Cycle lengths and minimum degree of graphs

Liu, Chun Hung; Ma, Jie

doi:10.1016/j.jctb.2017.08.002

Cited by 32 publications

(41 citation statements)

References 34 publications

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“…The conjecture received new attention recently. In particular Liu-Ma [41] settled the case when h is even, and Diwan [24] proved it for m = 2. To date, Sudakov and Verstraëte [52] hold the best known bound for the general case on this problem.…”

Section: Proof Of Lemma 11mentioning

confidence: 97%

The Number of Triple Systems Without Even Cycles

Mubayi

Wang

2019

Combinatorica

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For k 4, a loose k-cycle C k is a hypergraph with distinct edges e 1 , e 2 , . . . , e k such that consecutive edges (modulo k) intersect in exactly one vertex and all other pairs of edges are disjoint. Our main result is that for every even integer k 4, there exists c > 0 such that the number of triple systems with vertex set [n] containing no C k is at most 2 cn 2 .An easy construction shows that the exponent is sharp in order of magnitude. This may be viewed as a hypergraph extension of the work of Morris and Saxton, who proved the analogous result for graphs which was a longstanding problem. For r-uniform hypergraphs with r > 3, we improve the trivial upper bound but fall short of obtaining the order of magnitude in the exponent, which we conjecture is n r−1 .Our proof method is different than that used for most recent results of a similar flavor about enumerating discrete structures, since it does not use hypergraph containers. One novel ingredient is the use of some (new) quantitative estimates for an asymmetric version of the bipartite canonical Ramsey theorem.F as a (not necessarily induced) subgraph. Henceforth we will call G an F -free r-graph. Write Forb r (n, F ) for the set of F -free r-graphs with vertex set [n]. Since each subgraph of an F -free r-graph is also F -free, we trivially obtain |Forb r (n, F )| 2 ex r (n,F ) by taking subgraphs of an F -free r-graph on [n] with the maximum number of edges. On the other hand for fixed r and F , |Forb r (n, F )| i exr(n,F ) n r i = 2 O(exr(n,F ) log n) , so the issue at hand is the factor log n in the exponent. The work of Erdős-Kleitman-Rothschild [25] and Erdős-Frankl-Rödl [26] for graphs, and Nagle-Rödl-Schacht [45] for hypergraphs (see also [44] for the case r = 3) improves the upper bound above to obtain |Forb r (n, F )| = 2 ex r (n,F )+o(n r ) .Although much work has been done to improve the exponent above (see [1,6,7,8,31,34,48] for graphs and [10,11,21,47,13,50] for hypergraphs), this is a somewhat satisfactory state of affairs when ex r (n, F ) = Ω(n r ) or F is not r-partite.

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Section: Proof Of Lemma 11mentioning

confidence: 97%

The Number of Triple Systems Without Even Cycles

Mubayi

Wang

2019

Combinatorica

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“…Since then there has been an extensive research [15,32,26,28] on related topics. Very recently Liu and the second author [27] proved a tight result that every graph G with minimum degree at least 2k + 1 contains k cycles of consecutive even lengths; and if G is 2-connected and non-bipartite, then G also contains k cycles of consecutive odd lengths. Among others, one closely related problem is the study of cycle lengths modulo then immediately yields bounds on the Turán numbers of Berge cycles (for both even and odd cycles simultaneously).…”

Section: Introductionmentioning

confidence: 99%

Cycles of given lengths in hypergraphs

Jiang

2018

Journal of Combinatorial Theory, Series B

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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...

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“…• Thomassen [27] conjectured that every graph of minimum degree at least k + 1 contains cycles of all possible even lengths mod k. This conjecture was proved when k is even by Liu and Ma [19], and they showed further that there are cycles of k/2 consecutive even lengths in this case. Liu and Ma also showed that if k is odd and G is a graph of minimum degree at least k + 5, then G contains cycles of all even lengths mod k. It is natural to conjecture the following strengthening of Thomassen's conjecture [27]: Conjecture 8.…”

Section: Discussionmentioning

confidence: 99%

“…Thomassen [27] conjectured that every graph of minimum degree at least k + 1 contains cycles of all possible even lengths mod k. Since every graph with average degree a least 2k contains a subgraph of minimum degree at least k + 1, this conjecture implies that c k (ℓ) ≤ 2k when k is even. Recently, Liu and Ma [19] showed for k even that a graph of minimum degree at least k + 1 contains cycles of all possible even lengths mod k, answering a conjecture of Thomassen [27] for even values of k. Since any graph whose blocks are cliques of order k + 1 contains no cycle of length 2 modulo k, this is best possible.…”

Section: Introductionmentioning

confidence: 86%

The Extremal Function for Cycles of Length $\ell$ mod $k$

Sudakov

Verstraëte

2017

Electron. J. Combin.

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Burr and Erdős conjectured that for each k, ℓ ∈ Z + such that kZ + ℓ contains even integers, there exists c k (ℓ) such that any graph of average degree at least c k (ℓ) contains a cycle of length ℓ mod k. This conjecture was proved by Bollobás, and many successive improvements of upper bounds on c k (ℓ) appear in the literature. In this short note, for 1 ≤ ℓ ≤ k, we show that c k (ℓ) is proportional to the largest average degree of a C ℓ -free graph on k vertices, which determines c k (ℓ) up to an absolute constant. In particular, using known results on Turán numbers for even cycles, we obtain c k (ℓ) = O(ℓk 2/ℓ ) for all even ℓ, which is tight for ℓ ∈ {4, 6, 10}. Since the complete bipartite graph K ℓ−1,n−ℓ+1 has no cycle of length 2ℓ mod k, it also shows c k (ℓ) = Θ(ℓ) for ℓ = Ω(log k).

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Cycle lengths and minimum degree of graphs

Cited by 32 publications

References 34 publications

The Number of Triple Systems Without Even Cycles

The Number of Triple Systems Without Even Cycles

Cycles of given lengths in hypergraphs

The Extremal Function for Cycles of Length $\ell$ mod $k$

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