SStar chromatic index
Zdenˇek Dvoˇr´ak ∗ KAM & ITICharles UniversityPrague Bojan Mohar †‡ Dept. of MathematicsSimon Fraser UniversityBurnabyRobert ˇS´amal § KAM & ITICharles UniversityPrague
Abstract
The star chromatic index χ (cid:48) s ( G ) of a graph G is the minimum numberof colors needed to properly color the edges of the graph so that no pathor cycle of length four is bi-colored. We obtain a near-linear upper boundin terms of the maximum degree ∆ = ∆( G ). Our best lower bound on χ (cid:48) s in terms of ∆ is 2∆(1 + o (1)) valid for complete graphs. We also considerthe special case of cubic graphs, for which we show that the star chromaticindex lies between 4 and 7 and characterize the graphs attaining the lowerbound. The proofs involve a variety of notions from other branches ofmathematics and may therefore be of certain independent interest. Edge-colorings of graphs have long tradition. Although the chromatic index ofa graph with maximum degree ∆ is either equal to ∆ or ∆ + 1 (Vizing [16]), it ishard to decide when one or the other value occurs. This is a consequence of thefact that distinguishing between graphs whose chromatic index is ∆ or ∆ + 1 is ∗ Email: [email protected] . Institute for Theoretical Computer Science is sup-ported as project 1M0545 by Ministry of Education of the Czech Republic. Partially supportedby a Czech-Slovenian bilateral project MEB 091037 and BI-CZ/10-11-004. † Email: [email protected] . Supported in part by an NSERC Discovery Grant (Canada), by theCanada Research Chair program, and by the Research Grant P1–0297 of ARRS (Slovenia). ‡ On leave from: IMFM & FMF, Department of Mathematics, University of Ljubljana,Ljubljana, Slovenia. § Email: [email protected] . Institute for Theoretical Computer Science is supportedas project 1M0545 by Ministry of Education of the Czech Republic. Partially supported bygrant GA ˇCR P201/10/P337. Partially supported by a Czech-Slovenian bilateral project MEB091037 and BI-CZ/10-11-004. a r X i v : . [ m a t h . C O ] D ec P-hard (Holyer [10]). This is true even for the special case when ∆ = 3 (cubicand subcubic graphs).Two special parameters concerning vertex colorings of graphs under someadditional constraints have received lots of attention. The first kind is that ofan acyclic coloring (see [8, 2]), where we ask not only that every color class isan independent vertex set but also that any two color classes induce an acyclicsubgraph. The second kind is obtained when we request that any two colorclasses induce a star forest — this variant is called star coloring (see [1, 15] formore details). These types of colorings give rise to the notions of the acyclicchromatic number and the star chromatic number of a graph, respectively.A proper k -edge-coloring of a graph G is a mapping ϕ : E ( G ) → C , where C is a set (of colors ) of cardinality k , and for any two adjacent edges e, f of G ,we have ϕ ( e ) (cid:54) = ϕ ( f ). A subgraph F of G is said to be bi-colored (under theedge-coloring ϕ ) if | ϕ ( E ( F )) | ≤
2. A proper k -edge-coloring ϕ is an acyclic k -edge-coloring if there are no bi-colored cycles in G , and is a star k -edge-coloring if there are neither bi-colored 4-cycles nor bi-colored paths of length 4 in G (bylength of a path we mean its number of edges). The star chromatic index of G ,denoted by χ (cid:48) s ( G ), is the smallest integer k such that G admits a star k -edge-coloring. Note that the above definition of acyclic/star edge-coloring of a graph G is equivalent with acyclic/star vertex coloring of the line-graph L ( G ).If one considers the class of graphs G ∆ of maximum degree at most ∆,Brooks’ Theorem shows that the usual chromatic number is O (∆). The maxi-mum acyclic chromatic number on G ∆ is Ω(∆ / / log / ∆) and O (∆ / ) (Alon,McDiarmid, and Reed [2]). The maximum star chromatic number on G ∆ isΩ(∆ / / log / ∆) and O (∆ / ) (Fertin, Raspaud, and Reed [6]).In contrast with the aforementioned ∆ / behaviour in the class of all graphsof maximum degree ∆, the acyclic chromatic index is linear in terms of themaximum degree. Alon et al. [2] proved that it is at most 64∆, and Molloyand Reed [13] improved the upper bound to 16∆. One would expect a similarphenomenon to hold for star edge-colorings. However, the only previous work[12] just improves the constant in the bound O (∆ / ) from vertex coloring.In this paper we show a near-linear upper bound for the star chromaticindex in terms of the maximum degree (Theorem 3.1). Additionally, we providesome lower bounds (Theorem 4.1) and consider the special case of cubic graphs(Theorem 5.1). The proofs involve a variety of notions from other branches ofmathematics and are therefore of certain independent interest. χ (cid:48) s ( K n ) We shall first treat the special case of complete graphs. The study of their starchromatic index is motivated by the results presented in Section 3 since theygive rise to general upper bounds on the star chromatic index.2 heorem 2.1
The star chromatic index of the complete graph K n satisfies χ (cid:48) s ( K n ) ≤ n · √ o (1)) √ log n (log n ) / . In particular, for every ε > there exists a constant c such that χ (cid:48) s ( K n ) ≤ cn ε for every n ≥ . Proof.
Let A be an n -element set of integers, to be chosen later. We willassume that the vertices of K n are exactly the elements of A , V ( K n ) = A , andcolor the edge ij by color i + j .Obviously, this defines a proper edge-coloring. Suppose that ijklm is abi-colored path (or bi-colored 4-cycle). By definition of the coloring we have i + j = k + l and j + k = l + m , implying i + m = 2 k . Thus, if we ensuredthat the set A does not contain any solution to i + m = 2 k with i, m (cid:54) = k wewould have found a star edge-coloring of K n . It is easy to see, that such triple( i, k, m ) forms a 3-term arithmetic progression; luckily, a lot is known aboutsets without these progressions.We will use a construction due to Elkin [5] (see also [7] for a shorter exposi-tion) who has improved an earlier result of Behrend [4]. As shown by Elkin [5],there is a set A ⊂ { , , . . . , N } of cardinality at least c N (log N ) / / √ √ log N such that A contains no 3-term arithmetic progression.The defined coloring uses only colors 1 , , . . . , N (possibly not all of them),thus we have shown that χ (cid:48) s ( K n ) ≤ N . We still need to get a bound on N interms of n . In the following, c , c , . . . are absolute constants.For every ε > n = | A | ≥ c N (log N ) / √ √ log N ≥ c N − ε (1)Since we also have N ≤ c n ε for every ε >
0, we may plug this in (1) and usethe fact that (log N ) / − √ √ log N is a decreasing function of N for large N to conclude that n ≥ c N ((1 + ε ) log n ) / − √ √ (1+ ε ) log n . Thus we get N ≤ c n √ √ (1+ ε ) log n (log n ) − / . One more round of this ‘boot-strapping’ yields the desired inequality N ≤ n √ o (1)) √ log n (log n ) / . emark. A tempting possibility for modification is to use a set A (in anarbitrary group) that contains no 3-term arithmetic progression and | A + A | is small. Any such set could serve for our construction, with the same proof.Even more generally, we only need a symmetric function p : A × A → N , where A = { , , . . . , n } , such that p ( a, · ) : A → N is a 1-1 function for each fixed a ∈ A , N is small, and p does not yield bi-colored paths (for all i, j, k, l, m weeither have p ( i, j ) (cid:54) = p ( k, l ) or p ( j, k ) (cid:54) = p ( l, m )). We have been unable, however,to find a set that would yield a better bound than that of Theorem 2.1. The purpose of this section is to present a way to find star edge-coloring of anarbitrary graph G , using a star edge-coloring of the complete graph K n with n = ∆( G ) + 1.We will use the concept of frugal colorings as defined by Hind, Molloy andReed [9]. A proper vertex coloring of a graph is called β -frugal if no more than β vertices of the same color appear in the neighbourhood of a vertex. Molloy andReed [13, 14] proved that every graph has an O (log ∆ / log log ∆)-frugal coloringusing ∆+1 colors. If ∆ is large enough, one may use 50 for the implicit constantin the O (log ∆ / log log ∆) asymptotics. Theorem 3.1
For every graph G of maximum degree ∆ we have χ (cid:48) s ( G ) ≤ χ (cid:48) s ( K ∆+1 ) · O (cid:16) log ∆log log ∆ (cid:17) (2) and therefore χ (cid:48) s ( G ) ≤ ∆ · O (1) √ log ∆ . Proof.
Using the above-mentioned result of Molloy and Reed [14], we find a β -frugal (∆ + 1)-coloring f with β = O (log ∆ / log log ∆). We assume the colorsused by f are the vertices of K ∆+1 , so that the frugal coloring is f : V ( G ) → V ( K ∆+1 ). Let c be a star edge-coloring of K ∆+1 . A natural attempt is tocolor the edge uv of G by c ( f ( u ) f ( v )). This coloring, however, may not even beproper: if a vertex v has neighbours u and w of the same color, then the edges vu and vw will be of the same color. To resolve this, we shall produce anotheredge-coloring, with the aim to distinguish these edges; then we will combine thetwo colorings.We define an auxiliary coloring g of E ( G ) using 2 β colors. Let us first set V i = { v ∈ V ( G ) : f ( v ) = i } , i ∈ V ( K ∆+1 )and define the induced subgraphs G ij = G [ V i ∪ V j ]. For each pair { i, j } weshall define the coloring g on the edges of G ij ; in the end this will define g ( e )for every edge e of G . Recall that the frugality of f implies that the maximumdegree in G ij is at most β . Consequently, the maximum degree in the (distance)square of L ( G ij ) is at most 2 β ( β − < β . Therefore, we can find a coloring4f E ( G ij ) using 2 β colors so that no two edges of this graph have the samecolor, if their distance in the line graph is 1 or 2.Now we can define the desired star edge-coloring of G : we color an edge uv by the pair h ( uv ) = ( c ( f ( u ) f ( v )) , g ( uv )) . First, we show this coloring is proper. Consider adjacent (distinct) edges vu and vw . If f ( u ) (cid:54) = f ( w ), then f ( u ) f ( v ) and f ( v ) f ( w ) are two distinct adjacentedges of K ∆+1 , hence c assigns them distinct colors. On the other hand, if f ( u ) = f ( w ) = i (say), we put j = f ( v ) and notice that uv and vw are twoadjacent edges of G ij , hence the coloring g distinguishes them.It remains to show that G has no 4-path or cycle colored with two alternatingcolors. Let us call such object simply a bad path (considering C as a closedpath). Suppose for a contradiction that the path uvwxy is bad. By lookingat the first coordinate of h we observe that the c -color of the edges of the trail f ( u ) f ( v ) f ( w ) f ( x ) f ( y ) assumes either just one value or two alternating ones.As c is a star edge-coloring of K ∆+1 , this trail cannot be a path (nor a 4-cycle).A simple case analysis shows that in fact f ( u ) = f ( w ) = f ( y ) and f ( v ) = f ( x ).Put i = f ( u ), j = f ( v ) and consider again the g coloring of G ij . By construction, g ( uv ) (cid:54) = g ( wx ), showing that uvwxy is not a bad path, a contradiction.As we saw in this section, an upper bound on the star chromatic index of K n yields a slightly weaker result for general bounded degree graphs. We wishto note that, if convenient, one may start with other special graphs in place of K n , in particular with K n,n . It is easy to see that χ (cid:48) s ( K n,n ) ≤ χ (cid:48) s ( K n ) + n (if the vertices of K n,n are a i , b i ( i = 1 , . . . , n ) then we color edges a i b j and a j b i using the color of the edge ij in K n , while each edge a i b i gets a unique color).On the other hand, a simple recursion yields an estimate χ (cid:48) s ( K n ) ≤ (cid:100) log n (cid:101) (cid:88) i =1 i − χ (cid:48) s ( K (cid:100) n/ i (cid:101) , (cid:100) n/ i (cid:101) ) . From this it follows that if χ (cid:48) s ( K n,n ) is O ( n ) (or n (log n ) O (1) , n o (1) , respec-tively) then χ (cid:48) s ( K n ) is O ( n log n ) (or n (log n ) O (1) , n o (1) , respectively). χ (cid:48) s ( K n ) Our best lower bound on χ (cid:48) s ( K n ) is provided below and is linear in terms of n .The upper bound from Theorem 2.1 is more than a polylogarithmic factor awayfrom this. So, even the asymptotic behaviour of χ (cid:48) s ( K n ) remains a mystery. Theorem 4.1
The star chromatic index of the complete graph K n satisfies χ (cid:48) s ( K n ) ≥ n (1 + o (1)) . roof. Assume there is a star edge-coloring of K n using b colors. Let a i bethe number of edges of color i , let b i,j be the number of 3-edge paths colored i, j, i . We set up a double-counting argument. Note that all sums over i , j areassumed to be over all available colors (that is, from 1 to b ). As every edge getsone color, we have (cid:88) i a i = (cid:18) n (cid:19) . (3)Fixing i , we have a matching M i with a i edges and each edge sharing both endswith an edge from M i contributes to some b i,j . Consequently, (cid:88) j b i,j = 4 (cid:18) a i (cid:19) . (4)Finally, we fix color j and observe that each 3-edge path colored i, j, i (forsome i ) uses two edges among the 2 a j · ( n − a j ) edges connecting a vertex of M j to a vertex outside of M j . This leads to (cid:88) i b i,j ≤ a j ( n − a j ) . (5)Now we use (4) and (5) to evaluate the double sum (cid:80) i,j b i,j in two ways, getting4 (cid:88) i (cid:18) a i (cid:19) ≤ (cid:88) j a j ( n − a j ) . This inequality reduces to 4 (cid:88) i a i ≤ ( n + 2) (cid:88) i a i . By the Cauchy-Schwartz inequality, ( (cid:80) a i ) ≤ b · (cid:80) a i , and then using (3), weobtain 4 (cid:18) n (cid:19) ≤ b ( n + 2) . Therefore, b ≥ n ( n − / ( n + 2) = (2 + o (1)) n . A regular graph of degree three is said to be cubic . A graph of maximum degreeat most three is subcubic . A graph G is said to cover a graph H if there is agraph homomorphism from G to H that is locally bijective. Explicitly, thereis a mapping f : V ( G ) → V ( H ) such that whenever uv is an edge of G , theimage f ( u ) f ( v ) is an edge of H , and for each vertex v ∈ V ( G ), f is a bijectionbetween the neighbours of v and the neighbours of f ( v ).6 heorem 5.1 (a) If G is a subcubic graph, then χ (cid:48) s ( G ) ≤ . (b) If G is a simple cubic graph, then χ (cid:48) s ( G ) ≥ , and the equality holds ifand only if G covers the graph of the -cube. For the part (a) of this theorem we will need the following lemma. It seemsto be possible to use this lemma for other classes of graphs, therefore it mightbe of certain independent interest.
Lemma 5.2
Let f : E ( G ) → { , . . . , k } be a k -edge-coloring. (a) Let e be an edge of G . Suppose that the restriction of f to E ( G ) \ { e } is a star edge-coloring of G − e and that f ( e ) is distinct from f ( e (cid:48) ) whenever d ( e, e (cid:48) ) ≤ (that is, either e, e (cid:48) share a vertex, or a common adjacent edge).Then f is a star edge-coloring of G . (b) Let A be a set of vertices of G , let B = V ( G ) \ A , and let X be the setof edges with one end in A and the other in B . Suppose that1. (a restriction of ) f is a star edge-coloring of G [ A ] ;2. (a restriction of ) f is a star edge-coloring of G [ B ] ;3. no edges e , e in X share a common vertex in A or a common adjacentedge in G [ A ] ;4. for every edge e ∈ X and every edge e (cid:48) in G [ B ] ∪ X such that d ( e, e (cid:48) ) ≤ we have f ( e ) (cid:54) = f ( e (cid:48) ) (distance is measured in G [ B ] ∪ X , not in G );5. for every edge e ∈ X and every edge e (cid:48) in G [ A ] we have f ( e ) (cid:54) = f ( e (cid:48) ) .Then f is a star edge-coloring of G . Proof (of the lemma). (a) Since f is a star edge-coloring of G − e , no 4-path (or 4-cycle) in G − e is bi-colored. If P is a bi-colored 4-path (4-cycle)containing e , then P contains an edge of the same color as e at distance ≤ e , a contradiction.(b) Conditions (3), (4), (5) imply that for every edge e ∈ X and every edge e (cid:48) ∈ E ( G ), if d ( e, e (cid:48) ) ≤
2, then f ( e ) (cid:54) = f ( e (cid:48) ). Therefore, we can repeatedly applypart (a), starting with the graph G [ A ] ∪ G [ B ] and adding one edge of X at atime.To explain a bit the conditions of part (b) in the above lemma: the pointhere is that in the condition 5, we do not check what is the distance of e and e (cid:48) .In our applications, A will be a particular small subgraph of G (such as those inFigure 1) and B the ‘unknown’ rest of the graph. We do not want to distinguishwhether some edges in X share a vertex in B . This, however, may create new4-paths, henceforth the particular formulation of this lemma. Proof (of the theorem). (a) Trying to get a contradiction, let us assume that G is a subcubic graph with the minimum number of edges for which χ (cid:48) s ( G ) > G (connectivity, absence of various small sub-graphs). This will eventually allow us to construct the desired 7-edge-coloring7y decomposing G into a collection of cycles connected by paths of length 1or 2.Clearly, G G is connected. Suppose that G contains a cut-edge xy . Let G x and G y be the components of G − xy which contain the vertex x and y , respectively. By the minimality of G , each of G x and G y admits a star 7-edge-coloring. In G x there are at most 6 edges that are incident to a neighborof x . By permuting the colors, we may assume that color 7 is not used on theseedges. Similarly, we may assume that color 7 is not used on the edges in G y that are incident with neighbors of y . Then we can color the edge xy by usingcolor 7 and obtain a star 7-edge-coloring of G (we use Lemma 5.2(a)). Thiscontradiction shows that G is 2-connected. If G contained a path wxyz , where x and y are degree 2 vertices, thenwe could color G − xy by induction, and extend the coloring to a star 7-edge-coloring of G by using Lemma 5.2(a). (For the edge e = xy we use a color thatdoes not appear on the at most six edges incident to w or z .) Thus, such path wxyz does not exist. In particular, G is not a cycle.Suppose next that G contains a degree 2 vertex z whose neighbors x and y are adjacent. We will use Lemma 5.2(a) for e = xz . By inductionwe may find a star edge-coloring of G − e , and as there are at most six edgesin G at distance ≤ e , we can extend the coloring to e to satisfy thecondition of the lemma. So the graph G can be star edge-colored using 7 colors,a contradiction. This shows that the neighbors of a degree 2 vertex cannotbe adjacent in G . Further suppose that G contains parallel edges. Three parallel edges wouldconstitute the whole (easy to color) graph, so suppose there are two paralleledges between vertices u and v . Unless G contains a bridge, or G has at mostthree vertices (and is easy to color), there are neighbors u (cid:48) of u , v (cid:48) of v and u (cid:48) (cid:54) = v (cid:48) . By induction we can color G \ { u, v } . Next, we extend this coloringto the edges uu (cid:48) , vv (cid:48) , so that each of them has different color than the ≤ ≤ uu (cid:48) and vv (cid:48) have different colors, say a and b , then it is enough to use on the two paralleledges any two distinct colors that are different from a and b . If uu (cid:48) and vv (cid:48) havethe same color, then there are at most 5 colors of edges at distance ≤ G does notcontain parallel edges. Next we suppose that G contains one of the first three graphs in Fig-ure 1 as a subgraph, where other edges of G attach only at the vertices de-noted by the empty circles, and some of these vertices may be identified. Weuse Lemma 5.2, part (b). We let A be the set of vertices of the subgraph in thefigure that are denoted by full circles, so X is the set of the three thick edges.By induction, G [ B ] is star 7-edge-colorable. This coloring can be extended to G [ B ] ∪ X so that color of each edge e in X differs from the color of all edgesat distance ≤ e (there are at most 6 such edges). We assume that thecolors used on X are in { , , } . For G [ A ] we use the coloring as shown in thefigure. This satisfies conditions of Lemma 5.2, part (b), and therefore G can bestar 7-edge-colored. 8ext suppose that G contains the fourth graph in Figure 1 as a subgraph(again, other edges can only attach at the ‘empty’ vertices, some of which maybe identified). We use Lemma 5.2, part (b) to show that G − e is 7-edge-coloredin a particular way that allows us to use Lemma 5.2, part (a) to extend thecoloring on e . We let A be the vertices of the pentagon, so that X is the set ofthe three thick edges. Note that the conditions of the part (b) are satisfied for G − e , but not for G itself. By induction there is a star 7-edge-coloring of G [ B ],and we again extend it to X so that the edges in X have distinct color fromedges at distance ≤
2. Observe that there are at most six edges in G [ B ] ∪ X that are at distance ≤ e , so there is a color, say C , not used on any ofthose. We shall reserve C to be used at e . First, however, we apply part (b) tocolor the graph G − e . We use the coloring of G [ A ] shown in the figure, assumingthat C (cid:54)∈ { , , } and that none of the colors 1 , , X . Finally, weuse part (a) to extend the coloring on G , letting the color of e be C .As the last reduction, we show that G does not contain a path wxyz ,where w and z are degree 2 vertices. If G did contain such a path, wecould color H = G − { w, x, y, z } . Next, we describe how to extend this coloringto a star 7-edge-coloring of G . We will denote the edges as in Figure 2; to easethe notation we will use a to denote both the edge and its color.We may assume that all vertices among w, x, y, z , and their neighbours aredistinct (*), as otherwise G contains one of the previously handled subgraphs —those in Figure 1, triangle with a degree 2 vertex, parallel edges or two adjacentdegree 2 vertices (the straightforward checking is left to the reader). It may,however, happen that, e.g., edges s and u have an edge adjacent to both ofthem. This has no effect on the proof, we only will have, say, b = e .The edges a , . . . , h are part of H , so they are colored already. Similarly asin the previous cases, we choose a color for s , t , u , v so that none of these edgesshares a color with an edge of G − { p, q, r } at distance at most 2; using 7 colors,this is easy to achieve. Condition (*) in the previous paragraph implies, thatevery 3-edge path starting at w , x , y or z by an edge s , t , u or v avoids edges p , q , r — consequently, no such path has first and last edge of the same color, andno such path can be part of a bi-colored 4-path or 4-cycle. This greatly reducesthe number of path and cycles we need to take care of.Next, we pick a color for q that differs from c , d , e , f , t , and u .1 234 1 1 2 34 1 4 231 e b c d e f ghs t u vp q rw x y z Figure 2: Illustration of the proof that minimal counterexample to Theorem 5.1does not contain path wxyz as depicted in the figure.Now, we distinguish several cases based on colors of s , q , and v . We againassume the colors are 1, . . . , 7; up to symmetry we have only the following cases. Case 1. s = 1 , q = 2 , v = 3We only need to avoid bi-colored paths aspt , bspt , sptc , sptd , and the foursymmetrical paths in the right part of the figure. If t = 1, we choose p to bedifferent from 1 , , a, b, c, d . If t (cid:54) = 1, it suffices to make p different from s, q, t .The procedure for r is analogous. Case 2. s = 1 , q = 1 , v = 2In this case t (cid:54) = 1, so we only need to avoid bi-colored paths aspq, bspq, spqr, spqu and urvh, urvg , eurv, f urv . If u = 2, we make sure that r differs from 1 , , e, f, g, h .Otherwise, it suffices to make r different from q, u, v . Then we choose p to differfrom a, b, , t, u, r . Case 3. s = 1 , q = 2 , v = 1This is handled in exactly the same way as Case 1. Case 4. s = 1 , q = 1 , v = 1Now t, u (cid:54) = 1, so we only need to avoid the paths aspq, bspq, spqr, pqrv, qrvg, qrvh ,and spqu, tqrv . To do this, we only need to ensure, that p (cid:54) = 1 , a, b, t, u, r and r (cid:54) = 1 , h, g, u, t, p , which is easily possible. This finishes the proof of the claimthat minimal counterexample G does not contain a path wxyz , where w and z are degree 2-vertices.This finished the first part of the proof. Next we will use the above-derivedproperties of the supposed minimal counterexample G to find its star 7-edge-coloring and thus reach a contradiction. We will use only the boldface claimsfrom the above part of the proof.Let G (cid:48) be the graph obtained from G by suppressing all degree 2 vertices,i.e., replacing each path xzy , where z is a degree 2 vertex, by a single edge xy .Clearly, G (cid:48) is a cubic graph. It is bridgeless (as G is bridgeless) and containsno parallel edges – as G contains no parallel edges, no triangle with a degree 2vertex and no 4-cycle with two opposite degree 2 vertices.By a result of Kaiser and ˇSkrekovski [11], G (cid:48) contains a perfect matching M (cid:48) such that M (cid:48) does not contain all edges of any minimal 3-cut or 4-cut. Notethat each edge in G (cid:48) corresponds either to a single edge in G or to a path oflength two. Let M denote the set of edges of G corresponding to an edge of M (cid:48) .10ur goal is to use four colors (say 4, 5, 6, 7) on M , and three colors (say 1, 2, 3)on the other edges that form a disjoint union of circuits. We form an auxiliarygraph K , whose vertices are the edges in M . We make two of these edges e , f adjacent in K if either they form a 2-edge path corresponding to an edge in M (cid:48) or there is an edge in G joining an end of e with an end of f . Observe that K isa graph of maximum degree at most four. Also note that if K is disconnected,then each component contains a vertex of degree at most three. By the BrooksTheorem, K is 4-colorable unless it contains a connected component isomorphicto K . It is easy to see that the latter case occurs if and only if K = K and G = G (cid:48) .Let us first consider the case when K is 4-colorable. In this case we will notneed the fact that M (cid:48) does not contain minimal 3-cuts or 4-cuts. The 4-coloringof the vertices of K determines a 4-coloring of the edges in M with the propertythat every color class is an induced matching in G . We shall show that we canstar 3-color the edges in G − M unless G − M contains a 5-cycle; this case willbe treated separately. By extending that 3-edge coloring to a 7-edge-coloringof G (by using the 4-coloring of edges in M ) we obtain a star 7-edge-coloringsince none of the four colors used on the edges in M can give rise to a bi-colored4-path or a cycle (Lemma 5.2(a)). Thus it suffices to find a star 3-edge-coloringof G − M . This is not hard unless G − M contains a 5-cycle. Recall that G − M is the union of disjoint cycles and every k -cycle, where k (cid:54) = 5, admits a star3-edge-coloring: This is easy if k ∈ { , } or if k is divisible by three. If k ≡ k >
5, we can use the colors in the following order 1232123 · · · k ≡ k >
5, we can use the colors 12132123 · · · G − M . To color them, we shallchoose an edge e = e C in each 5-cycle C and a color c = c C , that is otherwiseused as a color for M . Then we color e with color c and color the 4-path C − e as 1 , , ,
1. We pick c and e in such a way that no edge of M at distance atmost 2 from e has color c (we will show below that this is possible). It is easyto check that this, together with the fact that K is properly colored, preventsall 4-paths and 4-cycles from being bi-colored (Lemma 5.2(a) again). So, thisfinishes the proof of the case when K is 4-colorable—provided we show how topick e and c for each 5-cycle C . To do this, we let F be the set of edges of M that are incident with a vertex of C but not part of C . Further, we let X be the(possibly empty) set of edges of M adjacent with some edge of F . Easily, | X | isthe number of 2-edge paths in M that are adjacent to C . (A 2-edge path withboth ends at C counts twice; in this case X and F intersect.) As G contains no3-edge path with both ends of degree 2, we have | X | ≤
2. We distinguish twocases based on the color pattern on edges of F . These cases cover all possibilitiesup to renaming the colors. Case 1.
Edges of F use in some order colors 4, 4, 5, 6, and 7 (that is, onecolor appears twice, the other colors once). If X = ∅ , there are three possiblechoices for edge e : for each color c among 5, 6, and 7 we may choose the edgeof C opposite to the edge of F colored c . Edges of X may be at distance 2 tosome of these edges of C . However, there are at most two such edges, hence atmost two colors are affected. So, one of colors 5, 6, and 7 is still valid.11 ase 2. Edges of F use in some order colors 4, 4, 5, 5, and 6 (that is, twocolors twice, one once, one not at all). In this case, if X = ∅ , all five edges of C can be colored 7. Each edge of X (if such an edge exists) is at distance 2 fromtwo edges of C , so one edge of C is far from edges of X and we can let this edgebe e and c be 7.Finally, let us consider the case when K does not admit a 4-coloring, i.e., K = K . As argued before, this implies that G = G (cid:48) is a cubic graph containingprecisely 10 vertices. Note that G − M is a 2-regular graph with no 3-cyclesor 4-cycles (due to the choice of M (cid:48) ). Thus G − M is isomorphic either to a10-cycle, or to the union of two 5-cycles.If G − M is the union of two 5-cycles, then it is easy to check that G is thePetersen graph, and hence χ (cid:48) s ( G ) = 5. (A star 5-edge-coloring is easy to find,and the star 4-edge-coloring does not exist as shown in part (b) below.)The final case is that G − M is a 10-cycle. Color its chords with colors 1,2, 3, 4, 5. Then color the first, fourth, and seventh edges by colors from 1, 2,3, 4, 5 so that no two edges at distance two share a color. Finally, color theremaining edges with 6 and 7. This completes the proof.(b) Every 3-edge-coloring of a cubic graph has bi-colored cycles, thus χ (cid:48) s ( G ) ≥
4. In Figure 3 there is a 4-edge-coloring of the cube Q . It is easy to verifythat this is indeed a star edge-coloring. Perhaps the fastest way to see this isto observe that for each i (cid:54) = j , there is (a unique) 3-edge path colored i, j, i between the two vertices colored j . Consider now a graph G that covers Q anduse the covering map to lift the edge-coloring of Q to an edge-coloring of G .From the definition of covering projections we see that a path of length 2 in G is mapped to a path of length 2 in Q . It follows that the defined edge-coloringis proper. It also follows that a path of length 4 in G is mapped to a path oflength 4 in Q or to a 4-cycle in Q , and a 4-cycle in G is mapped to a 4-cyclein Q . It follows that we have a star edge-coloring of G .For the reverse implication, suppose that G has a star 4-edge-coloring c . Letus first define a (vertex) 4-coloring f by letting f ( v ) be the (unique) color thatis missing on edges incident with v . f is a proper coloring. For a contradiction, suppose that f ( u ) = f ( v )for an edge uv of G . Let u , u be the other neighbors of u , and v , v bethe other neighbors of v . By symmetry we may assume that f ( u ) = f ( v ) = 4, c ( uv ) = 3, f ( uu i ) = f ( vv i ) = i (for i = 1 , u i uvv i imply that 3 is neither used on edges incident with v nor on those incidentwith v . This, however, implies that there is an edge-colored 2 incident with v and an edge-colored 1 incident with v , which create a bi-chromatic 4-edgepath (or 4-cycle), a contradiction. Note that the cases where uv is contained ina triangle ( u = v , u = v or both) are also covered by the above. f is a covering map G → K . Suppose for a contradiction that thereis a vertex v with neighbors v , v such that f ( v ) = f ( v ). By symmetry wemay assume that f ( v ) = 4, f ( v ) = f ( v ) = 3, c ( vv i ) = i . Now v must be12
234 2 143 1234 123 412 34
Figure 3: Cube Q with star edge-coloring by four colors. The vertex labels areused in the proof of Theorem 5.1.incident with an edge of color 2 and v must be incident with an edge of color 1,producing again a bi-chromatic 4-edge path (or cycle). f together with c define a covering G → Q . Let i, j, k, l denote 1 , , , v of G has f ( v ) = i then the c -colors of its incidentedges are j , k , l and the same holds for the f -colors of its adjacent vertices.There are exactly two possibilities: either the edges incident with v colored j, k, l lead to vertices colored k, l, j (respectively), or to vertices colored l, j, k .We refer to these two possibilities as the local color pattern at v .Observe that in Q as depicted in Figure 3, there are for each i two verticescolored i and they use different local color patterns. This implies there is aunique vertex mapping F : V ( G ) → V ( Q ) such that for each v ∈ V ( G ) thefollowing conditions hold:1. we have f ( v ) = f ( F ( v )) (we use f also for the vertex coloring of Q ), and2. v and F ( v ) use the same local color pattern.To show that F is a covering map, we need to observe that for each v ∈ V ( G ),the three neighbours of v in G map by F to the three neighbours of F ( v ) in Q .As we already know that f is a covering map to K , it suffices to show, thata neighbour u of v is indeed mapped to the neighbour of F ( v ) with color f ( u )(and not to the other vertex with the same color). For this we observe that thelocal coloring pattern at a vertex v determines the local coloring pattern at eachneighbour of v , in any cubic graph that is star 4-edge-colored. As this holdsboth in G and in Q , our definition of F yields a covering map, which finishesthe proof.There are cubic graphs whose star chromatic index is equal to 6. One exam-ple is K , . To see this, let us suppose that we have a star edge-coloring of K , ,and let F be a color class. If | F | = 3, then every other color class contains at13ost one edge and hence there are at least seven colors all together. So, we mayassume that every color class contains one or two edges only. If F = { ab, cd } isa color class, then one of the edges ad or cb forms a singleton color class sincethe second edge in the color class of ad (and the same for cb ) would need to bethe edge of K , disjoint from a, b, c, d . This implies that there are at least twosingleton color classes. Hence, the total number of colors is at least 6. Finally,a star 6-edge-coloring of K , is easy to construct, proving that χ (cid:48) s ( K , ) = 6. As we saw in Sections 2 and 4, establishing the star chromatic index is nontrivialeven for complete graphs. We established bounds(2 + o (1)) · n ≤ χ (cid:48) s ( K n ) ≤ n · √ o (1)) √ log n (log n ) / . Question 1
What is the true order of magnitude of χ (cid:48) s ( K n ) ? Is χ (cid:48) s ( K n ) = O ( n ) ? In the previous section we obtained the bound χ (cid:48) s ( G ) ≤ G . We also saw that χ (cid:48) s ( K , ) = 6. A bipartite cubic graph that wethought might require seven colors is the Heawood graph (the incidence graphof the points and lines of the Fano plane). However, it turned out that also itsstar chromatic index is at most 6. After some additional thoughts, we proposethe following. Conjecture 2 If G is a subcubic graph, then χ (cid:48) s ( G ) ≤ . It would be interesting to understand the list version of star edge-coloring:by an edge k -list for a graph G we mean a collection ( L e ) e ∈ E ( G ) such that each L e is a set of size k . We shall say that G is k -star edge choosable if for everyedge k -list ( L e ) there is a star edge-coloring c such that c ( e ) ∈ L e for everyedge e . We let ch (cid:48) s ( G ) be the minimum k such that G is k -star edge choosable.All of the results in this paper may have extension to list colorings. Let us askspecifically two questions: Question 3
Is it true that ch (cid:48) s ( G ) ≤ for every subcubic graph G ? (Perhapseven ≤ ?) Question 4
Is it true that ch (cid:48) s ( G ) = χ (cid:48) s ( G ) for every graph G ? Acknowledgement
We thank to the anonymous referees for helpful comments, in particular forsuggesting an improvement of the proof of Theorem 5.1.14 eferences [1] M. O. Albertson, G. G. Chappell, H. A. Kiersted, A. K¨undgen, and R.Ramamurthi, Coloring with no 2-colored P ’s, Electr. J. Combinatorics 1(2004), ≥
7, J. Lanzhou University (Nat. Sci.) 44 (2008), 94–95.[13] M. Molloy, B. Reed, Further algorithmic aspects of Lov´asz local lemma,Proceedings of the 30th Annual ACM Symposium on Theory of Computing,Dallas, Texas, 1998, pp. 524–529.[14] M. Molloy, B. Reed, Asymptotically optimal frugal colouring, In
Proceed-ings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algo-rithms (New York, New York, January 04–06, 2009) , SIAM, 2009, pp. 106–114.[15] J. Neˇsetˇril, P. Ossona de Mendez, Colorings and homomorphisms of minorclosed classes, in