- Email: [email protected]
Minimum rainbow H-decompositions of graphs
Available online at www.sciencedirect.com
Electronic Notes in Discrete Mathematics 61 (2017) 955–961 www.elsevier.com/locate/endm
Minimum rainbow H-decompositions of graphs 2 ¨ Lale Ozkahya
Department of Computer Engineering Hacettepe University Beytepe 06800 Ankara, Turkey
Yury Person 1,3 Institute of Mathematics Goethe University Robert-Mayer-Str. 10 60325 Frankfurt am Main, Germany
Abstract Given graphs G and H, we consider the problem of decomposing a properly edgecolored graph G into few parts consisting of rainbow copies of H and single edges. We establish a close relation to the previously studied problem of minimum Hdecompositions, where an edge coloring does not matter and one is merely interested in decomposing graphs into copies of H and single edges. Keywords: graph decomposition, rainbow copies, edge-critical, Tur´ an graph, stability approach
1 2 3
YP was supported by DFG grant PE 2299/1-1. Email: [email protected] Email: [email protected]
http://dx.doi.org/10.1016/j.endm.2017.07.059 1571-0653/© 2017 Elsevier B.V. All rights reserved.
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L. Özkahya, Y. Person / Electronic Notes in Discrete Mathematics 61 (2017) 955–961
Introduction and new results
For two graphs G and H and a proper edge coloring χ of G, a rainbow Hdecomposition of G is a partition of the edges of G into G1 ,. . . , Gt such that every Gi is either a single edge or is rainbow-colored and isomorphic to H, where a rainbow coloring of H assigns distinct colors to all edges of H. Rainbow H-decompositions present a variation on a well-studied topic of H-decompositions. Before we state our main results, we present the history of this general problem. For two graphs G and H, an H-decomposition of G is a partition of the edges of G into G1 ,. . . , Gt such that every Gi is either a single edge or is isomorphic to H. An H-decomposition of G with smallest possible t is called minimum and φ(G, H) = t denotes its cardinality. The following relation φ(G, H) = e(G) − (e(H) − 1)N (G, H) clearly holds, where N (G, H) denotes the maximum number of edge-disjoint copies of H in G. We write φ(n, H) := maxG∈Gn φ(G, H) where Gn denotes all graphs on n vertices. This function was studied first by Erd˝os, Goodman and P´osa [3], who were motivated by the problem of representing graphs by set intersections. They showed that φ(n, K3 ) = ex(n, K3 ), where ex(n, F ) denotes the maximum size of a graph on n vertices, that does not contain H as a subgraph. Moreover, these authors proved in [3] that the only graph that maximizes this function is the complete balanced bipartite graph. Consequently, they conjectured that φ(n, Kr ) = ex(n, Kr ) and the only optimal graph is the Tur´an graph Tr−1 (n), the complete balanced (r − 1)-partite graph on n vertices, where the sizes of the partite sets differ from each other by at most one. Bollob´as [2] verified this conjecture by showing that φ(n, Kr ) = ex(n, Kr ) for all n ≥ r ≥ 3. Pikhurko and Sousa [12] proved that for any fixed graph H with chromatic number r ≥ 3, φ(n, H) = ex(n, H) + o(n2 ) and they made the following conjecture. Conjecture 1.1 For any graph H with chromatic number at least 3, there is an n0 = n0 (H) such that φ(n, H) = ex(n, H) for all n ≥ n0 . This conjecture has been verified by Sousa for clique extensions of order r ≥ 4 (n ≥ r) [15], the cycles of length 5 (n ≥ 6) and 7 (n ≥ 10) [14,16]. In a previous work [11] we verified Conjecture 1.1 for edge-critical graphs, where a graph H is edge-critical if there exists an edge e ∈ E(H) such that χ(H) > χ(H − e). Later Allen, B¨ottcher and Person [1] obtained general upper bounds for all graphs H that improve the error term in the result of Pikhurko and Sousa [12] and extend the result on the edge-critical case.
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Recently, Liu and Sousa [10] studied the following colored variant of the H-decomposition problem. Let φk (G, H) be the smallest number t such that any graph G of order n and any coloring of its edges with k colors admits a monochromatic H-decomposition with at most t parts, where H is a complete graph. Later, Liu, Pikhurko and Sousa [9] generalized this by investigating φk (G, H), which is the smallest number t such that any graph G of order n and any coloring of its edges with k colors admits a monochromatic H = {H1 , . . . , Hk }-decomposition in t parts such that each part is either a single edge or forms a monochromatic copy of Hi in color i, for some 1 ≤ i ≤ k. Extending the results of Liu and Sousa [10], they solve this problem when each graph in H is a clique and n ≥ n0 (H) is sufficiently large. 1.1
Main Theorem
We call a rainbow H-decomposition of G under the proper edge coloring χ into t parts (i.e. each part is either a rainbow H or single edge) with the smallest possible t minimum and φR χ (G, H) = t denotes its cardinality. It is R R not difficult to see that φR (G, H) = e(G)−(e(H)−1)N χ χ (G), where Nχ (G, H) denotes the maximum number of edge-disjoint rainbow copies of H under this proper coloring χ of G. In this paper, we study the function φR (n, H) := max max φR χ (G, H), G∈Gn
χ
where we maximize over all graphs G from Gn and over all proper colorings χ of G. We will refer to decompositions that attain φR (n, H) as minimum rainbow H-decompositions of graphs. Observe that φR χ (G, H) ≥ φ(G, H) for any graph G and any proper edge coloring χ of G, otherwise it means that there are more than N (G, H) edgedisjoint rainbow copies of H in G under χ, which is a contradiction. Therefore, we have φR (n, H) ≥ φ(n, H). On the other hand, maxχ φR χ (G, H) ≤ e(G), where equality is achieved when there is no copy of H in G or there is a proper edge coloring of G without rainbow copies of H. We prove the following result for any (non-bipartite) clique Kr . Theorem 1.2 For any r ≥ 3 there is an n0 such that any graph G on n ≥ n0 vertices with some proper edge coloring χ that satisfies φR χ (G, Kr ) ≥ ex(n, Kr ) must in fact be isomorphic to the Tur´ an graph Tr−1 (n). In particular, φR (n, Kr ) = ex(n, Kr ) for all n ≥ n0 , and the only graph attaining φR (n, Kr ) is the Tur´ an graph Tr−1 (n).
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We also obtain generalizations of the result of Pikhurko and Sousa [12] on φ(n, H) = ex(n, H) + o(n2 ) (for non-bipartite H) and on our result from [11] about φ(n, H) = ex(n, H) for edge-critical graphs H, which read as follows. Theorem 1.3 For any edge-critical graph H with chromatic number at least 3, there is an n0 = n0 (H) such that φR (n, H) = ex(n, H) for all n ≥ n0 . Moreover, the only graph attaining φR (n, H) is the Tur´ an graph Tχ(H)−1 (n). Theorem 1.4 Let H be any graph of chromatic number at least three. Then we have φR (n, H) = ex(n, H) + o(n2 ). The proofs use essentially the same techniques as Theorem 1.2.
2
Proof sketches
To obtain Theorem 1.2, we first prove the stability for cliques for the function φ(G, Kr ) and then we amplify it to the stability for cliques for the rainbow function φR ori and Tuza [5] and of Hoi [6], we obtain χ (G, Kr ). By results of Gy˝ the following stability result. Proposition 2.1 For every r ≥ 3 and for every γ > 0 there exist ε > 0 and n0 ∈ N such that for every graph G on n ≥ n0 vertices the following is true. If φ(G, Kr ) ≥ ex(n, Kr ) − εn2 2 ˙ r−1 with r−1 then there exists a partition of V (G) = V1 ∪˙ . . . ∪V i=1 e(Vi ) < γn . We remark, that the stability result that we proved in [11, Lemma 4] was applicable only to graphs H with χ(H) ≥ 3 that are not cliques. The proof of Theorem 1.2 relies on the following stability theorem for proper edge colorings and is our main contribution. Theorem 2.2 For every r ≥ 3 and for every γ > 0 there exist ε > 0 and n0 ∈ N such that for every graph G on n ≥ n0 vertices with any proper coloring χ of its edges the following is true. If 2 φR χ (G, Kr ) ≥ ex(n, Kr ) − εn
˙ r−1 with then there exists a partition of V (G) = V1 ∪˙ . . . ∪V
r−1 i=1
e(Vi ) < γn2 .
Notice that the case r = 3 is covered by Proposition 2.1, as any proper edge coloring colors a triangle with different colors.
L. Özkahya, Y. Person / Electronic Notes in Discrete Mathematics 61 (2017) 955–961
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Before giving some details of the proof of Theorem 1.2, we state the main auxiliary lemma that we use. This lemma shows that for any graph G on n vertices and any proper edge coloring χ, the numbers (of edge-disjoint copies) N (G, Kr ) and NχR (G, Kr ) differ by at most o(n2 ). Lemma 2.3 For all r ≥ 3 and any c > 0 there exists an n0 such that the following holds. In any proper edge coloring χ of a graph G on n ≥ n0 vertices we have N (G, Kr ) ≤ NχR (G, Kr ) + cn2 . Proof sketch. We fix a collection of edge-disjoint (not necessarily rainbow) copies of Kr of maximum cardinality. The proof proceeds by an application of the regularity lemma of Szemer´edi [17]. Then we identify various ε-regular pairs of not too small density that build r-partite graphs, where most edgedisjoint copies of Kr ‘live’. We need to argue then that we can find roughly that many rainbow copies instead. To do so, we would like to apply a Theorem of Pippenger and Spencer [13] and before we appropriately split ε-regular pairs, similarly as is done for example in [12]. This time however, the density of these ‘split’ ε-regular pairs may be much lower than ε, which would make the usual approach to apply the counting lemma (and then to apply a Theorem of Pippenger and Spencer) impossible. Our solution is to count the rainbow copies of Kr before splitting the ε-regular pairs (of sufficiently high density) and only then to split them randomly. All we need to estimate is then the number of rainbow copies that will be in (random) subgraphs of ε-regular rpartite graphs, which we do by applying a concentration result of Kim and Vu [8]. 2 Proof of Theorem 2.2. For given r ≥ 3 and γ > 0 let ε = εT 2.1 be as asserted by Proposition 2.1. We choose c := ε /e(Kr ), we also set ε := c and assume that n is large enough so that Lemma 2.3 is applicable. Let now G be a graph on n vertices with a proper coloring χ of its edges 2 such that φR By Lemma 2.3, we have χ (G, Kr ) ≥ ex(n, Kr ) − εn holds. R 2 N (G, Kr ) ≤ Nχ (G, Kr ) + cn (for n large enough). Therefore, φR χ (G, Kr ) = e(G) − (e(Kr ) − 1)NχR (G, Kr ) ≥ ex(n, Kr ) − εn2 implies φ(G, Kr ) = e(G) − (e(Kr ) − 1)N (G, Kr ) ≥ 2 e(G) − (e(Kr ) − 1)(NχR (G, Kr ) + cn2 ) ≥ φR χ (G, Kr ) − (e(Kr ) − 1)cn ≥ ex(n, Kr ) − εn2 − (e(Kr ) − 1)cn2 = ex(n, Kr ) − e(Kr )cn2 = ex(n, Kr ) − ε n2 , and Proposition 2.1 yields the desired partition.
2
Proof sketch of Theorem 1.2 We will apply Theorem 2.2 in the form
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when ε = 0, and we choose γ sufficiently small. We assume that there is a graph G on n vertices (n large enough) such that there exists a proper edge coloring of G with φR χ (G, Kr ) ≥ ex(n, Kr ) and G is not isomorphic to the Tur´an graph Tr−1 (n). By following the steps of [11], we apply first Claim 7 from [11] and assume that φR χ (G) = ex(n, H) + m for some m ≥ 0. We obtain a subgraph G of G on n vertices such that δ(G ) ≥ δ(Tr−1 (n )) and φR χ (G , Kr ) ≥ ex(n , Kr ) + m. Then, Theorem 2.2 asserts the existence of a partition of G into r − 1 parts that maximizes the number of edges between different parts so that the number m2 of edges within the partition classes is not zero but is at most γn2 . It is observed that the orders of these r − 1 parts are almost equal due to the maximality condition (Claim 8 from [11, Theorem 3]). This implies that e(G )≤ ex(n , Kr ) + m2 . m2 + 1 many rainbow copies of Kr . (r2)−1 We find these copies iteratively (cf. Claim 9 from [11, Theorem 3]). The only difference to the embedding in [11] is that we need to find at each iteration a copy of Kr , which under the edge coloring χ is rainbow. This is done by finding first a complete (r − 1)-partite graph F with parts of size at least 8(r − 1)3 , so that one of the parts contains an edge (Claim 9 from [11, Theorem 3], this basically follows from an application of Theorem of Erd˝os and Stone [4]). A rainbow copy of Kr is guaranteed to exist in such a graph F by a result of Keevash, Mubayi, and Verstra¨ete [7, Lemma 2.2]. This allows us Sudakov,
In the final step, we find at least
m2 + 1 many rainbow copies of Kr in G under the edge (r2)−1 coloring χ. Clearly, this implies that
to find at least
m r 2 r
−1 + 1 < ex(n , Kr ), φR χ (G , Kr ) ≤ ex(n , Kr ) + m2 − 2 − 1 2 a contradiction.
2
References [1] Allen, P., J. B¨ottcher and Y. Person, An improved error term for minimum H-decompositions of graphs, J. Combin. Theory Ser. B 108 (2014), pp. 92–101. [2] Bollob´ as, B., On complete subgraphs of different orders, Math. Proc. Cambridge Philos. Soc. 79 (1976), pp. 19–24.
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[3] Erd˝os, P., A. Goodman and L. P´ osa, The representation of a graph by set intersections, Canad. J. Math. 18 (1966), pp. 106–112. [4] Erd˝os, P. and A. H. Stone, On the structure of linear graphs, Bull. Amer. Math. Soc. 52 (1946), pp. 1087–1091. [5] Gy˝ ori, E. and Z. Tuza, Decompositions of graphs into complete subgraphs of given order, Studia scientiarum mathematicarum Hungarica (1987), pp. 315– 320. [6] Hoi, N. H., On the problems of edge disjoint cliques in graphs, Master’s thesis, E¨ otv¨ os Lor´and University, Hungary (2005). [7] Keevash, P., D. Mubayi, B. Sudakov and J. Verstra¨ete, Rainbow Tur´ an problems, Combin. Probab. Comput. 16 (2007), pp. 109–126. [8] Kim, J. and V. Vu, Concentration of multivariate polynomials and its applications, Combinatorica 20 (2000), pp. 417–434. [9] Liu, H., O. Pikhurko and T. Sousa, Monochromatic clique decompositions of graphs, J. Graph Theory 80 (2015), pp. 287–298. [10] Liu, H. and T. Sousa, Monochromatic Kr -decompositions of graphs, J. Graph Theory 76 (2014), pp. 89–100. ¨ [11] Ozkahya, L. and Y. Person, Minimum H-decompositions of graphs: edge-critical case, J. Combin. Theory Ser. B 102 (2012), pp. 715–725. [12] Pikhurko, O. and T. Sousa, Minimum H-decompositions of graphs, J. Combin. Theory Ser. B 97 (2007), pp. 1041–1055. [13] Pippenger, N. and J. Spencer, Asymptotic behavior of the chromatic index for hypergraphs, J. Combin. Theory Ser. A 51 (1989), pp. 24–42. [14] Sousa, T., Decompositions of graphs into 5-cycles and other small graphs, Electron. J. Combin. 12 (2005), p. 7. [15] Sousa, T., Decompositions of graphs into a given clique-extension, Ars Combin. C (2011), pp. 465–472. [16] Sousa, T., Decomposition of graphs into cycles of length seven and single edges, Ars Combin. 119 (2015), pp. 321–329. [17] Szemer´edi, E., Regular partitions of graphs, in: Probl`emes combinatoires et th´eorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976), Colloq. Internat. CNRS 260, CNRS, Paris, 1978 pp. 399–401.
Electronic Notes in Discrete Mathematics 61 (2017) 955–961 www.elsevier.com/locate/endm
Minimum rainbow H-decompositions of graphs 2 ¨ Lale Ozkahya
Department of Computer Engineering Hacettepe University Beytepe 06800 Ankara, Turkey
Yury Person 1,3 Institute of Mathematics Goethe University Robert-Mayer-Str. 10 60325 Frankfurt am Main, Germany
Abstract Given graphs G and H, we consider the problem of decomposing a properly edgecolored graph G into few parts consisting of rainbow copies of H and single edges. We establish a close relation to the previously studied problem of minimum Hdecompositions, where an edge coloring does not matter and one is merely interested in decomposing graphs into copies of H and single edges. Keywords: graph decomposition, rainbow copies, edge-critical, Tur´ an graph, stability approach
1 2 3
YP was supported by DFG grant PE 2299/1-1. Email: [email protected] Email: [email protected]
http://dx.doi.org/10.1016/j.endm.2017.07.059 1571-0653/© 2017 Elsevier B.V. All rights reserved.
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L. Özkahya, Y. Person / Electronic Notes in Discrete Mathematics 61 (2017) 955–961
Introduction and new results
For two graphs G and H and a proper edge coloring χ of G, a rainbow Hdecomposition of G is a partition of the edges of G into G1 ,. . . , Gt such that every Gi is either a single edge or is rainbow-colored and isomorphic to H, where a rainbow coloring of H assigns distinct colors to all edges of H. Rainbow H-decompositions present a variation on a well-studied topic of H-decompositions. Before we state our main results, we present the history of this general problem. For two graphs G and H, an H-decomposition of G is a partition of the edges of G into G1 ,. . . , Gt such that every Gi is either a single edge or is isomorphic to H. An H-decomposition of G with smallest possible t is called minimum and φ(G, H) = t denotes its cardinality. The following relation φ(G, H) = e(G) − (e(H) − 1)N (G, H) clearly holds, where N (G, H) denotes the maximum number of edge-disjoint copies of H in G. We write φ(n, H) := maxG∈Gn φ(G, H) where Gn denotes all graphs on n vertices. This function was studied first by Erd˝os, Goodman and P´osa [3], who were motivated by the problem of representing graphs by set intersections. They showed that φ(n, K3 ) = ex(n, K3 ), where ex(n, F ) denotes the maximum size of a graph on n vertices, that does not contain H as a subgraph. Moreover, these authors proved in [3] that the only graph that maximizes this function is the complete balanced bipartite graph. Consequently, they conjectured that φ(n, Kr ) = ex(n, Kr ) and the only optimal graph is the Tur´an graph Tr−1 (n), the complete balanced (r − 1)-partite graph on n vertices, where the sizes of the partite sets differ from each other by at most one. Bollob´as [2] verified this conjecture by showing that φ(n, Kr ) = ex(n, Kr ) for all n ≥ r ≥ 3. Pikhurko and Sousa [12] proved that for any fixed graph H with chromatic number r ≥ 3, φ(n, H) = ex(n, H) + o(n2 ) and they made the following conjecture. Conjecture 1.1 For any graph H with chromatic number at least 3, there is an n0 = n0 (H) such that φ(n, H) = ex(n, H) for all n ≥ n0 . This conjecture has been verified by Sousa for clique extensions of order r ≥ 4 (n ≥ r) [15], the cycles of length 5 (n ≥ 6) and 7 (n ≥ 10) [14,16]. In a previous work [11] we verified Conjecture 1.1 for edge-critical graphs, where a graph H is edge-critical if there exists an edge e ∈ E(H) such that χ(H) > χ(H − e). Later Allen, B¨ottcher and Person [1] obtained general upper bounds for all graphs H that improve the error term in the result of Pikhurko and Sousa [12] and extend the result on the edge-critical case.
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Recently, Liu and Sousa [10] studied the following colored variant of the H-decomposition problem. Let φk (G, H) be the smallest number t such that any graph G of order n and any coloring of its edges with k colors admits a monochromatic H-decomposition with at most t parts, where H is a complete graph. Later, Liu, Pikhurko and Sousa [9] generalized this by investigating φk (G, H), which is the smallest number t such that any graph G of order n and any coloring of its edges with k colors admits a monochromatic H = {H1 , . . . , Hk }-decomposition in t parts such that each part is either a single edge or forms a monochromatic copy of Hi in color i, for some 1 ≤ i ≤ k. Extending the results of Liu and Sousa [10], they solve this problem when each graph in H is a clique and n ≥ n0 (H) is sufficiently large. 1.1
Main Theorem
We call a rainbow H-decomposition of G under the proper edge coloring χ into t parts (i.e. each part is either a rainbow H or single edge) with the smallest possible t minimum and φR χ (G, H) = t denotes its cardinality. It is R R not difficult to see that φR (G, H) = e(G)−(e(H)−1)N χ χ (G), where Nχ (G, H) denotes the maximum number of edge-disjoint rainbow copies of H under this proper coloring χ of G. In this paper, we study the function φR (n, H) := max max φR χ (G, H), G∈Gn
χ
where we maximize over all graphs G from Gn and over all proper colorings χ of G. We will refer to decompositions that attain φR (n, H) as minimum rainbow H-decompositions of graphs. Observe that φR χ (G, H) ≥ φ(G, H) for any graph G and any proper edge coloring χ of G, otherwise it means that there are more than N (G, H) edgedisjoint rainbow copies of H in G under χ, which is a contradiction. Therefore, we have φR (n, H) ≥ φ(n, H). On the other hand, maxχ φR χ (G, H) ≤ e(G), where equality is achieved when there is no copy of H in G or there is a proper edge coloring of G without rainbow copies of H. We prove the following result for any (non-bipartite) clique Kr . Theorem 1.2 For any r ≥ 3 there is an n0 such that any graph G on n ≥ n0 vertices with some proper edge coloring χ that satisfies φR χ (G, Kr ) ≥ ex(n, Kr ) must in fact be isomorphic to the Tur´ an graph Tr−1 (n). In particular, φR (n, Kr ) = ex(n, Kr ) for all n ≥ n0 , and the only graph attaining φR (n, Kr ) is the Tur´ an graph Tr−1 (n).
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We also obtain generalizations of the result of Pikhurko and Sousa [12] on φ(n, H) = ex(n, H) + o(n2 ) (for non-bipartite H) and on our result from [11] about φ(n, H) = ex(n, H) for edge-critical graphs H, which read as follows. Theorem 1.3 For any edge-critical graph H with chromatic number at least 3, there is an n0 = n0 (H) such that φR (n, H) = ex(n, H) for all n ≥ n0 . Moreover, the only graph attaining φR (n, H) is the Tur´ an graph Tχ(H)−1 (n). Theorem 1.4 Let H be any graph of chromatic number at least three. Then we have φR (n, H) = ex(n, H) + o(n2 ). The proofs use essentially the same techniques as Theorem 1.2.
2
Proof sketches
To obtain Theorem 1.2, we first prove the stability for cliques for the function φ(G, Kr ) and then we amplify it to the stability for cliques for the rainbow function φR ori and Tuza [5] and of Hoi [6], we obtain χ (G, Kr ). By results of Gy˝ the following stability result. Proposition 2.1 For every r ≥ 3 and for every γ > 0 there exist ε > 0 and n0 ∈ N such that for every graph G on n ≥ n0 vertices the following is true. If φ(G, Kr ) ≥ ex(n, Kr ) − εn2 2 ˙ r−1 with r−1 then there exists a partition of V (G) = V1 ∪˙ . . . ∪V i=1 e(Vi ) < γn . We remark, that the stability result that we proved in [11, Lemma 4] was applicable only to graphs H with χ(H) ≥ 3 that are not cliques. The proof of Theorem 1.2 relies on the following stability theorem for proper edge colorings and is our main contribution. Theorem 2.2 For every r ≥ 3 and for every γ > 0 there exist ε > 0 and n0 ∈ N such that for every graph G on n ≥ n0 vertices with any proper coloring χ of its edges the following is true. If 2 φR χ (G, Kr ) ≥ ex(n, Kr ) − εn
˙ r−1 with then there exists a partition of V (G) = V1 ∪˙ . . . ∪V
r−1 i=1
e(Vi ) < γn2 .
Notice that the case r = 3 is covered by Proposition 2.1, as any proper edge coloring colors a triangle with different colors.
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Before giving some details of the proof of Theorem 1.2, we state the main auxiliary lemma that we use. This lemma shows that for any graph G on n vertices and any proper edge coloring χ, the numbers (of edge-disjoint copies) N (G, Kr ) and NχR (G, Kr ) differ by at most o(n2 ). Lemma 2.3 For all r ≥ 3 and any c > 0 there exists an n0 such that the following holds. In any proper edge coloring χ of a graph G on n ≥ n0 vertices we have N (G, Kr ) ≤ NχR (G, Kr ) + cn2 . Proof sketch. We fix a collection of edge-disjoint (not necessarily rainbow) copies of Kr of maximum cardinality. The proof proceeds by an application of the regularity lemma of Szemer´edi [17]. Then we identify various ε-regular pairs of not too small density that build r-partite graphs, where most edgedisjoint copies of Kr ‘live’. We need to argue then that we can find roughly that many rainbow copies instead. To do so, we would like to apply a Theorem of Pippenger and Spencer [13] and before we appropriately split ε-regular pairs, similarly as is done for example in [12]. This time however, the density of these ‘split’ ε-regular pairs may be much lower than ε, which would make the usual approach to apply the counting lemma (and then to apply a Theorem of Pippenger and Spencer) impossible. Our solution is to count the rainbow copies of Kr before splitting the ε-regular pairs (of sufficiently high density) and only then to split them randomly. All we need to estimate is then the number of rainbow copies that will be in (random) subgraphs of ε-regular rpartite graphs, which we do by applying a concentration result of Kim and Vu [8]. 2 Proof of Theorem 2.2. For given r ≥ 3 and γ > 0 let ε = εT 2.1 be as asserted by Proposition 2.1. We choose c := ε /e(Kr ), we also set ε := c and assume that n is large enough so that Lemma 2.3 is applicable. Let now G be a graph on n vertices with a proper coloring χ of its edges 2 such that φR By Lemma 2.3, we have χ (G, Kr ) ≥ ex(n, Kr ) − εn holds. R 2 N (G, Kr ) ≤ Nχ (G, Kr ) + cn (for n large enough). Therefore, φR χ (G, Kr ) = e(G) − (e(Kr ) − 1)NχR (G, Kr ) ≥ ex(n, Kr ) − εn2 implies φ(G, Kr ) = e(G) − (e(Kr ) − 1)N (G, Kr ) ≥ 2 e(G) − (e(Kr ) − 1)(NχR (G, Kr ) + cn2 ) ≥ φR χ (G, Kr ) − (e(Kr ) − 1)cn ≥ ex(n, Kr ) − εn2 − (e(Kr ) − 1)cn2 = ex(n, Kr ) − e(Kr )cn2 = ex(n, Kr ) − ε n2 , and Proposition 2.1 yields the desired partition.
2
Proof sketch of Theorem 1.2 We will apply Theorem 2.2 in the form
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when ε = 0, and we choose γ sufficiently small. We assume that there is a graph G on n vertices (n large enough) such that there exists a proper edge coloring of G with φR χ (G, Kr ) ≥ ex(n, Kr ) and G is not isomorphic to the Tur´an graph Tr−1 (n). By following the steps of [11], we apply first Claim 7 from [11] and assume that φR χ (G) = ex(n, H) + m for some m ≥ 0. We obtain a subgraph G of G on n vertices such that δ(G ) ≥ δ(Tr−1 (n )) and φR χ (G , Kr ) ≥ ex(n , Kr ) + m. Then, Theorem 2.2 asserts the existence of a partition of G into r − 1 parts that maximizes the number of edges between different parts so that the number m2 of edges within the partition classes is not zero but is at most γn2 . It is observed that the orders of these r − 1 parts are almost equal due to the maximality condition (Claim 8 from [11, Theorem 3]). This implies that e(G )≤ ex(n , Kr ) + m2 . m2 + 1 many rainbow copies of Kr . (r2)−1 We find these copies iteratively (cf. Claim 9 from [11, Theorem 3]). The only difference to the embedding in [11] is that we need to find at each iteration a copy of Kr , which under the edge coloring χ is rainbow. This is done by finding first a complete (r − 1)-partite graph F with parts of size at least 8(r − 1)3 , so that one of the parts contains an edge (Claim 9 from [11, Theorem 3], this basically follows from an application of Theorem of Erd˝os and Stone [4]). A rainbow copy of Kr is guaranteed to exist in such a graph F by a result of Keevash, Mubayi, and Verstra¨ete [7, Lemma 2.2]. This allows us Sudakov,
In the final step, we find at least
m2 + 1 many rainbow copies of Kr in G under the edge (r2)−1 coloring χ. Clearly, this implies that
to find at least
m r 2 r
−1 + 1 < ex(n , Kr ), φR χ (G , Kr ) ≤ ex(n , Kr ) + m2 − 2 − 1 2 a contradiction.
2
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