aa r X i v : . [ m a t h . C O ] J un Making a K -free graph bipartite Benny Sudakov ∗ Abstract
We show that every K -free graph G with n vertices can be made bipartite by deleting at most n / n/
3. This proves an old conjecture of P. Erd˝os.
The well-known Max Cut problem asks for the largest bipartite subgraph of a graph G . This problemhas been the subject of extensive research, both from the algorithmic perspective in computer scienceand the extremal perspective in combinatorics. Let n be the number of vertices and e be the numberedges of G and let b ( G ) denote the size of the largest bipartite subgraph of G . The extremal part of MaxCut problem asks to estimate b ( G ) as a function of n and e . This question was first raised almost fortyyears ago by P. Erd˝os [8] and attracted a lot of attention since then (see, e.g., [3, 2, 4, 1, 16, 11, 10, 5, 7]).It is well known that every graph G with e edges can be made bipartite by deleting at most e/ b ( G ) ≥ e/
2. To see this just consider a random partition of vertices of G into twoparts V , V and estimate the expected number of edges in the cut ( V , V ). A complete graph K n on n vertices shows that the constant 1 / forbidden subgraph H . We call such graphs H -free. Indeed, using sparserandom graphs one can easily construct a graph G with e edges such that it has no short cycles butcan not be made bipartite by deleting less than e/ − o ( e ) edges. Such G is clearly H -free for everyforbidden graph H which is not a forest. It is a natural question to estimate the error term b ( G ) − e/ G ranges over all H -free graph with e edges. We refer interested reader to [3, 2, 1, 16], where suchresults were obtained for various forbidden subgraphs H .In this paper we restrict our attention to dense ( e = Ω( n )) H -free graphs for which it is possibleto prove stronger bounds for Max Cut. According to a long-standing conjecture of Erd˝os [9], everytriangle-free graph on n vertices can be made bipartite by deleting at most n /
25 edges. This bound,if true, is best possible (consider an appropriate blow-up of a 5-cycle). Erd˝os, Faudree, Pach and ∗ Department of Mathematics, Princeton University, Princeton, NJ 08544. E-mail: [email protected] supported in part by NSF CAREER award DMS-0546523, NSF grant DMS-0355497, USA-Israeli BSF grant,and by an Alfred P. Sloan fellowship. G of order n it is enough to delete (1 / − ǫ ) n edges to make itbipartite. They also verify the conjecture for all graphs with at least n / K -free graphs. His old conjecture (see e.g., [10]) asserts that it isenough to delete at most (1 + o (1)) n / K -free graph on n vertices. Herewe confirm this in the following strong form. Theorem 1.1
Every K -free graph G with n vertices can be made bipartite by deleting at most n / edges. Moreover, the only extremal graph which requires deletion of that many edges is a complete3-partite graph with parts of size n/ . This result can be used to prove the following asymptotic generalization.
Corollary 1.2
Let H be a fixed graph with chromatic number χ ( H ) = 4 . If G is a graph on n verticesnot containing H as a subgraph, then we can delete at most (1 + o (1)) n / edges from G to make itbipartite. Another old problem of Erd˝os, that is similar in spirit, is to determine the best local density in K r -free graphs for r ≥ G on n vertices should contain a set of n/ n /
50 edges. The blow-up of a 5-cycle in which we replace each vertex byan independent set of size n/ r > n/
2. In particular, their conjecture implies thatevery K -free graph on n vertices should contain a set of n/ n /
18 edges.Krivelevich [15] noticed that for regular graphs a bound in the local density problem implies abound for the problem of making the graph bipartite. Indeed, suppose n is even, G is a d -regular K -free graph on n vertices and S is a set of n/ dn/ P s ∈ S d ( s ) = 2 e ( S ) + e ( S, ¯ S )and dn/ P s / ∈ S d ( s ) = 2 e ( ¯ S ) + e ( S, ¯ S ), i.e. e ( S ) = e ( ¯ S ). Deleting the 2 e ( S ) edges within S or ¯ S makes the graph bipartite, so if we could find S spanning at most n /
18 edges we would delete atmost n / G bipartite. Unfortunately, the converse reasoning does not work. Nevertheless,we believe that the result of Theorem 1.1 provides some supporting evidence for conjecture of Chungand Graham.The rest of this short paper is organized as follows. The proof of our main theorem appears in thebeginning of next section. Next we show how to obtain Corollary 1.2 using this theorem together withwell known Szemer´edi’s Regularity Lemma [17] (see also [14]). The last section of the paper containssome concluding remarks and open questions. Notation.
We usually write G = ( V, E ) for a graph G with vertex set V = V ( G ) and edge set E = E ( G ), setting n = | V | and e = e ( G ) = | E ( G ) | . If X ⊂ V is a subset of the vertex set then G [ X ] denotes the restriction of G to X , i.e. the graph on X whose edges are those edges of G with2oth endpoints in X . We will write e ( X ) = e ( G [ X ]) and similarly, we write e ( X, Y ) for the numberof edges with one endpoint in X and the other in Y . N ( v ) is the set of vertices adjacent to a vertex v and d ( v ) = | N ( v ) | is the degree of v . For any two vertices u, v we denote by N ( u, v ) the set ofcommon neighbors of u and v , i.e., all the vertices adjacent to both of them. We will also write d ( u, v ) = | N ( u, v ) | . Finally if three vertices u, v, and w are all adjacent then they form a triangle in G and we denote this by △ = { u, v, w } . In this subsection we present the proof of our main theorem. We start with the following well knownfact (see, e.g., [1]), whose short proof we include here for the sake of completeness.
Lemma 2.1
Let G be a -partite graph with e edges. Then G contains a bipartite subgraph with atleast e/ edges. Proof.
Let V , . . . , V be a partition of vertices of G into four independent sets. Partition these setsrandomly into two classes, where each class contains exactly two of the sets V i . Consider a bipartitesubgraph H of G with these color classes. For each fixed edge ( u, v ) of G the probability that u and v will lie in the different classes is precisely (2 · / (cid:0) (cid:1) = 2 /
3. Therefore, by linearity of expectation,the expected number of edges in H is 2 e/
3, completing the proof. ✷ Next we need two simple lemmas.
Lemma 2.2
Let G be a graph with e edges and m triangles. Then it contains a triangle { u, v, w } such that d ( u, v ) + d ( u, w ) + d ( v, w ) ≥ me . Proof.
A simple averaging argument, using that P ( x,y ) ∈ E ( G ) d ( x, y ) = 3 m and Cauchy-Schwartzinequality, shows that there is a triangle { u, v, w } in G with d ( u, v ) + d ( u, w ) + d ( v, w ) ≥ m X { x,y,z } = △ (cid:16) d ( x, y ) + d ( x, z ) + d ( y, z ) (cid:17) = 1 m X ( x,y ) ∈ E ( G ) d ( x, y ) ≥ em P ( x,y ) ∈ E ( G ) d ( x, y ) e ! = (3 m ) me = 9 me . ✷ Lemma 2.3
Let G be a graph on n vertices with e edges and m triangles. Then G contains a bipartitesubgraph of size at least e /n − m/n . Proof.
Let v be a vertex of G and let e v denotes the number of edges spanned by the neighborhood N ( v ). Consider the bipartite subgraph of G whose parts are N ( v ) and its complement V ( G ) \ N ( v ).3t is easy to see that number of edges in this subgraph is P u ∈ N ( v ) d ( u ) − e v . Thus averaging over allvertices v we have that b ( G ) ≥ n X v (cid:18) X u ∈ N ( v ) d ( u ) − e v (cid:19) = 1 n X v d ( v ) − n X v e v (1) ≥ (cid:18) P v d ( v ) n (cid:19) − m/n = 4 e /n − m/n. Here we used Cauchy-Schwartz inequality together with identities P v e v = 3 m , P v d ( v ) = 2 e . ✷ Now we can obtain our first estimate on the Max Cut in K -free graphs. This result can be usedto prove the conjecture for graphs with ≤ n / Lemma 2.4
Let G be a K -free graph on n vertices with e edges. Then it contains a bipartite subgraphof size at least e/ e / (7 n ) . Proof.
Let v be a vertex of G and denote by e v the number of edges spanned by the neighborhoodof v . Consider a subgraph of G induced by the set N ( v ). This subgraph G [ N ( v )] has d ( v ) vertices, e v edges and contains no triangles, since G is K -free. Therefore by previous lemma (with m = 0) it hasa bipartite subgraph H of size at least 4 e v /d ( v ). Let ( A, B ) , A ∪ B = N ( v ) be the bipartition of H .Consider a bipartite subgraph H ′ of G with parts ( A ′ , B ′ ), where A ⊂ A ′ , B ⊂ B ′ and we place eachvertex v ∈ V ( G ) \ N ( v ) in A ′ or B ′ randomly and independently with probability 1 /
2. All edges of H are edges of H ′ , and each edge incident to a vertex in V ( G ) \ N ( v ) appears in H ′ with probability1 /
2. As the number of edges incident to vertices V ( G ) \ N ( v ) is e − e v , by linearity of expectation,we have b ( G ) ≥ E (cid:2) e ( H ′ ) (cid:3) ≥ ( e − e v ) / e v /d ( v ). By averaging over all vertices vb ( G ) ≥ e + 1 n X v (cid:16) e v /d ( v ) − e v / (cid:17) . (2)To finish the proof we take a convex combination of inequalities (1) and (2) with coefficients 3 / / b ( G ) ≥ n X v d ( v ) − n X v e v ! + 47 e + 1 n X v (cid:16) e v /d ( v ) − e v / (cid:17)! = 27 e + 17 n X v (cid:16) d ( v ) − e v + 16 e v /d ( v ) (cid:17) = 27 e + 17 n X v d ( v ) (cid:16) − (cid:0) e v /d ( v ) (cid:1) + 16 (cid:0) e v /d ( v ) (cid:1) (cid:17) ≥ e + 27 n X v d ( v ) ≥ e + 27 (cid:18) P v d ( v ) n (cid:19) = 27 e + 87 e /n , where we used that 3 − t + 16 t = (4 t − + 2 ≥ t , P v d ( v ) = 2 e and Cauchy-Schwartzinequality. ✷ emark. The above result is enough for our purposes, but one can get a slightly better inequalityby taking a convex combination of (1) and (2) with coefficients 1 / (1 + a ) and a/ (1 + a ) with a = 1 . Lemma 2.5
Let f ( t ) = t/
18 + (cid:0) / − t − /t (cid:1) . Then f ( t ) ≤ / for all t ∈ [3 / , and equalityholds only when t = 2 . Proof.
Note that f (2) = 1 / f ( t ) − / t − t + 31 t − t + 418 t = ( t − t − t + 9 t − t . Consider g ( t ) = 4 t − t + 9 t − / , g ′ ( t ) =12 t − t + 9 is zero when t = ±√ , so the largest root of g ′ ( t ) is less than 3 /
2. Therefore g ( t )is strictly increasing function for t ≥ / g ( t ) > g (3 /
2) = 1 / > t ∈ [3 / , t > t − t < f ( t ) − / < t ∈ [3 / , ✷ Having finished all the necessary preparations we are now in a position to complete the proof ofour main result.
Proof of Theorem 1.1.
It is easy to see that complete 3-partite graph with parts of size n/ n/ = n /
27 triangles and that every edge of this graph is contained in exactly n/ n/ n / n/ = n / ≤ n / K -free graph bipartite.Let G be a K -free graph on n vertices with e edges. Tur´an’s theorem [18] says that e ≤ n /
3, withequality only when G is a complete 3-partite graph with parts of size n/
3. By Lemma 2.4, we needto delete at most e − b ( G ) ≤ e/ − e / (7 n ) = (cid:0) ( e/n ) − ( e/n ) (cid:1) n edges to make G bipartite.The function g ( t ) = 5 t/ − t / t ≤ / g ( t ) ≤ g (1 /
4) = 3 / e ≤ n / n / < n / G bipartite.Next, consider the case when n / ≤ e ≤ n / m be the number of triangles in G . ByLemma 2.3, we can delete at most e − b ( G ) ≤ e − (cid:0) e /n − m/n (cid:1) edges to make G bipartite. Sowe can assume that e − e /n + 6 m/n ≥ n / G satisfies m ≥ n (cid:0) n / e /n − e (cid:1) and Lemma 2.2 implies that G contains a triangle △ = { u, v, w } with d ( u, v ) + d ( u, w ) + d ( v, w ) ≥ me ≥ e/n + n / (6 e ) − n/ . Let V = N ( u, v ) , V = N ( u, w ), V = N ( v, w ) and let X = V ( G ) \ ( ∪ i =1 V i ). Since G is K -freeand ( u, v ) , ( u, w ) , ( v, w ) are edges of G we have that sets V i , ≤ i ≤ G ′ of G with parts V , V , V and X . This graph has e ( G ′ ) = e − e ( X )edges where e ( X ) is the number of edges spanned by X . By Tur´an’s theorem e ( X ) ≤ | X | / | X | = n − X i | V i | = n − (cid:0) d ( u, v ) + d ( u, w ) + d ( v, w ) (cid:1) ≤ n/ − e/n − n / (6 e ) . G ′ is 4-partite we can now use Lemma 2.1 to deduce that b ( G ) ≥ b ( G ′ ) ≥ e ( G ′ ) / (cid:0) e − e ( X ) (cid:1) . Therefore the number of edges we need to delete to make G bipartite is bounded by e − b ( G ) ≤ e − (cid:0) e − e ( X ) (cid:1) / e/ e ( X ) / ≤ e/ | X | / ≤ e/ (cid:16) n/ − e/n − n / (6 e ) (cid:17) = (cid:18)
118 (6 e/n ) + 29 (cid:16) / − e/n − (6 e/n ) − (cid:17) (cid:19) n = f (cid:0) e/n (cid:1) · n , where f ( t ) = t/
18 + (cid:0) / − t − /t (cid:1) . As n / ≤ e ≤ n / / ≤ t = 6 e/n ≤
2. Then,by Lemma 2.5, f (6 e/n ) ≤ / e = n /
3. This shows that we can delete at most n / G bipartite and we need to delete that many edges only when e ( G ) = n /
3, i.e., G is a complete 3-partite graph with parts of size n/
3. . ✷ -chromatic subgraph In this short subsection we show how to use Theorem 1.1 to deduce a similar statement about graphswith any fixed forbidden 4-chromatic subgraph. The proof is a standard application of Szemer´edi’sRegularity Lemma and we refer the interested reader to the excellent survey of Koml´os and Simonovits[14], which discusses various results proved by this powerful tool.We start with a few definitions, most of which follow [14]. Let G = ( V, E ) be a graph, and let A and B be two disjoint subsets of V ( G ). If A and B are non-empty, define the density of edges between A and B by d ( A, B ) = e ( A,B ) | A || B | . For ǫ > A, B ) is called ǫ -regular if for every X ⊂ A and Y ⊂ B satisfying | X | > ǫ | A | and | Y | > ǫ | B | we have | d ( X, Y ) − d ( A, B ) | < ǫ . An equitablepartition of a set V is a partition of V into pairwise disjoint classes V , · · · , V k of almost equal size,i.e., (cid:12)(cid:12) | V i | − | V j | (cid:12)(cid:12) ≤ i, j . An equitable partition of the set of vertices V of G into the classes V , · · · , V k is called ǫ -regular if | V i | ≤ ǫ | V | for every i and all but at most ǫk of the pairs ( V i , V j )are ǫ -regular. The above partition is called totally ǫ -regular if all the pairs ( V i , V j ) are ǫ -regular. Thefollowing celebrated lemma was proved by Szemer´edi in [17]. Lemma 2.6
For every ǫ > there is an integer M ( ǫ ) such that every graph of order n > M ( ǫ ) hasan ǫ -regular partition into k classes, where k ≤ M ( ǫ ) . In order to apply the Regularity Lemma we need to show the existence of a complete multipartitesubgraph in graphs with a totally ǫ -regular partition. This is established in the following lemma whichis a special case of a well-known result, see, e.g., [14]. Lemma 2.7
For every δ > and integer t there exist an < ǫ = ǫ ( δ, t ) and n = n ( δ, t ) with thefollowing property. If G is a graph of order n > n and ( V , · · · , V ) is a totally ǫ -regular partition ofvertices of G such that d ( V i , V j ) ≥ δ for all i < j , then G contains a complete -partite subgraph K ( t ) with parts of size t . roof of Corollary 1.2. Let H be a fixed 4-chromatic graph of order t and let G be a graph on n vertices not containing H as a subgraph. Suppose δ > ǫ = min (cid:0) δ, ǫ ( δ, t ) (cid:1) , where ǫ ( δ, t )is defined in the previous statement. Then, by Lemma 2.6, for sufficiently large n there exists an ǫ -regular partition ( V , · · · , V k ) of vertices of G .Consider a new graph G ′ on the vertices { , . . . , k } in which ( i, j ) is an edge iff ( V i , V j ) is an ǫ -regular pair with density at least δ . We claim that G ′ contains no K . Indeed, any such clique in G ′ corresponds to 4 parts in the partition of G such that any pair of them is ǫ -regular and has densityat least δ . This contradicts our assumption on G , since by Lemma 2.7, the union of these parts willcontain a copy of complete 4-partite graph K ( t ) which clearly contains H .By applying Theorem 1.1 to graph G ′ , we deduce that there is a set D of at most k / G ′ whose deletion makes it bipartite. Now delete all the edges of G between the pairs ( V i , V j ) with( i, j ) ∈ D . Delete also the edges of G that lie within classes of the partition, or that belong to a non-regular pair, or that join a pair of classes of density less than δ . It is easy to see that the remaininggraph is bipartite and the number of edges we deleted is at most( k / n/k ) + ǫn + δn ≤ (cid:0) / δ (cid:1) n = (1 + o (1)) n / . ✷ How many edges do we need to delete to make a K r -free graph G of order n bipartite? For r = 3 , n /
25 edges is always enough and that extremal example is a blow-up of a 5-cycle. In this paper weanswered the question for r = 4 and proved that the unique extremal construction in this case is acomplete 3-partite graph with equal parts. Our result suggests that a complete ( r − n with equal parts is worst example also for all remaining values of r . Therefore we believethat it is enough to delete at most ( r − r − n edges for even r ≥ r − r − n edges for odd r ≥ K r -free graph G of order n . It seems that some of the ideas presentedhere can be useful to make a progress on this problem for even r . Acknowledgment.
I would like to thank J´ozsef Balogh and Peter Keevash for interesting discussionson the early stages of this project.
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