aa r X i v : . [ m a t h . C O ] S e p Improved bounds for the extremal number ofsubdivisions
Oliver Janzer ∗ Abstract
Let H t be the subdivision of K t . Very recently, Conlon and Lee have provedthat for any integer t ≥
3, there exists a constant C such that ex( n, H t ) ≤ Cn / − / t . In this paper, we prove that there exists a constant C ′ such thatex( n, H t ) ≤ C ′ n / − t − . For a graph H , the extremal function ex( n, H ) is defined to be the maximal numberof edges in an H -free graph on n vertices. This function is well understood for graphs H with chromatic number at least three by the Erd˝os-Stone-Simonovits theorem. [3,5]However, for bipartite graphs H , much less is known. For a survey on the subject,see [7]. One of the few general results, proved by F¨uredi [6], and reproved by Alon,Krivelevich and Sudakov [1] is the following. Theorem 1 (F¨uredi, Alon-Krivelevich-Sudakov) . Let H be a bipartite graph such thatin one of the parts all the degrees are at most r . Then there exists a constant C suchthat ex ( n, H ) ≤ Cn − /r . Conlon and Lee [2] have conjectured that the only case when this is tight up tothe implied constant is when H contains a K r,r (it is conjectured [8] that ex( n, K r,r ) =Ω( n − /r )), and that for other graphs H there exists some δ > n, H ) = O ( n − /r − δ ).The subdivision of a graph L is the bipartite graph with parts V ( L ) and E ( L ) (thevertex set and the edge set of the graph L , respectively) where v ∈ V ( L ) is joined to e ∈ E ( L ) if v is an endpoint of e . It is easy to see that any C -free bipartite graph inwhich every vertex in one part has degree at most two is a subgraph of H t for somepositive integer t , where H t is the subdivision of K t . Conlon and Lee have verifiedtheir conjecture in the r = 2 case by proving the following result. Theorem 2 (Conlon and Lee [2, Theorem 5.1]) . For any integer t ≥ , there exists aconstant C t such that ex ( n, H t ) ≤ C t n / − / t . They have observed the lower bound ex( n, H t ) ≥ c t n / − t − / t − t − coming from theprobabilistic deletion method, and have asked for an upper bound of the form ex( n, H ) ≤ C t n / − δ t , where 1 /δ t is bounded by a polynomial in t . We can prove such a boundeven for a linear δ t . ∗ Department of Pure Mathematics and Mathematical Statistics, University of Cambridge. E-mail:[email protected] heorem 3. For any integer t ≥ , there exists a constant C t such that ex ( n, H t ) ≤ C t n t − t − = C t n / − t − . It would be very interesting to know whether or not this bound is tight up to theimplied constant. It certainly is tight for t = 3 as ex( n, C ) = Θ( n / ).We can in fact prove a slightly stronger result. For integers s ≥ t ≥
3, let L s,t be the graph which is a K s + t − with the edges of a K s removed. That is, thevertex set of L s,t is S ∪ T where S ∩ T = ∅ , | S | = s and | T | = t −
1, and xy is an edgeif and only if x ∈ T or y ∈ T . Let L ′ s,t be the subdivision of L s,t . Theorem 4.
For any two integers s ≥ and t ≥ , there exists a constant C s,t suchthat ex ( n, L ′ s,t ) ≤ C s,t n / − t − . This result certainly implies Theorem 3 as L ,t = K t . Moreover, we can applyTheorem 4 to obtain good bounds on the extremal number of the subdivision of thecomplete bipartite graph K a,b as well. Let us write H a,b for the subdivision of K a,b .Conlon and Lee [2, Theorem 4.2] have proved that for any 2 ≤ a ≤ b there existsa constant C such that ex( n, H a,b ) ≤ Cn / − b . They have also observed the lowerbound ex( n, H a,b ) = Ω a,b ( n / − a + b − / ab − ) (which follows from the probabilistic deletionmethod). Hence their upper bound is reasonably close to best possible when a = b ,but is weak when b is much larger then a .Since K a,b is a subgraph of L b,a +1 , Theorem 4 implies the following result, by taking s = b and t = a + 1. Corollary 5.
For any two integers ≤ a ≤ b , there exists a constant C a,b , suchex ( n, H a,b ) ≤ C a,b n / − a − . We shall use the following lemma of Conlon and Lee [2, Lemma 2.3], which is a slightmodification of a result of Erd˝os and Simonovits [4]. Let us say that a graph G is K-almost-regular if max v ∈ V ( G ) deg( v ) ≤ K min v ∈ V ( G ) deg( v ). Moreover, followingConlon and Lee, we say that a bipartite graph G with a bipartition A ∪ B is balanced if | B | ≤ | A | ≤ | B | . Lemma 6.
For any positive constant α < , there exists n such that if n ≥ n , C ≥ and G is an n -vertex graph with at least Cn α edges, then G has a K -almost-regular balanced bipartite subgraph G ′ with m vertices such that m ≤ n α (1 − α )2(1+ α ) , | E ( G ′ ) | ≥ C m α and K = 60 · /α . This reduces Theorem 4 to the following.
Theorem 7.
For every K ≥ , and positive integers s ≥ , t ≥ , there exists aconstant c = c ( s, t, K ) with the following property. Let n be sufficiently large and let G be a balanced bipartite graph with bipartition A ∪ B , | B | = n such that the degree ofevery vertex of G is between δ and Kδ , for some δ ≥ cn t − t − . Then G contains a copyof L ′ s,t . G with bipartition A ∪ B , the neighbourhood graph is theweighted graph W G on vertex set A where the weight of the pair uv is d G ( u, v ) = | N G ( u ) ∩ N G ( v ) | . Here and below N G ( v ) denotes the neighbourhood of the vertex v in the graph G . For a subset U ⊂ A , we write W ( U ) for the total weight in U , ie. W ( U ) = P uv ∈ ( U ) d G ( u, v ).We shall use the following simple lemma of Conlon and Lee [2, Lemma 2.4]. Lemma 8.
Let G be a bipartite graph with bipartition A ∪ B , | B | = n , and minimumdegree at least δ on the vertices in A . Then for any subset U ⊂ A with δ | U | ≥ n , X uv ∈ ( U ) d G ( u, v ) ≥ δ n (cid:18) | U | (cid:19) In other words, the conclusion of Lemma 8 is that W ( U ) ≥ δ n (cid:0) | U | (cid:1) .In the next definition, and in the rest of this paper, for a weighted graph W onvertex set A , if u, v ∈ A , then W ( u, v ) stands for the weight of uv . Moreover, we shalltacitly assume throughout the paper that s ≥ t ≥ Definition 9.
Let W be a weighted graph on vertex set A and let u, v ∈ A be distinct.We say that uv is a light edge if 1 ≤ W ( u, v ) < (cid:0) s + t − (cid:1) and that it is a heavy edge if W ( u, v ) ≥ (cid:0) s + t − (cid:1) .Note that if there is a K s + t − in W G formed by heavy edges, then clearly there isan L s,t in W G formed by heavy edges, therefore there is an L ′ s,t in G .The next lemma is one of our key observations. Lemma 10.
Let G be an L ′ s,t -free bipartite graph with bipartition A ∪ B , | B | = n andsuppose that W ( A ) ≥ s + t ) n . Then the number of light edges in W G is at least W ( A )4( s + t ) . Proof.
Let B = { b , . . . , b n } . Let k i = | N G ( b i ) | and suppose that k i ≥ s + t − i . As G is L ′ s,t -free, there is no K s + t − in W [ N G ( b i )] formed by heavy edges.Thus, by Tur´an’s theorem, the number of light edges in N G ( b i ) is at least( s + t − (cid:18) k i s + t − (cid:19) = 12 k i (cid:0) k i s + t − − (cid:1) ≥ k i s + t − . But X i : k i < s + t − (cid:18) k i (cid:19) < s + t ) n ≤ W ( A )2 , so X i : k i ≥ s + t − (cid:18) k i (cid:19) ≥ W ( A )2 . Since every light edge is present in at most (cid:0) s + t − (cid:1) of the sets N G ( b i ), it follows thatthe total number of light edges is at least 3 (cid:0) s + t − (cid:1) X i : k i ≥ s + t − k i s + t − ≥ W ( A )4( s + t ) . Corollary 11.
Let G be an L ′ s,t -free bipartite graph with bipartition A ∪ B , | B | = n ,and minimum degree at least δ on the vertices in A . Then for any subset U ⊂ A with | U | ≥ s + t ) nδ and | U | ≥ , the number of light edges in W G [ U ] is at least δ s + t ) n (cid:0) | U | (cid:1) . Proof.
By Lemma 8, we have W ( U ) ≥ δ n (cid:0) | U | (cid:1) ≥ δ n | U | ≥ s + t ) n . Now the resultfollows by applying Lemma 10 to the graph G [ U ∪ B ].We are now in a position to complete the proof of Theorem 7. Proof of Theorem 7 . Let c be specified later and suppose that n is sufficiently large.Assume, for contradiction, that G is L ′ s,t -free. We shall find distinct u , . . . , u t − ∈ A with the following properties.(i) Each u i u j is a light edge in W G (ii) If i, j, k are distinct, then N G ( u i ) ∩ N G ( u j ) ∩ N G ( u k ) = ∅ (iii) For each 1 ≤ i ≤ t −
1, the number of v ∈ A with the property that for every j ≤ i , u j v is a light edge is at least ( δ s + t ) n ) i · | A | As n is sufficiently large, we have | A | ≥ n/ ≥ s + t ) nδ , therefore by Corollary 11there are at least δ s + t ) n (cid:0) | A | (cid:1) light edges in A , so we may choose u ∈ A such that thenumber of light edges u v is at least δ s + t ) n ( | A | − ≥ δ s + t ) n | A | .Now suppose that 2 ≤ i ≤ t −
1, and that u , . . . , u i − have been constructedsatisfying (i),(ii) and (iii). Let U be the set of vertices v ∈ A with the property that u j v is a light edge for every j ≤ i −
1. By (iii), we have | U | ≥ ( δ s + t ) n ) i − | A | .Now let U consist of those v ∈ U for which N G ( u j ) ∩ N G ( u k ) ∩ N G ( v ) = ∅ holdsfor all 1 ≤ j < k ≤ i −
1. Since u j u k is a light edge for any 1 ≤ j < k ≤ i − d G ( u j , u k ) < (cid:0) s + t − (cid:1) . But the degree of every b ∈ B is at most Kδ ,therefore the number of v ∈ A for which N G ( u j ) ∩ N G ( u k ) ∩ N G ( v ) = ∅ is at most (cid:0) s + t − (cid:1) Kδ , so | U \ U | ≤ (cid:0) i − (cid:1)(cid:0) s + t − (cid:1) Kδ . But note that for sufficiently large n , we have( δ s + t ) n ) i − | A | ≥ (cid:0) i − (cid:1)(cid:0) s + t − (cid:1) Kδ because δ = o (( δ /n ) t − n ) and δ = o (( δ /n ) n ).Thus, | U | ≥ | U | ≥ (cid:0) δ s + t ) n (cid:1) i − | A | . But for sufficiently large c = c ( s, t, K ), we have ( δ s + t ) n ) i − | A | ≥ s + t ) nδ . Indeed,this is obvious when δ ≥ s + t ) n , and otherwise, using δ ≥ cn t − t − , we have12 (cid:0) δ s + t ) n (cid:1) i − | A | ≥ (cid:0) δ s + t ) n (cid:1) t − | A | ≥ s + t ) ) t − · δ t − n t − ≥ s + t ) nδ Thus, by Corollary 11, there exists some u i ∈ U with at least δ s + t ) n ( | U | − ≥ ( δ s + t ) n ) i | A | light edges adjacent to it in U . This completes the recursive constructionof the vertices { u j } ≤ j ≤ t − . 4y (iii) for i = t −
1, there is a set V ⊂ A consisting of at least ( δ s + t ) n ) t − | A | vertices v such that for every j ≤ t − u j v is a light edge. We shall now prove thatthere exist distinct v , . . . , v s ∈ V such that N G ( u i ) ∩ N G ( u j ) ∩ N G ( v k ) = ∅ for all i = j , and N G ( u i ) ∩ N G ( v j ) ∩ N G ( v k ) = ∅ for all j = k . It is easy to see that thissuffices since then there is a copy of L ′ s,t in G , which is a subdivision of the copy of L s,t in W G whose vertices are v , . . . , v s , u , . . . , u t − .We shall now choose v , . . . , v s one by one. Since every u i u j is a light edge, thenumber of those v ∈ A with N G ( u i ) ∩ N G ( u j ) ∩ N G ( v ) = ∅ for some i = j is at most (cid:0) t − (cid:1)(cid:0) s + t − (cid:1) Kδ . Moreover, given any choices for v , . . . , v k − ∈ V , as each u i v j is alight edge, the number of those v ∈ A with N G ( u i ) ∩ N G ( v j ) ∩ N G ( v ) = ∅ for some i, j isat most ( t − k − (cid:0) s + t − (cid:1) Kδ . Therefore as long as | V | > (cid:0) t − (cid:1)(cid:0) s + t − (cid:1) Kδ +( t − s − (cid:0) s + t − (cid:1) Kδ , suitable choices for v , . . . , v s can be made. Since | V | ≥ ( δ s + t ) n ) t − | A | ,this last inequality holds for large enough c = c ( s, t, K ). References [1] Noga Alon, Michael Krivelevich, and Benny Sudakov. Tur´an numbers of bipar-tite graphs and related Ramsey-type questions.
Combinatorics, Probability andComputing , 12:477–494, 2003.[2] David Conlon and Joonkyung Lee. On the extremal number of subdivisions. arXivpreprint arXiv:1807.05008 , 2018.[3] Paul Erd˝os and Mikl´os Simonovits. A limit theorem in graph theory. In
StudiaSci. Math. Hung , 1965.[4] Paul Erd˝os and Mikl´os Simonovits. Some extremal problems in graph theory. In
Combinatorial theory and its applications , 1969.[5] Paul Erd˝os and Arthur H Stone. On the structure of linear graphs.
Bull. Amer.Math. Soc , 52(1087-1091):1, 1946.[6] Zolt´an F¨uredi. On a Tur´an type problem of Erd˝os.
Combinatorica , 11(1):75–79,1991.[7] Zolt´an F¨uredi and Mikl´os Simonovits. The history of degenerate (bipartite) ex-tremal graph problems. In
Erd˝os Centennial , pages 169–264. Springer, 2013.[8] Tam´as K˝ov´ari, Vera S´os, and P´al Tur´an. On a problem of K. Zarankiewicz. In