Two counterexamples to a conjecture about even cycles

TW O COUNTEREXAMPLES TO A CONJECTURE ABOUT EVEN CYCLES D A VID CONLON, EION MULRENIN, AND COSMIN POHO A T A Abstract. A conjecture of V erstra¨ ete states that for an y fixed ℓ < k there exists a positive constan t c suc h that any C 2 k -free graph G con tains a C 2 ℓ -free subgraph with at least c | E ( G ) | edges. F or ℓ = 2, this conjecture was verified b y K¨ uhn and Osth us in 2004. W e identify t wo coun terexamples to this conjecture for ℓ = 4 and k = 5: the first comes from a recen t construction of a dense C 10 -free subgraph of the h ypercub e and the second from W enger’s construction for extremal C 10 -free graphs. 1. Introduction F or a family F of graphs and a p ositiv e integer n , the extr emal numb er ex( n, F ) is the maxim um n um b er of edges in an n -v ertex graph which con tains no member of F as a subgraph. T rivially , ex( n, F ) ≤ ex( n, F ) for all F ∈ F . Con v ersely , the c omp actness c onje ctur e of Erd˝ os and Si- mono vits [ 8 ] asserts that if F is a finite family which do es not contain a forest or, equiv alen tly , if ex( n, F ) = Ω( n 1+ ε ) for some ε > 0 and all F ∈ F , 1 then there is some F ∈ F suc h that ex( n, F ) = O (ex( n, F )). This conjecture is of most interest when the family F includes bipartite graphs, since a celebrated result of Erd˝ os and Stone [ 9 ] asymptotically determines the b eha vior of the extremal num b er for all non-bipartite graphs. In particular, one migh t ask whether the conjecture holds for the family C 2 k := { C 3 , C 4 , . . . , C 2 k } of all cycles of length at most 2 k , where it is essentially asking if ex( n, C 2 k ) = O (ex( n, C 2 k )) — see, for example, [ 14 , Conjecture IV] for an explicit statement of this question as a conjecture. Despite the fact [ 3 ] that b oth of these functions are known to b e O ( n 1+1 /k ), this simple question of whether they agree up to a constant has only been resolv ed for k ∈ { 2 , 3 , 5 } , with the corresp onding low er b ounds in these cases witnessed b y the so-called gener alize d p olygons [ 13 ]. Inspired b y this problem, a n umber of natural extremal questions about the relationships betw een graphs av oiding particular ev en cycles hav e b een studied. One of the first results in this v ein was a theorem of Gy˝ ori [ 11 ], who prov ed that ev ery C 6 -free bipartite graph G contains a C 4 -free subgraph with at least 1 2 | E ( G ) | edges. Several y ears later, K ¨ uhn and Osthus [ 12 ] extended Gy˝ ori’s result b y The first author was supp orted b y NSF Aw ard DMS-2348859 and the third author was supported by NSF Award DMS-2246659. 1 Without this h yp othesis, families such as F = { K 1 , 2 , 2 K 2 } , where 2 K 2 is a matc hing with t wo edges, give simple coun terexamples. See for example [ 16 ] for more details. 1 pro ving that for all k ≥ 3, every C 2 k -free bipartite 2 graph G has a C 4 -free subgraph with at least 1 k − 1 e ( G ) edges. In the same pap er [ 12 ], K ¨ uhn and Osthus stated the following conjecture, which they attribute to V erstra ¨ ete (cf. [ 14 , Conjecture VI II]) and which w ould easily imply that ex( n, C 2 k ) = O (ex( n, C 2 k )), sa ying that their theorem should extend to all pairs of even cycles. Conjecture 1.1. F or al l inte gers 2 ≤ ℓ < k , ther e exists a p ositive c onstant c such that every C 2 k -fr e e bip artite gr aph G has a C 2 ℓ -fr e e sub gr aph F with | E ( F ) | ≥ c | E ( G ) | . K ¨ uhn and Osth us also verified this conjecture for other v alues of ℓ and k , including, for any giv en ℓ , infinitely man y v alues of k . Despite this evidence for the conjecture, our first result sa ys that arbitrarily large counterexamples exist for C 8 and C 10 . Theorem 1.2. F or any p ositive c onstant c and any p ositive inte ger N 0 , ther e exists an inte ger N ≥ N 0 and a C 10 -fr e e bip artite gr aph G on N vertic es with the pr op erty that every sub gr aph with at le ast c | E ( G ) | e dges c ontains a C 8 . W e cannot claim muc h credit for this construction, as it is a straightforw ard consequence of a b eautiful recen t result by Greb ennik ov and Marciano [ 10 ] of a dense C 10 -free subgraph of the h yp ercube. How ever, as the graphs G in Theorem 1.2 ha v e only Θ( N log N ) edges, it is natural to w onder whether denser counterexamples exist. Our second result, which is our main contribution, confirms this, sho wing that there is a family of counterexamples with essen tially the maximum p ossible n umber of edges. Theorem 1.3. F or any p ositive c onstant c and any p ositive inte ger N 0 , ther e exists an inte ger N ≥ N 0 and a C 10 -fr e e bip artite gr aph W with N vertic es on e ach side, N 6 / 5 e dges and the pr op erty that every sub gr aph with at le ast c | E ( W ) | e dges c ontains a C 8 . W e will tak e the graphs W to be those from W enger’s construction [ 15 ] for extremal C 10 -free graphs, whic h we describ e in Section 3 b elo w. 2. Proof of Theorem 1.2 Here and throughout, Q n denotes the n -dimensional h yp ercub e with 2 n v ertices and n 2 n − 1 edges and, for graphs G and H , ex( G, H ) denotes the maxim um num b er of edges in an H -free subgraph of G . Theorem 1.2 is an immediate consequence of the following tw o theorems. Theorem 2.1 (Greb ennik ov and Marciano, 2025 [ 10 ]) . F or every p ositive inte ger n , ther e is a sub gr aph F n of Q n which is C 10 -fr e e and has | E ( F n ) | > 0 . 024 | E ( Q n ) | . Theorem 2.2 (Ch ung, 1992 [ 4 ]) . ex( Q n , C 8 ) = o ( | E ( Q n ) | ) . 2 By the elementary fact that any graph can be made bipartite b y deleting at most half the edges, this theorem remains true for non-bipartite G , but at the cost of an extra factor of 1 / 2. 2 Pr o of of The or em 1.2 . Supp ose for the sak e of con tradiction that there is c > 0 suc h that ev ery C 10 - free bipartite graph G has a C 8 -free subgraph with at least c | E ( G ) | edges. T aking suc h a subgraph of the graph F n , whic h is a subgraph of Q n and so necessarily bipartite, from Theorem 2.1 for eac h n , we obtain a sequence of C 8 -free subgraphs of Q n con taining at least 0 . 024 c | E ( Q n ) | edges of Q n , con tradicting Theorem 2.2 . □ F or the interested reader, w e include a brief sketc h of the construction of the graphs F n in Theorem 2.1 , which was itself inspired b y the recen t w ork of Ellis, Iv an, and Leader [ 7 ] on the T ur´ an densities of daisies. The main task is to construct a subgraph of Q n of p ositiv e relativ e densit y with no copy of C − 6 , the graph obtained by removing an edge from a cop y of C 6 in Q n . Note that ev ery C − 6 is a path of length fiv e, but not ev ery path of length fiv e is a C − 6 . This is sufficien t b ecause of the inequality ex( Q n , C 10 ) ≥ 1 3 · ex ∗ ( Q n , C − 6 ) noted by Axeno vich, Martin, and Win ter [ 1 ], where ex ∗ ( Q n , C − 6 ) denotes the maximum num b er of edges in a subgraph of Q n with no C − 6 . Grebenniko v and Marciano construct the required subgraph as follows: • First iden tify the r th level of Q n for each 0 ≤ r ≤ n with [ n ] ( r ) and consider the lay ers b et ween lev els r − 1 and r for all o dd r . • Observe that e v ery induced subgraph of a lay er that contains a C − 6 also contains a C 6 , so it is enough to construct a dense C 6 -free induced subgraph G r of the r th lay er for eac h o dd r . • Fix a v ector v 0 ∈ F r 2 \ { 0 } and, for each i ∈ [ n ], choose a vector v i ∈ F r 2 \ { 0 } uniformly at random. • Now let G r b e the induced subgraph b et ween the sets B r := { S ∈ [ n ] ( r ) : { v i : i ∈ S } forms a basis for F r 2 } , B r − 1 := { S ∈ [ n ] ( r − 1) : { v 0 } ∪ { v i : i ∈ S } forms a basis for F r 2 } . • Each of these random graphs G r is dense in the r th lay er in exp ectation and is (determin- istically) C 6 -free. As a matter of fact, Chung’s result is slightly stronger than what is stated ab o ve. She prov ed that ex( Q n , C 8 ) = O ( n − 1 / 4 e ( Q n )) , obtaining a p olylogarithmic sa ving in terms of the num b er of v ertices. In the next section, we will pro v e a stronger p o w er-saving result for C 8 -free subgraphs of W enger graphs. 3. Proof of Theorem 1.3 W e first recall W enger’s construction [ 15 ] for C 2 k -free graphs, follo wing the simplified in terpre- tation given in [ 6 ]. W e denote by W k ( q ) the bipartite incidence graph on P ∪ L , where P = F k q 3 and L is the set of affine lines in F k q with directions of the form (1 , a, . . . , a k − 1 ) for a ∈ F q . More precisely , for each a ∈ F q , we define L a to b e the set of all lines of the form { v + t · (1 , a, a 2 , . . . , a k − 1 ) : t ∈ F q } for some v ∈ F k q and then we set 3 L = [ a ∈ F q L a . It can b e shown via a (simple) case analysis that, pro vided k ≥ ℓ , W k ( q ) is C 2 ℓ -free for ℓ = 2 , 3 , 5 and con tains copies of C 2 ℓ when ℓ / ∈ { 2 , 3 , 5 } — see [ 6 , Theorem 1]. Theorem 1.3 thus follo ws immediately by applying the following more general theorem to the C 10 -free W enger graphs W 5 ( q ). Recall that for graphs G and H , ex( G, H ) denotes the maxim um n umber of edges in an H -free subgraph of G . Theorem 3.1. F or a prime p ower q , let L 1 , L 2 , . . . , L q b e q p ar al lel classes of lines in F 5 q , e ach of size q 4 , and let G L ( q ) b e the bip artite incidenc e gr aph b etwe en P = F 5 q and L = S i ∈ [ q ] L i . Then, for every δ > 0 , ther e exists q 0 ( δ ) such that, for al l prime p owers q ≥ q 0 ( δ ) , every sub gr aph H ⊆ G L ( q ) with at le ast δ q 6 e dges c ontains a c opy of C 8 . In fact, as q → ∞ , ex ( G L ( q ) , C 8 ) = O ( q 23 / 4 ) . Pr o of. Let q b e a large prime pow er, fix parallel classes of lines L 1 , . . . , L q in F 5 q and let L = S i ∈ [ q ] L i and G = G L ( q ). Throughout the pro of, any asymptotic notation will b e used in the regime where q → ∞ . W e b egin with the following lemma, which says that the intersection graph on L has a very rigid structure. Lemma 3.2. F or two distinct p ar al lel classes L i , L j ⊆ L , the interse ction gr aph b etwe en L i and L j is a vertex-disjoint union of q 3 c opies of K q ,q , indexe d by the affine 2 -planes in F 5 q which c ontain lines fr om b oth classes. Pr o of. Tw o lines ℓ i ∈ L i and ℓ j ∈ L j in tersect only if they are coplanar, so fix any suc h pair ℓ i and ℓ j and let Π b e the (unique) 2-plane which con tains them. Then Π contains q − 1 other lines from eac h of L i and L j , since an y parallel class in an affine 2-plane ov er F q has q lines in it. Moreov er, since an y t w o non-parallel lines in an affine 2-plane m ust in tersect, these lines form a K q ,q in the in tersection graph b et w een L i and L j . Using the observ ation that all lines in L i whic h are not con tained in Π are necessarily skew to ℓ j and vice versa, it follows that these copies of K q ,q are v ertex-disjoin t and partition the edge set of the in tersection graph b et ween L i and L j . Finally , since the copies of K q ,q are indexed by affine 2-planes containing lines from b oth L i and L j , we ma y coun t them b y coun ting the num b er of such 2-planes. In order for an affine 2-plane to contain 3 More generally , w e can pick an y set of lines with the prop erty that an y k of the directions determined by the parallel classes are linearly independent; here, the c hoice of directions is the moment curve { (1 , a, a 2 , . . . , a k − 1 ) : a ∈ F q } . 4 a line from L i and a line from L j , this plane must b e either Π or an affine 2-plane parallel to Π. The num b er of these 2-planes is precisely q 5 /q 2 = q 3 . □ No w, fix a subgraph H ⊆ G . F or a plane Π and a set of lines K , we define K (Π) to b e the set of lines from K whic h lie entirely in Π. F or distinct parallel classes L i and L j in L and a plane Π containing lines from b oth L i and L j , we define a bipartite graph G i,j Π ( H ) as follo ws: • the left vertex set is L i (Π), the q lines from L i whic h lie in Π; • the right v ertex set is L j (Π), the q lines from L j whic h lie in Π; • ℓ i ∈ L i (Π) and ℓ j ∈ L j (Π) are adjacent if, for x = ℓ i ∩ ℓ j their unique common p oin t, b oth incidences x ∈ ℓ i and x ∈ ℓ j remain edges in H . Note that eac h suc h common p oin t x is well defined by the fact that tw o non-parallel lines in an affine plane must in tersect. Here comes the crucial p oin t. Lemma 3.3. If H is C 8 -fr e e, then every auxiliary gr aph G i,j Π ( H ) is C 4 -fr e e. Pr o of. A 4-cycle in G i,j Π ( H ) consists of tw o lines in L i (Π) and tw o lines in L j (Π) with all four cross-in tersections active. These four lines then form a cop y of C 8 in H , as the four intersection p oin ts are distinct b ecause lines with the same direction are parallel and distinct. Therefore, a C 4 in some G i,j Π ( H ) would pro duce a C 8 in H . □ W e are now ready to complete the pro of of Theorem 3.1 . Let H ⊆ G b e C 8 -free and write m = e ( H ). The pro of pro ceeds by double counting the total num b er Ψ of cherries in H consisting of a p oin t and tw o lines. More precisely , let Ψ b e the num b er of triples ( x, ℓ i , ℓ j ) where x ∈ F 5 q , ℓ i ∈ L i , ℓ j ∈ L j for distinct i, j ∈ [ q ] and where x = ℓ i ∩ ℓ j with the incidences ( x, ℓ i ), ( x, ℓ j ) b oth b eing in E ( H ). On the one hand, Ψ = P x ∈ F 5 q  deg H ( x ) 2  . W e kno w P x ∈ F 5 q deg H ( x ) = m , so the conv exity of the function t 7→  t 2  yields that Ψ = X x ∈ F 5 q  deg H ( x ) 2  ≥ q 5  m/q 5 2  = m 2 2 q 5 − m 2 . (1) On the other hand, observe that the cherries we wan t to count are in one-to-one corresp ondence with edges in the auxiliary intersection graphs G i,j Π ( H ), i.e., Ψ = X i  = j ∈ [ q ] X Π | E  G i,j Π ( H )  | , (2) where the inside sum runs ov er the q 3 planes Π whic h contain lines from b oth L i and L j . Since H is C 8 -free, Lemmas 3.2 and 3.3 show that each G i,j Π ( H ) is a C 4 -free subgraph of K q ,q ; therefore, the K˝ ov´ ari–S´ os–T ur´ an theorem (see, e.g., [ 2 , Theorem VI.2.2]) gives | E  G i,j Π ( H )  | ≤ q 3 / 2 + q. 5 Summing ov er the q 3 planes and plugging back in to ( 2 ), we get that Ψ ≤ X i  = j ∈ [ q ] q 3 · ( q 3 / 2 + q ) =  q 2  · q 3 · ( q 3 / 2 + q ) = O ( q 13 / 2 ) . Com bining this with ( 1 ), it now follows that m 2 2 q 5 − m 2 ≤ Ψ = O ( q 13 / 2 ) and so m = O ( q 23 / 4 ) . In particular, m = o ( q 6 ). Therefore, for every fixed δ > 0, the C 8 -free subgraph H cannot hav e δ q 6 edges once q is sufficien tly large. □ 4. Concluding remarks Giv en that we deduce Theorem 1.3 from the more general Theorem 3.1 on the robustness of C 8 ’s in p oin t-line incidence graphs ov er F 5 q , it might b e tempting to conjecture that all constructions of extremal C 10 -free graphs share this prop erty . This is, how ever, not true: indeed, as stated in the introduction, the so-called gener alize d hexagons [ 13 ] sho w that ex( n, C 10 ) = Ω( n 6 / 5 ). W e thus b eliev e that our result illustrates an in teresting difference b etw een the W enger construction and other constructions of dense C 10 -free graphs. By follo wing the same steps as in the argument in Section 3 , one may also show that for any k ≥ 2 the incidence graph G L ( q ) betw een the p oin ts of F k q and q parallel classes of lines has the prop ert y that ex( G L ( q ) , H ) = o ( e ( G L ( q ))) for every fixed graph H that is a sub division of a bipartite graph. In particular, no edge-sampling argumen t in W 2 k ( q ) can pro duce a C 4 k -free graph with 2 q 2 k v ertices and Ω( q 2 k +1 ) edges for k ≥ 2. Regarding more general pairs of ev en cycles, w e b eliev e that Conjecture 1.1 should not hold for C 4 k and C 4 k +2 for any k ≥ 2. The tw o constructions in this pap er confirm this for k = 2, but in general w e exp ect that ex( Q n , C 4 k ) = o (ex( Q n , C 4 k +2 )), which, as in the pro of of Theorem 1.2 , w ould yield the required counterexample. The in tuition here, coming from [ 5 ], is that the best kno wn upp er b ound for ex( Q n , C 4 k ) is in a sense inherited from the upp er b ound for ex( N , C 2 k ), while the upp er b ound for ex( Q n , C 4 k +2 ) comes from the upp er b ound for ex( N , H ) for an appro- priate 3-uniform h yp ergraph H . This then suggests that ex( Q n , C 4 k ) /e ( Q n ) should drop off more quic kly with k than ex( Q n , C 4 k +2 ) /e ( Q n ). If this is indeed the case, it is then lik ely that for an y p ositiv e in teger t ≡ 2 (mo d 4) there exists k 0 suc h that Conjecture 1.1 do es not hold for C 4 k and C 4 k + t for any k ≥ k 0 . References [1] M. Axeno vich, R. Martin, and C. Winter, On graphs em b eddable in a lay er of the h yp ercube and their extremal n umbers. Ann. Comb. 28 (2024), 1257–1283. MR 4827889 3 6 [2] B. Bollob´ as, Extr emal gr aph the ory , London Mathematical Society Monographs, 11, Academic Press, London-New Y ork, 1978. 5 [3] J. A. Bondy and M. Simonovits, Cycles of ev en length in graphs. J. Combin. The ory Ser. B 16 (1974), 97–105. MR 0340095 1 [4] F. R. K. Chung, Subgraphs of a h yp ercube con taining no small even cycles. J. Gr aph The ory 16 (1992), 273–286. MR 1168587 2 [5] D. Conlon, An extremal theorem in the h yp ercube. Ele ctron. J. Combin. 17 (2010), Research P ap er 111, 7 pp. MR 2679565 6 [6] D. Conlon, Extremal n umbers of cycles revisited. Amer. Math. Monthly 128 (2021), 464–466. MR 4249723 3 , 4 [7] D. Ellis, M.-R. Iv an, and I. Leader, T ur´ an densities for daisies and hypercub es. Bul l. L ond. Math. So c. 56 (2024), 3838–3853. MR 4835279 3 [8] P . Erd˝ os and M. Simonovits, Compactness results in extremal graph theory . Combinatoric a 2 (1982), 275–288. MR 0698653 1 [9] P . Erd˝ os and A. H. Stone, On the structure of linear graphs. Bul l. Amer. Math. Soc. 52 (1946), 1087–1091. MR 0018807 1 [10] A. Greb ennik ov and J. P . Marciano, C 10 has p ositiv e T ur´ an density in the hypercub e, J. Gr aph The ory 109 (2025), 31–34. MR 4879218 2 [11] E. Gy˝ ori, C 6 -free bipartite graphs and pro duct representation of squares. Discr ete Math. 165/166 (1997), 371– 375. MR 1439284 1 [12] D. K ¨ uhn and D. Osth us, F our-cycles in graphs without a giv en ev en cycle. J. Gr aph The ory 48 (2005), 147–156. MR 2110585 1 , 2 [13] F. Lazebnik, V. Ustimenk o and A. J. W oldar, Polarities and 2 k -cycle-free graphs, Discr ete Math. 197/198 (1999), 503–513. MR 1674884 1 , 6 [14] J. V erstra ¨ ete, Extremal problems for cycles in graphs. Recent trends in combinatorics, pp. 83–116, IMA V ol. Math. Appl., 159, Springer, Cham, 2016. MR 3526405 1 , 2 [15] R. W enger, Extremal graphs with no C 4 ’s, C 6 ’s, or C 10 ’s. J. Combin. The ory Ser. B 52 (1991), 113–116. MR 1109426 2 , 3 [16] Y. Wigderson, The Erd˝ os–Simonovits compactness conjecture needs more assumptions. Av ailable at https://ywigderson.math.ethz.ch/math/static/Compactness.pdf . 1 Dep ar tment of Ma thema tics, Calif ornia Institute of Technology, P asadena, CA 91125, USA Email addr ess : dconlon@caltech.edu Dep ar tment of Ma thema tics, Emor y University, A tlant a, GA 30322, USA Email addr ess : eion.mulrenin@emory.edu Dep ar tment of Ma thema tics, Emor y University, A tlant a, GA 30322, USA Email addr ess : cosmin.pohoata@emory.edu 7