Cycles of even lengths modulo <i>k</i>

Diwan, Ajit A.

doi:10.1002/jgt.20477

Cited by 13 publications

(12 citation statements)

References 9 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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“…In 1983, Thomassen [19] conjectured that for all natural numbers m and k, every graph of minimum degree at least k + 1 contains a cycle of length 2m modulo k. Thomassen showed that minimum degree 4m(k + 1) suffices. Cai and Shreve [3] showed that claw-free graphs of minimum degree k + 1 have cycles of all lengths modulo k. Diwan [7] proved that graphs of minimum degree 2k − 1 have cycles of all even lengths modulo k and that Thomassen's conjecture holds for m = 2. The currently best known result is by Liu and Ma [12] who verified Thomassen's conjecture if k is even and showed that minimum degree k + 4 suffices if k is odd.…”

Section: Cycle Lengths Modulo Kmentioning

confidence: 99%

Cycle Lengths Modulo k in Large 3-connected Cubic Graphs, Advances in Combinatorics

Lyngsie

Merker

2021

AiC

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The existence of cycles with a given length is classical topic in graph theory with a plethora of open problems. Examples related to the main result of this paper include a conjecture of Burr and Erdős from 1976 asked whether for every integer $m$ and a positive odd integer $k$, there exists $d$ such that every graph with average degree at least $d$ contains a cycle of length $m$ modulo $k$; this conjecture was proven by Bollobás in [Bull. London Math. Soc. 9 (1977), 97-98]( https://doi.org/10.1112/blms/9.1.97). Another example is a problem of Erdős from the 1990s asking whether there exists $A\subseteq\mathbb{N}$ with zero density and constants $n_0$ and $d_0$ such that every graph with at least $n_0$ vertices and the average degree at least $d_0$ contains a cycle with length in the set $A$, which was resolved by Verstraete in [J. Graph Theory 49 (2005), 151-167]( https://doi.org/10.1002/jgt.20072). In 1983, Thomassen conjectured that for all integers $m$ and $k$, every graph with minimum degree $k+1$ contains a cycle of length $2m$ modulo $k$. Note that the parity condition in the first and the third conjectures is necessary because of bipartite graphs. The current paper contributes to this long line of research by proving that for every integer $m$ and a positive odd integer $k$, every sufficiently large $3$-connected cubic graph contains a cycle of length $m$ modulo $k$. The result is the best possible in the sense that the same conclusion is not true for $2$-connected cubic graphs or $3$-connected graphs with minimum degree three.

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“…A theorem of Verstraëte [35] implies that for all k, every graph with average degree at least 8k contains cycles of all even lengths modulo k. For all odd k, a result of Fan [19] shows that minimum degree at least 3k − 2 suffices. Diwan [12] obtained a better bound for Conjecture 1.7 that for every positive integer k, every graph G with minimum degree at least 2k − 1 contains cycles of all even lengths modulo k, and every graph with minimum degree at least k + 1 contains a cycle of length 4 modulo k. For Conjecture 1.8, a recent result of [27] about consecutive odd cycles implies that minimum degree Ω(k) is suffices to ensure the existence of cycles of all lengths modulo k.…”

Section: Cycle Lengths Modulo Kmentioning

confidence: 99%

Cycle lengths and minimum degree of graphs

Liu

2018

Journal of Combinatorial Theory, Series B

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There has been extensive research on cycle lengths in graphs with large minimum degree. In this paper, we obtain several new and tight results in this area. Let G be a graph with minimum degree at least k + 1. We prove that if G is bipartite, then there are k cycles in G whose lengths form an arithmetic progression with common difference two. For general graph G, we show that G contains ⌊k/2⌋ cycles with consecutive even lengths and k − 3 cycles whose lengths form an arithmetic progression with common difference one or two. In addition, if G is 2-connected and non-bipartite, then G contains ⌊k/2⌋ cycles with consecutive odd lengths.Thomassen (1983) made two conjectures on cycle lengths modulo a fixed integer k: (1) every graph with minimum degree at least k + 1 contains cycles of all even lengths modulo k; (2) every 2-connected non-bipartite graph with minimum degree at least k + 1 contains cycles of all lengths modulo k. These two conjectures, if true, are best possible. Our results confirm both conjectures when k is even. And when k is odd, we show that minimum degree at least k + 4 suffices. This improves all previous results in this direction. Moreover, our results derive new upper bounds of the chromatic number in terms of the longest sequence of cycles with consecutive (even or odd) lengths.

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Cycles of even lengths modulo k

Cited by 13 publications

References 9 publications

The Number of Triple Systems Without Even Cycles

The Number of Triple Systems Without Even Cycles

Cycle Lengths Modulo k in Large 3-connected Cubic Graphs, Advances in Combinatorics

Cycle lengths and minimum degree of graphs

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