A note on the Turán function of even cycles

  title={A note on the Tur{\'a}n function of even cycles},
  author={Oleg Pikhurko},
The Tur´an function ex(n, F) is the maximum number of edges in an F-free graph on n vertices. The question of estimating this function for F = C2k, the cycle of length 2k, is one of the central open questions in this area that goes back to the 1930s. We prove that ex(n,C2k) ≤ (k − 1) n1+1/k + 16(k − 1)n, improving the previously best known general upper bound of Verstra¨ete [Combin. Probab. Computing 9 (2000), 369–373] by a factor 8 + o(1) when n � k. 
Tur\'an Numbers of Hypergraph Suspensions of Even Cycles
For fixed $k\ge 2$, determining the order of magnitude of the number of edges in an $n$-vertex bipartite graph not containing $C_{2k}$, the cycle of length $2k$, is a long-standing open problem. WeExpand
On 3-uniform hypergraphs without a cycle of a given length
We study the maximum number of hyperedges in a 3-uniform hypergraph on n vertices that does not contain a Berge cycle of a given length l. In particular we prove that the upper bound for C2k+1-freeExpand
Cycles of Even Lengths in Graphs
We see that this gives a better upper bound than that obtained by excluding Kk,k, best known to be O(n2−1/k) from the Zarankiewicz problem. It is conjectured that this result by Bondy-Simonovits isExpand
Supersaturation of even linear cycles in linear hypergraphs
A reduction lemma from general host graphs to almost-regular host graphs that can be used for other supersaturation problems, and may therefore be of independent interest, is developed. Expand
On 3-uniform hypergraphs without a cycle of a given length
It is proved that the upper bound for C_{2k+1}-free hypergraphs is of the order of O(k^2n^{1+1/k})$, improving theupper bound of Gyori and Lemons by a factor of $\Theta(k-2)$. Expand
A New Upper Bound on the Tuŕan Number of Even Cycles
  • Zhiyang He
  • 2021
In this paper, we prove ex(n,C2k) 6 (16 √ 5 √ k log k+o(1))·n1+1/k. This improves on a result of Bukh and Jiang from 2017, thereby reducing the best known upper bound by a factor of √ 5 log k.Expand
5-The Even Cycle Problem
For k ≥ 2 and l ≥ 0, let C (k, l) denote the family of all cycles of length l modulo k, and C (k) := C (l, 0). In proving the Even Cycle Theorem, we determine almost tight bounds on ex(n,C (k, l)) asExpand
On Some Cycles in Wenger Graphs
Let p be a prime, q be a power of p , and let F q be the field of q elements. For any positive integer n , the Wenger graph W n(q) is defined as follows: it is a bipartite graph with the vertexExpand
An analogue of the Erdős-Gallai theorem for random graphs
The Erdős–Gallai Theorem in random graphs is extended, complementing existing results, and the maximum number of edges in a P n -free subgraph of G ( N, p ) is determined, practically for all values of N, n and p. Expand
An analogue of the Erd\H{o}s-Gallai theorem for random graphs
Recently, variants of many classical extremal theorems have been proved in the random environment. We, complementing existing results, extend the Erd\H{o}s-Gallai Theorem in random graphs. InExpand


On Arithmetic Progressions of Cycle Lengths in Graphs
This paper proves that, for k > 2, a bipartite graph of average degree at least 4k and girth g contains cycles of (g/2 − 1)k consecutive even lengths. Expand
Cycles of even length in graphs
Abstract In this paper we solve a conjecture of P. Erdos by showing that if a graph G n has n vertices and at least 100kn 1+ 1 k edges, then G contains a cycle C 2 l of length 2 l for every integer lExpand
On the Turán number for the hexagon
Abstract A long-standing conjecture of Erdős and Simonovits is that ex ( n , C 2 k ) , the maximum number of edges in an n -vertex graph without a 2 k -gon is asymptotically 1 2 n 1 + 1 / k as nExpand
Polarities and 2k-cycle-free graphs
An infinite family {Gi} of C6–free graphs with |E(Gi)| ∼ 12 |V (Gi)| , i → ∞ is constructed, which improves the constant in the previous best lower bound on ex(v,C6) from 2/3 ≈ .462 to 1/2. Expand
Compactness results in extremal graph theory
The main purpose of this paper is to prove some compactness results for the case when L consists of cycles, and one of the main tools will be finding lower bounds on the number of pathsPk+1 in a graph ofn vertices andE edges. Expand
Properties of Certain Families of 2k-Cycle-Free Graphs
It is shown that the best known lower bound on the size of 2k-cycle-free extremal graphs for k = 3, 5, can be improved to ((k ? 1)·k? (k + 1)/k + o(1))v(k + 2)/k, and it is proved that all but finitely many such graphs must be either non-bipartite or have girth at most 2k ? 2. Expand
Graphs without quadrilaterals
  • Z. Füredi
  • Computer Science, Mathematics
  • J. Comb. Theory, Ser. B
  • 1983
The old conjecture f(q2 + q +1) = 12q(q + 1)2 is proved for infinitely many q (whenever q = 2k). Expand
Explicit Construction of Graphs with an Arbitrary Large Girth and of Large Size
This paper gives an explicit construction of a q -regular bipartite graph on v = 2 q k vertices with girth g ⩾ k + 5 which is the incidence graph of a flag-transitive semiplane. Expand
Minimal regular graphs of girths eight and twelve
In (3) Tutte showed that the order of a regular graph of degree d and even girth g > 4 is greater than or equal to Here the girth of a graph is the length of the shortest circuit. It was shown in (2)Expand
The sequence of primes is both an A and a B sequence . Our A and B sequences seem of be very much more general, but our theorems show that they cannot be very much more dense than the sequence of theExpand