The following sharpening of Tur\'an's theorem is proved. Let $T_{n,p}$ denote
the complete $p$--partite graph of order $n$ having the maximum number of
edges. If $G$ is an $n$-vertex $K_{p+1}$-free graph with $e(T_{n,p})-t$ edges
then there exists an (at most) $p$-chromatic subgraph $H_0$ such that
$e(H_0)\geq e(G)-t$.
  Using this result we present a concise, contemporary proof (i.e., one
applying Szemer\'edi's regularity lemma) for the classical stability result of
Simonovits.Comment: 4 pages plus reference

Füredi, Zoltán

English

arXiv

Let Tn,p denote the complete p-partite graph of order n having the maximum number of edges. The following sharpening of Turán's theorem is proved. Every Kp+1-free graph with n vertices and e(Tn,p)-t edges contains a p-partite subgraph with at least e(Tn,p)-2t edges. As a corollary of this result we present a concise, contemporary proof (i.e., one applying the Removal Lemma, a corollary of Szemerédi's regularity lemma) for the classical stability result of Simonovits [25]. © 2015 Elsevier Inc

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arXiv:1501.03129v1  [math.CO]  13 Jan 2015A proof of the stability of extremal graphs,Simonovits’ stability from Szemere´di’s regularityZolta´n Fu¨redi ∗Alfre´d Re´nyi Institute of Mathematics,13–15 Rea´ltanoda Street, 1053 Budapest, Hungary.E-mail: z-furedi@illinois.eduAbstractThe following sharpening of Tura´n’s theorem is proved. Let Tn,p denote the complete p–partite graph of order n having the maximum number of edges. If G is an n-vertex Kp+1-freegraph with e(Tn,p) − t edges then there exists an (at most) p-chromatic subgraph H0 suchthat e(H0) ≥ e(G)− t.Using this result we present a concise, contemporary proof (i.e., one using Szemere´di’sregularity lemma) for the classical stability result of Simonovits [21].1 The Tura´n problemGiven a graph G with vertex set V (G) and vertex set E(G) its number of edges is denoted by e(G).The neighborhood of a vertex x ∈ V is denoted by N(x), note that x /∈ N(x). For any A ⊂ Vthe restricted neighborhood NG(x|A) stands for N(x) ∩ A. Similarly, degG(x|A) := |N(x) ∩ A|.If the graph is well understood from the text we leave out subscripts. The Tura´n graph Tn,p isthe largest p-chromatic graph having n vertices, n, p ≥ 1. Given a partition (V1, . . . , Vp) of Vthe complete multipartite graph K(V1, . . . , Vp) has vertex set V and all the edges joining distinctpartite sets. A△B stands for the symmetric difference of the sets A and B. For further notationsand notions undefined here see, e.g., the monograph of Bolloba´s [4].Tura´n [23] proved that if an n vertex graph G has at least e(Tn,p) edges then it contains acomplete subgraph Kp+1, except if G = Tn,p. Given a class of graphs L, a graph G is called∗Research supported in part by the Hungarian National Science Foundation OTKA 104343, by the SimonsFoundation Collaboration Grant #317487, and by the European Research Council Advanced Investigators Grant267195.2010 Mathematics Subject Classifications: 05C35. furedi˙2015˙01˙12˙sim˙stabilityKey Words: Tura´n number, extremal graphs, stability. January 14, 20151Z. Fu¨redi: Stability of extremal graphs 2L-free if it does not contain any subgraph isomorphic to any member of L. The Tura´n numberex(n,L) is defined as the largest size of an n-vertex, L-free graph. Erdo˝s and Simonovits [11]gave a general asymptotic for the Tura´n number as follows. Let p + 1 := min{χ(L) : L ∈ L}.Thenex(n,L) =(1−1p)(n2)+ o(n2) as n→∞. (1)They also showed that if G is an extremal graph, i.e., e(G) = ex(n,L), then it can be obtainedfrom Tn,p by adding and deleting at most o(n2) edges. This result is usually called Erdo˝s–Stone–Simonovits theorem, although it was proved first in [11], but indeed (1) easily follows from aresult of Erdo˝s and Stone [12].The aim of this paper is to present a new proof for the following stronger version of (1), astructural stability theorem, originally proved by Erdo˝s and Simonovits [11], Erdo˝s [7, 8], andSimonovits [21]. For every ε > 0 and forbidden subgraph class L there is a δ > 0, and n0 suchthat if n > n0 and G is an n-vertex, L-free graph withe(G) ≥(1−1p)(n2)− δn2,then|E(Gn)△ E(Tn,p)| ≤ εn2. (2)I.e., one can change (add and delete) at most εn2 edges ofG and obtain a complete p-partite graph.In other words, if an n-vertex L-free graph G is almost extremal, min{χ(L) : L ∈ L} = p+1, thenthe structure of G is close to a p-partite Tura´n graph. This result is usually called Simonovits’stability of the extremum.Our main tool is a very simple proof for the case L = {Kp+1}.Stability results are usually more important than their extremal counterparts. That is whythere are so many investigations concerning the edit distance of graphs. Let G1 = (V, E1) andG2 = (V, E2) be two (finite, undirected) graphs on the same vertex set. The edit distance fromG1 to G2 is ed(G1, G2) := |E1△E2|. Let P denote a class of graphs and G be a fixed graph. Theedit distance from G to P is ed(G,P) = min{ed(G,F ) : F ∈ P, V (G) = V (F )}. This notion wasexplicitly introduced in [3], Alon and Stav [2] proved connections with Tura´n theory. For morerecent results see Martin [18].2 How to make a Kp+1-free graph p-chromaticEver since Erdo˝s [5] observed that one can always delete at most e/2 edges from any graph G tomake it bipartite there are many generalizations and applications of this (see, e.g., Alon [1] for amore precise form). Here we prove a version dealing with a narrower class of graphs. Recall thate(Tn,p) := max{e(K(V1, . . . , Vp)) :∑|Vi| = n}, the maximum size of a p-chromatic graph.Z. Fu¨redi: Stability of extremal graphs 3Theorem 1 Suppose that Kp+1 6⊂ G, |V (G)| = n, t ≥ 0, ande(G) = e(Tn,p)− t.Then there exists an (at most) p-chromatic subgraph H0, E(H0) ⊂ E(G) such thate(H0) ≥ e(G) − t.Corollary 2 (Stability of ex(n,Kp+1)) Suppose that G is Kp+1-free with e(G) ≥ e(Tn,p) − t.Then there is a complete p-chromatic graph K := K(V1, . . . , Vp) with V (K) = V (G), such that|E(G)△ E(K)| ≤ 3t.Indeed, delete t edges of G to obtain the p-chromatic H0. Since e(H0) ≥ e(Tn,p)− 2t one can addat most 2t edges to make it a complete p-partite graph. (Here Vi = ∅ is allowed). ✷There are other more exact stability results, e.g., Hanson and Toft [15] showed that fort < n/(2p) − O(1) the graph G itself is p-chromatic, there is no need to delete any edge. Someresults of E. Gyo˝ri [14] implies a stronger form, namely that e(H0) ≥ e(G) − O(t2/n2). Erdo˝s,Gyo˝ri, and Simonovits [10] considers only dense triangle-free graphs. The advantage of ourTheorem 1 is that it contains no ε, δ, n0, it is true for every n, p and t.The inequality in Corollary 2 is simple because we estimate the edit distance of G from a notnecessarily balanced p partite graph K. If we are interested in ed(G,Tn,p) then we can use thefollowing inequality. If e(K((V1, . . . , Vp)) ≥ e(Tn,p)− 2t, then a simple calculation shows that thesizes of Vi’s should be ’close’ to n/p (more exactly we get 4t ≥∑i(|Vi| − (n/p))2) and henceed(K,Tn,p) ≤ n√t/p (3)Proof of Theorem 1. We find the large p-partite subgraph H0 ⊂ G by analyzing Erdo˝s’degree majorization algorithm [6] what he used to prove Tura´n’s theorem. Our input is theKp+1-free graph G and the output is a partition V1, V2, . . . , Vp of V (G) such that∑i e(G|Vi) ≤ t.Let x1 ∈ V (G) be a vertex of maximum degree and let V1 := V \N(x1), V+1:= V \ V1. Notethat x1 ∈ V1 and deg(x) ≤ |V+1| for all x ∈ V1. Hence2e(G|V1) + e(V1, V+1) =∑x∈V1deg(x) ≤ |V1||V+1|.In general, define V +0:= V (G) and let xi be a vertex of maximum degree of the graph G|V+i−1,let Vi := V+i−1 \N(xi), V+i := V (G) \ (V1 ∪ · · · ∪ Vi). We have xi ∈ Vi, deg(xi, V+i−1) = |V+i | and2e(G|Vi) + e(Vi, V+i ) =∑x∈Videg(x|V +i−1) ≤ |Vi||V+i |. (4)The procedure stops in s steps when no more vertices left, i.e., if V1 ∪ · · · ∪Vs = V (G). Note thats ≤ p because {x1, x2, . . . , xs} span a complete graph.Z. Fu¨redi: Stability of extremal graphs 4Add up the left hand sides of (4) for 1 ≤ i ≤ s, we get e(G) + (∑i e(G|Vi)). The sum of theright hand sides is exactly e(K(V1, V2, . . . , Vs)). We obtaine(Tn,p)− t+(∑ie(G|Vi))= e(G) +(∑ie(G|Vi))≤ e(K(V1, V2, . . . , Vp)) ≤ e(Tn,p)implying∑i e(G|Vi) ≤ t. ✷3 Az application of the Removal LemmaWe only need a simple consequence of Szemere´di’s Regularity Lemma. Recall that the graphH contains a homomorphic image of F if there is a mapping ϕ : V (F ) → V (H) such that theimage of each F -edge is an H-edge. There is a ϕ : V (F )→ V (Ks) homomorphism if and only ifs ≥ χ(H). If there is no any ϕ : V (F )→ V (H) homomorphism then H is called hom(F )-free.Lemma 3 (A simple form of the Removal Lemma) For every α > 0 and graph F thereis an n1 such that if n > n1 and G is an n-vertex, F -free graph then it contains a hom(F )-freesubgraph H with e(H) > e(G)− αn2.This means that H does not contain any homomorphic image of F as a subgraph, especiallyif χ(F ) = p + 1 then H is Kp+1-free. The Removal Lemma can be attributed to Ruzsa andSzemere´di [20]. It appears in a more explicit form in [9] and [13]. For a survey of applica-tions of Szemere´di’s regularity lemma in graph theory see Komlo´s-Simonovits [16] or Komlo´s-Shokoufandeh-Simonovits-Szemere´di [17].Proof of (2) using Lemma 3 and Corollary 2. Suppose that F ∈ L, χ(F ) = p+ 1 and α > 0 anarbitrary real. Suppose that G is F -free with n > n1(F,α) and e(G) > e(Tn,p) − αn2. We haveto show that the edit distance of G to Tn,p is small. First we claim that the edit distance of Gto a complete p–partite graph K(V1, . . . , Vp) is at most 7αn2. Indeed, using the Removal Lemmawe obtain a Kp+1-free subgraph H of G such that e(H) > e(G) − αn2 > e(Tn,p) − 2αn2. ApplyTheorem 1 to H we get a p–partite H0 with e(H0) > e(Tn,p)− 4αn2. Then Corollary 2 yields aK := K(V1, . . . , Vp) with ed(K,H) < 6αn2, giving ed(K,G) ≤ 7αn2.Since e(K) ≥ e(H0) > e(Tn,p) − 4αn2, we can use (3) with t = 2αn2 to get ed(K,Tn,p) ≤n2√2α/p. This completes the proof that ed(G,Tn,p) ≤ (7α +√2α/p)n2. ✷Acknowledgments. The author is greatly thankful to M. Simonovits for helpful conversations.This result was first presented in a public lecture at Charles University, Prague, July 2006. Sincethen there were several references to it, e.g., in [19].Z. Fu¨redi: Stability of extremal graphs 5References[1] N. Alon: Bipartite subgraphs. Combinatorica 16 (1996), 301–311.[2] N. Alon and U. Stav: The maximum edit distance from hereditary graph properties. J. Combin.Theory, Ser. B 98 (2008), 672–697.[3] M. Axenovich and R. Martin: Avoiding patterns in matrices via a small number of changes. SIAMJ. Discrete Math. 20 (2006), 49–54.[4] B. Bolloba´s: Extremal Graph Theory. London Math. Soc. Monographs, Academic Press, 1978.[5] P. Erdo˝s: On even subgraphs of graphs. (Hungarian), Mat. Lapok 18 (1967), 283–288.[6] P. Erdo˝s: On the graph theorem of Tura´n. (Hungarian) Mat. Lapok 21 (1970), 249–251 (1971).[7] P. Erdo˝s: Some recent results on extremal problems in graph theory. Theory of Graphs, InternationalSymp. Rome, 1966, 118–123.[8] P. Erdo˝s: On some new inequalities concerning extremal properties of graphs. Theory of Graphs,Proc. Coll. Tihany, (Hungary) 1966, 77–81.[9] P. Erdo˝s, P. Frankl, and V. Ro¨dl: The asymptotic number of graphs not containing a fixed subgraphand a problem for hypergraphs having no exponent. Graphs Combin. 2 (1986), 113–121.[10] P. Erdo˝s, E. Gyo˝ri, and M. Simonovits: How many edges should be deleted to make a triangle-freegraph bipartite? Sets, graphs and numbers (Budapest, 1991), 239–263, Colloq. Math. Soc. Ja´nosBolyai, 60, North-Holland, Amsterdam, 1992.[11] P. Erdo˝s and M. Simonovits: A limit theorem in graph theory. Studia Sci. Math. Hungar. 1 (1966),51–57.[12] P. Erdo˝s and A. H. Stone: On the structure of linear graphs. Bull. Amer. Math. Soc. 52 (1946),1089–1091.[13] Z. Fu¨redi: Extremal hypergraphs and combinatorial geometry. Proc. Internat. Congress of Mathe-maticians, Vol. 1, 2 (Zu¨rich, 1994), 1343–1352, Birkha¨user, Basel, 1995.[14] E. Gyo˝ri: On the number of edge disjoint cliques in graphs of given size. Combinatorica 11 (1991),231–243.[15] D. Hanson and B. Toft: k-saturated graphs of chromatic number at least k. Ars Combin. 31 (1991),159–164.[16] J. Komlo´s and M. Simonovits: Szemere´di’s regularity lemma and its applications in graph theory.Combinatorics, Paul Erdo˝s is eighty, Vol. 2 (Keszthely, 1993), 295–352, Bolyai Soc. Math. Stud., 2,Budapest, 1996.[17] J. Komlo´s, Ali Shokoufandeh, M. Simonovits, and E. Szemere´di: The regularity lemma and itsapplications in graph theory. Theoretical aspects of computer science (Tehran, 2000), 84–112, LectureNotes in Comput. Sci., 2292, Springer, Berlin, 2002.[18] R. R. Martin: On the computation of edit distance functions. Discrete Math. 338 (2015), 291–305.[19] D. Mubayi: Books versus triangles, Journal of Graph Theory 70 (2012), 171–179.[20] I. Z. Ruzsa and E. Szemere´di: Triple systems with no six points carrying three triangles. Combina-torics Proc. Fifth Hungarian Colloq., Keszthely, 1976, Vol. II, pp. 939–945, Colloq. Math. Soc. Ja´nosBolyai, 18, North-Holland, Amsterdam-New York, 1978.Z. Fu¨redi: Stability of extremal graphs 6[21] M. Simonovits: A method for solving extremal problems in graph theory. Theory of Graphs, (P. Erdo˝sand G. Katona, eds.) Proc. Coll. Tihany (1966), 279–319.[22] E. Szemere´di: Regular partitions of graphs. Problemes Combinatoires et Theorie des Graphes (ed.I.-C. Bermond et al.), CNRS, 260 Paris, 1978, pp. 399–401.[23] P. Tura´n: On an extremal problem in graph theory. Matematikai Lapok, 48 (1941), 436–452 (inHungarian). Reprinted in English in: Collected papers of Paul Tura´n. Akade´miai Kiado´, Budapest,1989. Vol 1–3.

A proof of the stability of extremal graphs, Simonovits' stability from Szemer\'edi's regularity

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