Approximation bounds on maximum edge 2-coloring of dense graphs
L. Sunil Chandran, Rogers Mathew, Deepak Rajendraprasad, Nitin Singh
aa r X i v : . [ c s . D M ] O c t Approximation bounds on maximum edge2-coloring of dense graphs
L. Sunil Chandran , Rogers Mathew , Deepak Rajendraprasad ,and Nitin Singh Department of Computer Science and Automation, IndianInstitute of Science, Bangalore-560012, India ∗ , [email protected] Department of Computer Science and Engineering, IndianInstitute of Technology, Kharagpur-721301, India, [email protected] Department of Computer Science and Engineering, IndianInstitute of Technology, Palakkad, India, [email protected] IBM Research Lab, Manyata Embassy Business Park,Bangalore-560045, India, [email protected]
October 2, 2018
Abstract
For a graph G and integer q ≥
2, an edge q -coloring of G is an as-signment of colors to edges of G , such that edges incident on a vertexspan at most q distinct colors. The maximum edge q -coloring problemseeks to maximize the number of colors in an edge q -coloring of a graph G . The problem has been studied in combinatorics in the context of anti-Ramsey numbers. Algorithmically, the problem is NP-Hard for q ≥ /q . The case q = 2, is particu-larly relevant in practice, and has been well studied from the view pointof approximation algorithms. A 2-factor algorithm is known for generalgraphs, and recently a 5 / M of G . Give distinct colors to the edges of M . Let C , C , . . . , C t be theconnected components that results when M is removed from G. To alledges of C i give the ( | M | + i )th color.”In this paper, we first show that the approximation guarantee of thematching based algorithm is (1 + δ ) for graphs with perfect matching andminimum degree δ . For δ ≥
4, this is better than the approximationguarantee proved in [1]. For triangle free graphs with perfect matching,we prove that the approximation factor is (1 + δ − ), which is better than5 / δ ≥ ∗ part of this work was done while the author was visiting Max-Planck Institute for Infor-matik, Saarbruecken, Germany, under Alexandr von Humboldt fellowship. or the general case we show an approximation factor of (1 + ǫ ), where ǫ = κ +2 δ − . Here δ is the minimum degree and κ = n/ | M | is the ratio ofthe number of vertices to the cardinality of the maxium matching in G .When κ < δ −
3, this is better than the previously know 2 factor for thegeneral case. It may be noted that for a graph with maximum degree ∆and minimum degree δ , κ ≤ δ ) δ . We have the following immediatecorollaries:1. For d -regular graphs, ours is a (1 + d − ) factor approximation algo-rithm. This is better than the 2 factor known for the general case[4, 5, 6], when d ≥
8. (Note that all d -regular graphs need not havea perfect matching and thus 5/3 factor of [1] is not applicable.)2. For graphs of minimum degree h √ n + 1, we have an approximationfactor of (1+ h ). For h > √
6, this is better than the 2 factor knownfor the general case [4, 5, 6]We also show that when δ > ⌊ n ⌋ , the matching based algorithm is anoptimum algorithm for even n , and an additive 1 factor algorithm for odd n . For a graph G and an integer q ≥
2, the edge q -coloring problem seeks tomaximise the number of colors used to color the edges of G , subject to theconstraint that a vertex v ∈ V ( G ), is incident with edges of at most q differentcolors. We note that this problem differs from the classical edge coloring, whereall edges incident at a vertex have distinct color. The problem has also beenfound useful in modelling channel assignment in networks equipped with multi-channel wireless interfaces [8].In a combinatorial setting, the number of colors used in maximum edge q -coloring relates to anti-Ramsey number. For graphs G and H , the anti-Ramseynumber ar( G, H ) denotes the maximum number of colors k , such that in anedge coloring of G with k colors, all subgraphs of G isomorphic to H have atleast two edges of the same color. Then, it is seen that the number of colorsin maximum edge q -coloring of G is ar( G, K ,q +1 ) where K ,q +1 is a star with q + 2 vertices. Further details on anti-Ramsey numbers can be found in Erd¨oset. al [3]. Combinatorial bounds for edge q -coloring for several classes of graphsare obtained in [7].Recently, the maximum edge q -coloring problem has also been studied froman algorithmic perspective. Anna and Popa [1], proved that the problem is NP-Hard for every q ≥
2. Moreover, they proved that, assuming the unique gamesconjecture, it cannot be approximated within a factor less than 1 + 1 /q for every q ≥ P = N P , it cannot be approximated within a factor lessthan 1 + ( q − / ( q − for every q ≥ matching based algorithm . The same authors also showed that the problem ispolynomial time solvable for trees and complete graphs. Adamaszek and Popa[1] showed that the matching based algorithm of Feng et. al. improves to a5 / lgorithm 1 Matching Based Algorithm for edge 2-coloring [1]Let G be the input graph.Let M be a maximum matching in G .Assign distinct color to each edge of M .Assign a new color to each component of G \ M . Our contribution in this paper is to show that the approximation guarantee ofthe matching based algorithm (Algorithm 1) improves significantly for graphswith large minimum degree. Specifically, we obtain the following results:
Result 1:
For graphs with perfect matching, the matching based algorithmhas an approximation guarantee of (1 + δ ) where δ is the minimum degree.Recall that Adamaszek and Popa [1] proved an approximation guarantee of for this case. The approximation gurantee proved by us is better than that ofAdamaszek and Popa for δ ≥
4, and equals to theirs for δ = 3. Comment:
For graphs G with δ ≥ ⌊ n/ ⌋ , Algorithm 1 is almost optimal. Infact it is an additive 1 approximation algorithm. If we further assume that n is even, for δ > ⌊ n/ ⌋ the matching based algorithm is optimal. (This followsfrom result 1, noting that such graphs always have a perfect matching.) Result 2:
For triangle free graphs with perfect matching, we show a betterapproximation guarantee, namely (1 + δ − ). This is better than that of [1] for δ ≥ Result 3:
For the general case we show that the matching based algorithm isa (1 + κ +2 δ − )-factor approximation algorithm. Here κ = n/ | M | where n is thenumber of vertices in G and M is a maximum matching in G . Considering theattempts of Adamaszek and Popa to get a better approximation guarantee forgraphs with perfect matching, it is natural to consider this ratio. This factor isbetter than the previous known 2 factor for the general case, when κ < δ − κ + 2) / ( δ − • Corollary 1 of Result 3: It is easy to see that κ ≤ δ ) δ where ∆ is themaximum degree of G . (Consider the set of vertices V ( M ) spanned bythe edges of M . Since due to maximality of M , all edges of G should beadjacent to at least one vertex in V ( M ), we get 2 | M | ∆ ≥ ( n − | M | ) δ .)Therefore for d -regular graphs, the approximation guarantee is (1 + d − ).This is better than the 2 factor known for the general case [4, 5, 6], when k ≥
8. (Note that all d -regular graphs need not have a perfect matchingand thus 5/3 factor of [1] or the (1 + 2 /δ ) factor from our result 1, is notapplicable.) • Coroallary 2 of Result 3: For graphs of minimum degree h √ n + 1, we havean approximation factor of (1+ h ). To see this note that κ ≤ ∆+ δδ ≤ ∆ δ +1.Now substituting ∆ ≤ n and δ ≥ h √ n + 1, we see that the approximationfactor is at most (1 + h ). For h > √
6, this is better than the 2 factorknown for the general case [4, 5, 6].3
Notation and Preliminaries
Throughout this paper, we consider connected graphs with minimum degree δ ≥
3. An edge 2-coloring of a graph G with c colors is a map C : E ( G ) → [ c ],such that a vertex is incident with edges of at most two colors. Let ALG( G )denote the number of colors in the coloring returned by Algorithm 1 for thegraph G , and let OPT( G ) be the number of colors in the maximum edge 2-coloring of G . Let C be an edge 2-coloring of the graph G . A subgraph of G containing exactlyone edge of each color in C is called a characteristic subgraph of G with respectto coloring C . Note that the maximum degree of a characteristic graph is atmost two. Lemma 2.1.
For a graph G and an edge -coloring C of G , there exists acharacteristic graph χ which is disjoint union of paths.Proof. Let χ be a characteristic graph with minimum number of cycles. Weclaim that χ has no cycles. For sake of contradiction, suppose χ has a cycle.Let u be one of the vertices in the cycle, and v, w be its two neighbors in thecycle. Since δ ≥
3, there is neighbor z of u in G , which is not incident withan edge in χ . Now the edge uz must have the same color as uv or uw , say uv .Then χ − uv + uz is a characteristic subgraph with smaller number of cycles.Hence, χ did not have any cycles.In the remainder of the paper, we will tacitly assume that characteristic sub-graphs do not contain cycles. The components of the characteristic subgraphs(which are paths) will be called characteristic paths . In this section, we derive some useful bounds for graphs with perfect matching.The proofs here also help to illustrate key ideas in simpler setting, which willbe extended further in the proof of Theorem 4.1.
Theorem 3.1.
Let G be an n -vertex graph with perfect matching, where δ ( G ) ≥ . Then, we have: (i) OPT( G ) ≤ (1 + 2 /δ ) · ALG( G ) . Furthermore, if G is triangle-free then OPT( G ) ≤ (1 + 1 / ( δ − · ALG( G ) . (ii) OPT( G ) ≤ ALG( G ) + 1 for δ ≥ ⌊ n/ ⌋ . The above theorem yeilds the approximation factor of 5 / δ ≥ / Proof of Theorem 3.1.
Let C be an optimal edge 2-coloring of G , and let χ bea characteristic subgraph of G with respect to coloring C . Let N i , i ∈ { , , } G with degree i in χ . Let n i = | N i | . Clearly n + n + n = n . The number of colors c in C which is same as the number of edges in χ isgiven by c = n + n . For v ∈ N , let N ′ ( v ) denote the neighbors of v throughedges not in χ . Clearly | N ′ ( v ) | ≥ δ − v ∈ N . Claim : For u, v ∈ N , v / ∈ N ′ ( u ). Let C u and C v be the colors incident at vertices u and v respectively. If v ∈ N ′ ( u ) then the color of uv belongs to C u ∩ C v . Bythe definition of characteristic graph, if uv is not an edge in χ , then C u ∩ C v = ∅ .Thus edge uv cannot get a color if v ∈ N ′ ( u ). We infer that v / ∈ N ′ ( u ).From the above claim, it follows that N ′ ( v ) ⊆ N ∪ N for v ∈ N . Considerthe bipartite graph H with bipartition N ⊎ ( N ∪ N ) and edge set of H , E ( H )given by E ( H ) := ∪ v ∈ N E ( v, N ′ ( v )). We show that d H ( u ) ≤ u ∈ N and d H ( u ) ≤ u ∈ N . Assume u ∈ N . Now, there are at most two colorsincident at u in C , say colors a and b . If w is a neighbor of u in H , we musthave a ∈ C w or b ∈ C w . Since there are at most two vertices w ∈ N such that a ∈ C w , if follows that u has at most 4 neighbors in H . For u ∈ N , let z be theunique neighbor of u in χ . Let a be the color of edge uz . Notice that uz E ( H )since it is an edge of χ . Also a
6∈ C w for w ∈ N \{ z } . Thus u is not incidentwith a -colored edges in H . Thus edges incident on u in H must have the samecolor, and by the previous arguments, there can be at most two of them. Thus d H ( u ) ≤ u ∈ N . Counting the edges across the bipartition in two wayswe have: n ( δ − ≤ n + 2 n (1)The result now follows from some algebra, as shown below. c = 2 n + n
2= ( n + n + n ) + ( n − n )2= n n − n n + n + n = n we see that n δ ≤ n + 2 n . Therefore wehave ( n − n ) δ ≤ n δ − n ≤ n .Substituting, we get c ≤ n (cid:18) δ (cid:19) . (3)This proves the claimed approximation factor in part (i) of the theorem forgraphs with perfect matching. If the graph is further assumed to be triangle-free, we prove that a vertex u ∈ N ∪ N can have atmost one edge of a givencolor incident on it in H . Let u ∈ N ∪ N and let a be one of the colors incidentat u in C . Then, the only possible a -colored edges incident at u in H are edges uw where w ∈ N with a ∈ C w . Let x, y ∈ N be such that a ∈ C x ∩ C y . Then xy is an edge in χ . As G is triangle-free we conclude at most one of { x, y } is aneighbor of u . Thus u is incident with at most one edge of color a in H . Nowit follows that, in this case, we have d H ( u ) ≤ u ∈ N and d H ( u ) ≤ u ∈ N . The Equation (1) can now be written as: n ( δ − ≤ n + n . (4)5t follows that n ( δ − ≤ n + n . Making similar substitutions in Equation(2), we get c ≤ n (cid:18) δ − (cid:19) . (5)This proves the approximation factor claimed for triangle-free graphs in part (i)of the theorem.To prove part (ii), observe that by Dirac’s theorem, a graph G with δ ≥ ⌊ n/ ⌋ has a hamilton cycle, and hence a maximum matching of size ⌊ n/ ⌋ . ThusALG( G ) ≥ ⌊ n/ ⌋ + 1. Further from part (i), we have c ≤ n/ n/δ < ⌊ n/ ⌋ + 3.Thus since c is an integer we get c ≤ ⌊ n/ ⌋ + 2 ≤ ALG( G ) + 1. Note that if weassume that n is even, and δ > n/
2, this proves that the algorithm is optimal.
In this section, we prove our main result, which is the following:
Theorem 4.1.
Let G be an n -vertex connected graph with δ ( G ) ≥ . Let M be a maximum matching of G . Then OPT( G ) ≤ (1 + ε ) · ALG( G ) where ε = ( κ + 2) / ( δ − , with κ = n/ | M | being the ratio of number of vertices to thesize of maximum matching of the graph.Proof. Let C be an optimal coloring of G using c colors. Let χ be a character-sistic subgraph of G with respect to coloring C , with the maximum number ofcharacteristic paths. We will say a vertex v is an internal vertex of χ , if it isan internal vertex of one of the characteristic paths. Similarly, a vertex will becalled a terminal vertex of χ if it is a terminal vertex of one of the characteris-tic paths. In the proof of Theorem 3.1, we bound the number of edges in thecharacteristic subgraph (which is same as the number of colors in the coloring)by n/ n − n ) /
2. In the case of graphs with perfect matching, n/ n − n ) / n/ M ′ from within χ by selecting alternate edges in each characteristic path, startingwith the first edge in each path. Let t be the number of unselected edges. Then c = | M ′ | + t ≤ | M | + t .The remainder of the proof attempts to upper bound the excess term t . Infact we show that t ≤ | M |· (( κ +2) / ( δ − T be the set of vertices consisting of left endpoint of each unselectededge (see Figure 1). Thus we have | T | = t . First we note that T is an indepen-dent set in G . This is because vertices in T are mutually non-adjacent internalvertices of χ , and hence have mutually disjoint incident colors. For each v ∈ T ,choose a set of δ − v , which are not present in χ (this is pos-sible, as each vertex has at most two neighbors in χ ). Let us call these edges as special edges. Let H , H and H be sets of vertices of V \ T which are incidentwith 0, 1 and 2 of these special edges. Let h i = | H i | . Since, the vertices in T are6ncident with mutually disjoint sets of colors, a vertex in V \ T is incident withat most two special edges emanating from T . Thus V \ T = H ⊎ H ⊎ H , or h + h + h = n − t . Counting the special edges across the bipartition ( T, V \ T )we have: t ( δ −
2) = 2 h + h (6)Moreover, as h + h + h = n − t , we can rewrite the above equality as t ( δ −
2) = n − t + ( h − h ), and thus, t ( δ − ≤ n + h . (7) • • • • •• • • • •• • • • •• • • • •• × • ו × • ו × • ו × • × • •• Figure 1: Characterisitic paths are shown on the left. The unselected edges aremarked with × , and corresponding vertices in T are indicated by a box. Thespecial edges are indicated with dashed lines. The set H is indicated by theellipse.To obtain a bound on h , we consider the neighbors of vertices in H whichare not in T . Note that each vertex v ∈ H has at least δ − T . Let A be the set of terminal vertices of the characteristic subgraph χ and B be the set of internal vertices of χ , which are not in T . Let O denote the set ofvertices which are not incident with an edge in χ . Note that H ⊆ O . This isbecause a vertex incident with an edge in χ can only receive special edges of atmost one more color. But then it has at most one neighbor in T , and hence itis not a vertex in H . We show the following: Claim (a) : Let a be a color present in a characteristic path of length at leasttwo in χ . Then there is no a -colored edge between two vertices of O . We proveby contradiction. Let uv be an edge in a characteristic path of length at leasttwo, and let a be color of uv . Assume that u is not a terminal vertex of χ . Ifpossible let x and y be two vertices in O with edge xy having color a . Then χ − uv + xy is a characteristic subgraph with more characteristic paths than χ ,a contradiction. The claim follows. Claim (b) : Let a be a color present in a characteristic path of length at least twoin χ . Then there is no a -colored triangle formed by the a -colored edge in χ and avertex from O . To prove, again assume that uv is an a -colored edge in χ where u is not a terminal vertex. If possible let uvw be an a -colored triangle with w ∈ O . Again, χ − uv + vw is a characteristic subgraph with more characteristicpaths than χ , contradicting the choice of χ . The claim follows. Claim (c) : The neighbors of vertices in H , which are not in T , lie in A . Toprove, we observe that the colors incident on vertices in H appear in a charac-teristic path of length at least two (as vertices in T are on such paths). Then7s H ⊆ O , from Claim (a), we have that a vertex v ∈ H is not incident with avertex in O . Further, v is not incident with a vertex in B , as the only internalvertices in χ that are incident with colors at v are its neighbors in T . Hence,all the neighbors of v , not in T are in A .Consider the bipartite graph X with bipartition ( H , A ) with edge set con-sisting of A - H edges in G . Clearly d X ( v ) ≥ δ − v ∈ H . We now provethat d X ( u ) ≤ δ − u ∈ A . Let u ∈ A , and let uz be the edge of χ incidentat u , where a is the color of uz . Let v be a neighbor of u in H . We showthat uv is not of color a . For sake of contradiction suppose that color of uv is a . Then as v ∈ H , it has a neighbor w in T which is incident with edgeof color a . But then w = z as z is the only internal vertex in χ incident withedge of color a . Now uvz form an a -colored triangle, which contradicts Claim(b). Thus, all the edges of X incident at v must have the color different from a , and hence must have the same color (say b ). Clearly all the vertices in H incident with edge of color b , must be incident with a b -colored special edgefrom a vertex in T . As vertices in T are incident with mutually disjoint colors,we infer that all such vertices are incident with b -colored special edges from asingle vertex w ∈ T (see Figure 2). Since there are exactly δ − T , we conclude there are at most δ − H incident with color b . Hence d X ( u ) ≤ δ − u ∈ A . Now, we have h ( δ − ≤ | E ( H , A ) | ≤ ( δ − · | A | . Finally, observe that | A | ≤ | M | andhence h ≤ | M | . Substituting in Equation (7), we have: t ( δ − ≤ n + 2 | M | t ≤| M | · (cid:18) κ + 2 δ − (cid:19) . (8)where κ = n/ | M | . The claimed approximation now follows from the inequality c ≤ | M | + t . In this section we give a construction to show that the approximation factor inTheorem 4.1 is tight.
Theorem 5.1.
Given κ and δ , where κ < δ − , and a sufficiently large t with respect to δ ( say t > ( δ − ( δ +1 − κ ) ) , there exists a graph with number of vertices n = κt , and minimum degree δ such that it is possible to get a -edge coloringfor this graph that uses at least t (1 + κ − δ − ) colors.Proof. It is enough to construct a graph with the cardinality of maximum match-ing equals to t, n = κt minimum degree δ and demonstrate a 2-edge coloringfor this graph that uses at least t (1 + κ − δ − ) colors. Construct a bipartite graph G with set of vertices S on one side and T on the other side (see Figure 3 forschematic representation). Let S be the disjoint union of two sets of vertices S and S . Also let T be the disjoint union of two sets of vertices A and B . Let | A | = | S | = t , and | S | = δ .In G , we will first add a matching between A and S . In our coloring of G , the edges of this matching will get distinct colors, say from 1 to t . All the8 u • v • v • v • w • z a bb bbA H T Figure 2: The vertex u in A is incident with a -colored edge in χ . Its neighborsin H are through the “other” color at u , namely b . Moreover, these neighborsare also incident with b -colored special edges from a vertex in T .vertices in T = A ∪ B are made adjacent to all the vertices in S , so that all thevertices in T ∪ S has degree at least δ . These edges between T and S otherthan the matching edges will be given the color t + 1.Let t be chosen such that h = t ( δ +1 − κ ) / ( δ − − ( δ − t > ( δ − ( δ +1 − κ ) , so that h > S = u , u , . . . , u t − δ ,and let B = { v , v , . . . , v α } , where α = ( t − δ − h + 1)( δ − u i , 1 ≤ i ≤ h adjacent to { v , v , . . . , v δ − } , so that the degree of u i is δ , for1 ≤ i ≤ h . All the edges from u i to v j , 1 ≤ i ≤ h and 1 ≤ j ≤ δ − t + 2.For h < i ≤ t − δ , make u i adjacent to v ( i − h )( δ − , . . . , v ( i − h +1)( δ − . Thuseach of these vertices u i , h < i ≤ t − δ gets an exclusive neigbourhood of δ − B . All the edges from u h + j , ≤ j ≤ t − h − δ to the vertices of B ,will be given the color t + 2 + j .The total number of colors used is clearly 2 t − h − δ + 2. Note that by thisconstruction the degree of each u i is δ . Thus the minimum degree of G is δ .Moreover the number of vertices in G is 2 t + α = 2 t + ( t − δ − h + 1)( δ −
1) = t ( δ + 1) − h ( δ − − ( δ − . Since G is a bipartite graph with t vertices onone side, the cardinality of the maximum matching is t . It is easy to verify that n/t = κ , by substituting h = t ( δ + 1 − κ ) / ( δ − − ( δ −
1) in the expression for n . The number of colors used is 2 t − ( δ − − h ≥ t − t δ +1 − kδ − = t (1 + k − δ − ),as required. 9 • • • δ − δ − δ − δ − δtS : S : A : B :Figure 3: The figure illustrates the non-matching edges of the tight examplein Theorem 5.1. An ellipse represents a group of vertices, the number at thecenter being the cardinality of the group. A • represents a single vertex. Anedge between two groups indicates the complete set of edges between them. Let G be any d -regular graph on n vertices such that it can be properly edgecolored using d colors. (Clearly such graphs exist, for example all d -regularbipartite graphs are d -edge colorable). Properly edge color G using colors 1 to d . We then construct a new graph G ′ from G as follows: Replace each vertex v of G by a clique K v of d vertices v , . . . , v d . Thus G ′ has n ′ = nd vertices. Notethat n ′ is an even number.Add an edge u i v i in G ′ if uv ∈ E ( G ) and uv is colored with color i in theedge coloring of G . Clearly, G ′ is d -regular and has a perfect matching: The setof all edges M of G ′ which are not part of any clique K v , v ∈ V ( G ) clearly forma perfect matching of G ′ . More precisely, M = S ≤ i ≤ d { u i v i ∈ E ( G ′ ) : uv ∈ E ( G ) and uv is colored i in G } . But it is easy to see that removing all the n ′ edges of the perfect matching M of G ′ leaves n ′ d = n connected components,namely { K v : v ∈ V ( G ) } . Coloring the n ′ edges in the matching M with n ′ distinct colors and coloring the edges of each of the n ′ d components with a newcolor yields a 2-coloring using n ′ + n ′ d = n ′ (1 + 2 /d ) colors.On the other hand, it is easy to see that M is not the only perfect matchingavailable in G ′ . Suppose d is even. Then another simple way to get a perfectmatching of G ′ is as follows: From each clique K v , v ∈ V ( G ) pick a matchingof size d/
2. The union of all these matchings clearly is a perfect matching of G ′ . Let us name this perfect matching as M . Note that the matching basedalgorithm picks up an arbitrary perfect matching, colors its edges with distinctcolors and then gives new colors to the connected componets that results whenthat perfect matching is removed from the graph: one new color per component.Suppose the matching based algorithm picks up M instead of M to start with.It is obvious that if M is removed from G ′ , the resulting graph has only oneconnected component. Therefore the matching based algorithm (Algorithm 1)yields a 2-coloring of size n ′ + 1, where n ′ is the number of vertices of G ′ . (To10eal with the case when d is odd, we can assume that the graph G was a d -regular bipartite graph such that there exists a perfect matching F in it suchthat G \ F is still connected. When properly edge coloring G we can make surethat F forms the set of edges colored d . Now to get M we pick a d − sizedmatching of the first d − K v , and the set of edges { u d v d : uv ∈ F } . Clearly G ′ \ M remains connected.)Thus for G ′ , OPT( G ′ ) / ALG(G ′ ) = (cid:16) d − dn ′ +1 (cid:17) which is very close to(1 + d ) for large n . References [1] Anna Adamaszek and Alexandru Popa. Approximation and hardness re-sults for maximum edge q -coloring problem. ISAAC (2) , 132–143, 2010.[2] Anna Adamaszek and Alexandru Popa. Personal communication.[3] Paul Erd¨os, Mikl´os Simonovits and Vera T. S´os. Anti-Ramsey theorems.
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