Making $K_{r+1}$-Free Graphs $r$-partite

The Erd\H{o}s-Simonovits stability theorem states that for all \epsilon>0 there exists \alpha>0 such that if G is a K_{r+1}-free graph on n vertices with e(G)>ex(n,K_{r+1}) - \alpha n^2, then one can remove \epsilon n^2 edges from G to obtain an r-partite graph. F\"uredi gave a short proof that one can choose \alpha=\epsilon. We give a bound for the relationship of \alpha and \varepsilon which is asymptotically sharp as \epsilon \to 0.


Introduction
Erdős asked how many edges need to be removed in a triangle-free graph on n vertices in order to make it bipartite. He conjectured that the balanced blow-up of C 5 with class sizes n/5 is the worst case, and hence 1/25n 2 edges would always be sufficient. Together with Faudree, Pach and Spencer [5], he proved that one can remove at most 1/18n 2 edges to make a triangle-free graph bipartite. Further, Erdős, Győri and Simonovits [6] proved that for graphs with at least n 2 /5 edges, an unbalanced C 5 blow-up is the worst case. For r ∈ N, denote D r (G) the minimum number of edges which need to be removed to make G r-partite. Theorem 1.1 (Erdős, Győri and Simonovits [6]). Let G be a K 3 -free graph on n vertices with at least n 2 /5 edges. There exists an unbalanced C 5 blow-up of H with e(H) ≥ e(G) such that This proved the Erdős conjecture for graphs with at least n 2 /5 edges. A simple probabilistic argument (e.g. [6]) settles the conjecture for graphs with at most 2/25n 2 edges. A related question was studied by Sudakov; he determined the maximum number of edges in a K 4 -free graph which need to be removed in order to make it bipartite [13]. This problem for K 6 -free graphs was solved by Hu, Lidický, Martins, Norin and Volec [10]. We will study the question of how many edges in a K r+1 -free graph need at most to be removed to make it r-partite. For n ∈ N and a graph H, ex(n, H) denote the Turán number, i.e. the maximum number of edges of an H-free graph. The Erdős-Simonovits theorem [7] for cliques states that for every ε > 0 there exists Füredi [8] gave a nice short proof of the statement that a K r+1 -free graph G on n vertices with at least ex(n, K r+1 ) − t edges satisfies D r (G) ≤ t; thus providing a quantitative version of the Erdős-Simonovits theorem. In [10] Füredi's result was strengthened for some values of r. For small t, we will determine asymptotically how many edges are needed. For very small t, it is already known [3] that G has to be r-partite. Theorem 1.2 (Brouwer [3]). Let r ≥ 2 and n ≥ 2r + 1 be integers. Let G be a K r+1 -free graph on n vertices with e(G) ≥ ex(n, K r+1 ) − n r + 2. Then This result was rediscovered in [1,9,11,15]. We will study K r+1 -free graphs on fewer edges. Theorem 1.3. Let r ≥ 2 be an integer. Then for all n ≥ 3r 2 and for all 0 ≤ α ≤ 10 −7 r −12 the following holds. Let G be a K r+1 -free graph on n vertices with where t = αn 2 , then Note that we did not try to optimize our bounds on n and α in the theorem. One could hope for a slightly better error term of 30r 3 α 5/3 in Theorem 1.3, but the next natural step would be to prove a structural version. To state this structural version we introduce some definitions. The blow-up of a graph G is obtained by replacing every vertex v ∈ V (G) with finitely many copies so that the copies of two vertices are adjacent if and only if the originals are. For two graphs G and H, we define G⊗H to be the graph on the vertex set This conjecture can be seen as a generalization of Theorem 1.1. We will prove that Theorem 1.3 is asymptotically sharp by describing an unbalanced blow-up of K r−2 ⊗C 5 that needs at least that many edges to be removed to make it r-partite. This gives us a strong evidence that Conjecture 1.4 is true.
In Kang-Pikhurko's proof [11] of Theorem 1.2 the case e(G) = ex(n, K r+1 )−⌊n/r⌋+1 is studied. In this case they constructed a family of K r+1 -free non-r-partite graphs, which includes our extremal graph, for that number of edges. We recommend the interested reader to read the excellent survey [12] by Nikiforov. He gives a good overview on further related stability results, for example on guaranteeing large induced r-partite subgraphs of K r+1 -free graphs. We organize the paper as follows. In Section 2 we prove Theorem 1.3 and in Section 3 we give the sharpness example, i.e. we prove Theorem 1.5.
This also implies that t ≥ r because n ≥ 3r 2 . During our proof we will make use of Turán's theorem and a version of Turán's theorem for r-partite graphs multiple time.
Turán's theorem [14] determines the maximum number of edges in a K r+1 -free graph.
For a proof of this folklore result see for example [2,Lemma 3.3]. We denote the maximum degree of G by ∆(G). For two disjoint subsets U, W of V (G), write e(U, W ) for the number of edges in G with one endpoint in U and the other endpoint in W . We write e c (U, W ) for the number of non-edges between U and W , i.e. e c (U, W ) = |U ||W | − e(U, W ). Füredi [8] used Erdős' degree majorization algorithm [4] to find a vertex partition with some useful properties. We include a proof for completeness. (Füredi [8]). Let t, r, n ∈ N and G be an n-vertex K r+1 -free graph with e(G) ≥ ex(n, K r+1 ) − t. Then there exists a vertex partition

Lemma 2.3
(1) Since G is K r+1 -free this process stops at i ≤ r and thus gives a vertex partition In the proof of [8,Theorem 2], it is shown that the partition obtained from this algorithm satisfies Let H be the complete r-partite graph with vertex set V (G) and all edges between V i and V j for 1 ≤ i < j ≤ r. The graph H is r-partite and thus has at most ex(n, K r+1 ) edges. Finally, since G has at most t edges not in H and at least ex(n, K r+1 ) − t edges total, at most 2t edges of H can be missing from G, giving us and proving the last inequality.
For this vertex partition we can get bounds on the class sizes.
Proof. We know that Thus, we can conclude that Now, let x = |V 1 | − n/r. Then, Combining this with (2), we get |x| ≤ √ r + 4t ≤ 5 √ αn, and thus In a similar way we get the bounds on the sizes of the other classes.
Lemma 2.5. The graph G contains r vertices x 1 ∈ V 1 , . . . , x r ∈ V r which form a K r and for every i Proof. By Lemma 2.5 we can find vertices x 1 , . . . , x r forming a K r and having deg( Since G is K r+1 -free, the X i s are independent sets. Now we bound the size of X i using the bounds on the V i s. Since every x j has at most |V j | + 5rαn non-neighbors, we get A vertex v ∈ V (G) cannot be incident to all of the vertices x 1 , . . . , x r , because G is K r+1 -free. Further, every vertex from X is not incident to at least two of the vertices Combining (3) with (4), we conclude that |X| ≤ 5r 2 αn.
For the upper bound on the sizes of the sets X i we get We now bound the number of non-edges between X 1 , . . . , X r . Proof.
This gives the statement of the lemma.
Proof. By Lemma 2.7, For every vertex u ∈ X there is no K r in N X 1 (u) ∪ · · · ∪ N Xr (u). Thus, by applying Theorem 2.2 and Lemma 2.8, we get Bound (5) implies in particular that every vertex u ∈ X has degree at most C(α)n to one of the sets X 1 , . . . , X r , i.e.
Therefore, we can partitionX = A 1 ∪ . . . ∪ A r such that every vertex u ∈ A i has at most C(α)n neighbors in X i . By the following calculation, for every vertex u ∈X the second smallest neighborhood to the X i 's has size at least α 1/3 n.
where we used the definition ofX and Lemma 2.6. Combining the lower bound on the second smallest neighborhood with (5) we can conclude that for every u ∈X Hence, we can partitionX = B 1 ∪ . . . ∪ B r such that every vertex u ∈ B i has at most C(α)α −1/3 n neighbors in X i . Consider the partition A 1 ∪B 1 ∪X 1 ,A 2 ∪B 2 ∪X 2 , . . . , A r ∪ B r ∪ X r . By removing all edges inside the classes we end up with an r-partite graph. We have to remove at most edges. We have used (6), (7) and the fact that which can be seen by setting z = (1−d)k and finding the maximum of f (z) := z 1 − z r which is obtained at z = 2r/3.

Sharpness Example
In this section we will prove Theorem 1.5, i.e. that the leading term from Theorem 1.3 is best possible.
Proof of Theorem 1.5. Let G be the graph with vertex set V (G) = A ∪ X ∪ B ∪ C ∪ D ∪ X 1 · · · ∪ X r−2 , where all classes A, X, B, C, D, X 1 , . . . , X r−2 form independent sets; A, X, B, C, D form a complete blow-up of a C 5 , where the classes are named in cyclic order; and for each 1 ≤ i ≤ r − 2, every vertex from X i is incident to all vertices from V (G) \ X i . The sizes of the classes are |X| = 2r 3 αn, |A| = |B| = α 3 n, |C| = |D| = 1 − 2r 3 α r n− α 3 n, |X i | = 1 − 2r 3 α r n.