A note on the Turán function of even cycles

Pikhurko, Oleg

doi:10.1090/s0002-9939-2012-11274-2

Cited by 47 publications

(50 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Bondy and Simonovits [3] (and independently Erdős) showed that ex(n, C 2k ) = O(n 1+1/k ). Subsequent improvements on the leading coefficient were found by Verstraëte [23] and more recently by Pikhurko [20].…”

Section: Introductionmentioning

confidence: 90%

Turán Numbers of Subdivided Graphs

Jiang¹,

Seiver²

2012

SIAM J. Discrete Math.

View full text Add to dashboard Cite

Abstract. Given a positive integer n and a graph F , the Turán number ex(n, F ) is the maximum number of edges in an n-vertex simple graph that does not contain F as a subgraph. Let H be a graph and p a positive even integer. Let H (p) denote the graph obtained from H by subdividing each of its edges p − 1 times. We prove that ex(n, H (p) ) = O(n 1+ (16/p) ). This follows from a more general result that we establish, where different edges of H are allowed to be subdivided different numbers of times. Our result is closely related to the results of Jiang [J. Graph Theory, 67 (2011), pp. 139-152] and of Kostochka and Pyber [Combinatorica, 8 (1988), pp. 83-86] on topological minors.

show abstract

Section: Introductionmentioning

confidence: 90%

Turán Numbers of Subdivided Graphs

Jiang¹,

Seiver²

2012

SIAM J. Discrete Math.

View full text Add to dashboard Cite

show abstract

“…We start by mentioning the Turán numbers of even cycles in the graph case. A classic theorem of Bondy and Simonovits [4] shows that ex(n, C 2k ) ≤ 100k · n 1+1/k , and this bound was improved by several authors in [36,30,6]. The current best known upper bound is the following one obtained by Bukh and Jiang [6] ex(n, C 2k ) ≤ 80 k log k · n 1+1/k + O(n).…”

Section: Turán Numbers Of Berge Cycles In R-graphsmentioning

confidence: 99%

“…A classic theorem of Bondy and Simonovits [4] shows that ex(n, C 2k ) ≤ 100k · n 1+1/k . This bound was subsequently improved by several authors in [36,30,6]. The Turán problem for cycles in hypergraphs has been investigated for different notions of hypergraph cycles in [3,17,18,20,21,25] among others.…”

mentioning

confidence: 99%

Cycles of given lengths in hypergraphs

Jiang

2018

Journal of Combinatorial Theory, Series B

View full text Add to dashboard Cite

In this paper, we develop a method for studying cycle lengths in hypergraphs. Our method is built on earlier ones used in [21,22,18]. However, instead of utilizing the well-known lemma of Bondy and Simonovits [4] that most existing methods do, we develop a new and very simple lemma in its place. One useful feature of the new lemma is its adaptiveness for the hypergraph setting.Using this new method, we prove a conjecture of Verstraëte [37] that for r ≥ 3, every r-uniform hypergraph with average degree Ω(k r−1 ) contains Berge cycles of k consecutive lengths. This is sharp up to the constant factor. As a key step and a result of independent interest, we prove that every r-uniform linear hypergraph with average degree at least 7r(k + 1) contains Berge cycles of k consecutive lengths.In both of these results, we have additional control on the lengths of the cycles, which therefore also gives us bounds on the Turán numbers of Berge cycles (for even and odd cycles simultaneously). In relation to our main results, we obtain further improvements on the Turán numbers of Berge cycles and the Zarankiewicz numbers of even cycles. We will also discuss some potential further applications of our method. 1 a fixed integer k. This was proposed by Burr and Erdős [9] forty years ago and some conjectures were formalized by Thomassen [34] in 1983 (we refer interested readers to [33] for a thorough introduction).Our work is also closely related to the so-called Turán problem. Let F be a family of r-graphs. An r-graph is F-free if it does not contain any member of F as a subhypergraph. The Turán number ex r (n, F) of the family F denotes the maximum number of hyperedges contained in an n-vertex F-free r-graph. When r = 2, we will write it as ex(n, F). Studying the Turán function ex(n, F) for graphs and hypergraphs has been a central problem in extremal graph theory ever since the work of P. Turán [35]. For non-bipartite graphs, the problem is asymptotically solved by the celebrated Erdős-Stone-Simonovits Theorem [14] (see also [13]). However, the Turán problem for bipartite graphs remains mostly open, with the special case for even cycles C 2k receiving particular attention. A classic theorem of Bondy and Simonovits [4] shows that ex(n, C 2k ) ≤ 100k · n 1+1/k . This bound was subsequently improved by several authors in [36,30,6]. The Turán problem for cycles in hypergraphs has been investigated for different notions of hypergraph cycles in [3,17,18,20,21,25] among others. Our work in this paper focuses on so-called Berge cycles. The method we develop here also works for some other common notions of cycles, such as so-called linear cycles (or sometimes known as loose cycles). See Section 6 for discussion in that direction.A hypergraph H = (V, E) consists of a set V of vertices and a collection E of subsets of V. We call a member of E a hyperedge or simple an edge of H. A hypergraph H is r-uniform if all of its edges are r-subsets of V(H). We also simply call an r-uniform hypergraph an r-graph for brevity. A Berge path of length ℓ, is...

show abstract

“…In particular, for graphs (i.e., 2-uniform hypergraphs) the C 2 -free condition does not impose any restriction, and there is no difference between a (Berge) cycle C l and a linear cycle C lin l . Bondy and Simonovits [5] showed that for k ≥ 2, ex(n, C 2k ) ≤ c k n 1+ 1 k for all sufficiently large n. Improvements to the constant factor c k are made in [22,18,7]. The girth of a graph is the length of a shortest cycle contained in the graph.…”

Section: Introductionmentioning

confidence: 99%

Asymptotics for Turán numbers of cycles in 3-uniform linear hypergraphs

Ergemlidze¹,

Győri

Methuku³

2019

Journal of Combinatorial Theory, Series A

View full text Add to dashboard Cite

Let F be a family of 3-uniform linear hypergraphs. The linear Turán number of F is the maximum possible number of edges in a 3-uniform linear hypergraph on n vertices which contains no member of F as a subhypergraph.In this paper we show that the linear Turán number of the five cycle C 5 (in the Berge sense) is 1 3 √ 3 n 3/2 asymptotically. We also show that the linear Turán number of the four cycle C 4 and {C 3 , C 4 } are equal asmptotically, which is a strengthening of a theorem of Lazebnik and Verstraëte [17].We establish a connection between the linear Turán number of the linear cycle of length 2k + 1 and the extremal number of edges in a graph of girth more than 2k − 2. Combining our result and a theorem of Collier-Cartaino, Graber and Jiang [8], we obtain that the linear Turán number of the linear cycle of length 2k + 1 is Θ(n 1+ 1 k ) for k = 2, 3, 4, 6.

show abstract

A note on the Turán function of even cycles

Abstract: Abstract. The Turán 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 = C 2k , the cycle of length 2k, is one of the central open questions in this area that goes back to the 1930s. We prove that

Cited by 47 publications

References 17 publications

Turán Numbers of Subdivided Graphs

Turán Numbers of Subdivided Graphs

Cycles of given lengths in hypergraphs

Asymptotics for Turán numbers of cycles in 3-uniform linear hypergraphs

Contact Info

Product

Resources

About