversing” v . If the degree of v is even, the set W is a compatible decomposition of G as there is no restriction on how thetrails pass through the vertex v . Assume that the degree d v of v is odd.We now define an auxiliary abstract map H . The abstract map H con-tains a single vertex w of degree d v , and for ( i, j ) ∈ { , , . . . , d v } , we have ϕ ( w, i ) = ϕ ( w, j ) if and only if W contains a trail starting with the edge ϕ ( v, i ) and ending with ϕ ( v, j ). In other words, trails of W ′ that both startand finish with an edge incident with v correspond in H to loops incidentwith w , the loops incident with v are preserved and the trails starting orfinishing at v (but not both) correspond to half-edges. Trails containing noedge incident with v have no counterparts among the edges of H .By Lemma 2, the abstract map H has a compatible decomposition W H .We can now obtain a compatible decomposition W G of G as follows: all thetrails of W neither starting nor ending at v are added to W G . Each trail W of W H has a corresponding trail in W G that is obtained by replacing every9dge of W with the corresponding trail of W and linking these trails.It is straightforward to verify that W G is a set of open trails of G . Byinduction, the trails pass through vertices of G different from v in the wayrequired by the definition of a compatible decomposition. The trails also passthrough v in the required way because W H is a compatible decomposition of H . Hence, W G is a compatible decomposition of G . We are now ready to prove our main theorem.
Theorem 4.
The circular chromatic index of every cubic graph with a -factor composed of cycles of lengths different from and is at most / .Proof. Let G be a cubic graph and F a 2-factor of G composed of cycles oflengths different from 3 and 5, and let M be the perfect matching comple-mentary to F . The multi-graph obtained by contracting F can be viewedas an abstract map H : the vertices of H correspond to the cycles of the2-factor F , and the order in which the edges of M are incident with cyclesof F naturally defines the function ϕ . Note that H has no half-edges and itsloops correspond to chords of cycles of F .Since no cycle of F has length 3 or 5, no vertex of H has degree 1 , H has a compatible decomposition W .Color the edges of every trail of W with 0 and 1 in an alternating way. Sincethe edges of H correspond one-to-one to the edges of M , we have obtaineda coloring of the edges of M with 0 and 1.We now construct a (7 , G . Let C = v v · · · v ℓ − be a cycle of F and c i the color of the edge of M incident withthe vertex v i , for i ∈ { , , . . . , ℓ − } . If the length ℓ of C is even, we colorthe edges of C with 3 and 5 in an alternating way. Let us consider the casewhere ℓ is odd. Since W is a compatible decomposition, there exists an index k such that c k = c k +5 , c k +1 = c k +3 and c k +2 = c k +4 (indices are taken modulo ℓ ) as the colors of the edges of trails of W alternate.We now show that there exists an index k ′ such that c k ′ = c k ′ +1 = c k ′ +2 = c k ′ +3 . If c k +1 = c k +2 , then set k ′ = k + 1. Otherwise, c k +1 = c k +4 = c k +2 = c k +3 . Since either c k or c k +5 is equal to c k +1 = c k +4 , the index k ′ can be setto k or k + 2. 10 Figure 6: Coloring odd cycles in the proof of Theorem 4.By symmetry, we can assume in the remainder that k ′ = 1, c = c = 0and c = c = 1. Color the edge v v with 2, the edge v v with 4 andthe edge v v with 6. The remaining edges are colored with 3 and 5 in thealternating way (see Figure 6). We have obtained a proper coloring of C . Aswe can extend the coloring of the edges of M to all cycles of F , the resulting(7 , G does notexceed 7 / Corollary 5.
The circular chromatic index of every cubic bridgeless graphwith girth or more is at most / . Finally, we show that the assumption that the given graph is cubic canbe relaxed in Corollary 5.
Corollary 6.
The circular chromatic index of every subcubic graph with odd-girth or more is at most / .Proof. Let G be a subcubic graph with odd-girth 7 or more that has a circularchromatic index greater than 7 / G is at least2. Similarly, G is connected. The graph G is also bridgeless: otherwise, eachof the two graphs obtained from G by splitting along the bridge has a (7 , , G .If G has no vertices of degree 2, then G is a cubic bridgeless graph.Petersen’s theorem [11] ensures that G has a a 2-factor F . By our assumption,no cycle of F has length 3 or 5, and therefore G cannot be a counter-exampleby Theorem 4. So, assume that G has at least one vertex of degree 2. Let H be a 3-edge-connected cubic graph of odd-girth at least 7 from which weremove an edge. We construct the graph G ′ as follows. We take two disjointcopies of G . For each pair ( u, v ) of corresponding vertices of degree 2 (one11n each copy of G ), we add a copy of H , join u to a vertex of degree 2 of H ,and v to the other vertex of degree 2 in H . The resulting graph G ′ is cubicand has odd-girth at least 7. Moreover, since G is bridgeless, G ′ has at mosttwo bridges. More precisely, G ′ has two bridges if and only if G has exactlyone vertex of degree 2, and G ′ is bridgeless otherwise. Therefore, G ′ hasa perfect matching by Tutte’s theorem [9, 12]. Consequently, since G ′ hasodd-girth at least 7, Theorem 4 implies the existence of a (7 , G ′ . This edge-coloring restricted to G yields a (7 , G , acontradiction. Acknowledgment
The first and the last authors would like to thank Mohammad Ghebleh andLuke Postle for discussions and insights on circular edge-colorings of cubicbridgeless graphs.
References [1] P. Afshani, M. Ghandehari, M. Ghandehari, H. Hatami, R. Tusserkani,X. Zhu: Circular chromatic index of graphs of maximum degree 3, J.Graph Theory 49 (2005), 325–335.[2] M. Ghebleh, D. Kr´al’, S. Norine, R. Thomas: The circular chromaticindex of flower snarks, Electron. J. Combin. 13 (2006),
Matching theory , Volume 121 of North-Holland Mathematics Studies. North-Holland Publishing Co., Amster-dam, 1986. Annals of Discrete Mathematics, 29.[10] J. Maz´ak: Circular chromatic index of type 1 Blanuˇsa snarks, J. GraphTheory 59 (2008), 89–96.[11] J. Petersen: Die Theorie der regul¨aren graphs, Acta Math. 15 (1891),193–220.[12] W. T. Tutte: The Factorization of Linear Graphs, J. London Math. Soc.22 (1947), 107–111.[13] A. Vince: Star chromatic number, J. Graph Theory 12 (1988), 551–559.[14] V. G. Vizing: On an estimate of the chromatic class of a pp