A pair degree condition for Hamiltonian cycles in $3$-uniform hypergraphs

Schülke, Bjarne

doi:10.48550/arxiv.1910.02691

Cited by 5 publications

(9 citation statements)

References 12 publications

(28 reference statements)

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“…Finally, it is natural to ask whether these type of results generalise to hypergraphs. On the practical side, first steps in this direction were undertaken by Schülke [67] for tight cycles under codegree sequence conditions in 3-uniform hypergraphs and by and Bowtell and Hyde [9] for perfect matchings under minimum vertex degree conditions in 3-uniform graphs. On the theoretical side, Hamilton frameworks minimum degree conditions for tight Hamilton cycles have been proposed in our earlier work [53].…”

Section: Discussionmentioning

confidence: 99%

On sufficient conditions for Hamiltonicity in dense graphs

Lang¹,

Sanhueza‐Matamala²

2021

Preprint

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We study structural conditions in dense graphs that guarantee the existence of vertex-spanning substructures such as Hamilton cycles. It is easy to see that every Hamiltonian graph is connected, has a perfect fractional matching and, excluding the bipartite case, contains an odd cycle. Our main result in turn states that any large enough graph that robustly satisfies these properties must already be Hamiltonian. Moreover, the same holds for embedding powers of cycles and graphs of sublinear bandwidth subject to natural generalisations of connectivity, matchings and odd cycles.This solves the embedding problem that underlies multiple lines of research on sufficient conditions for Hamiltonicity in dense graphs. As applications, we recover and establish Bandwidth Theorems in a variety of settings including Ore-type degree conditions, Pósatype degree conditions, deficiency-type conditions, locally dense and inseparable graphs, multipartite graphs as well as robust expanders.structure, which are used as a black boxes. The proofs of these results can be found in Section 5.2.1. Ore-type conditions. Motivated by the Hajnal-Szemerédi Theorem, Kierstead and Kostochka [40] investigated optimal Ore-type conditions which ensure the existence of clique factors and proved the following result. Theorem 2.3 (Clique factors under Ore-type conditions). For n divisible by k, let G be a graph on n vertices with deg(x) + deg(y) 2 k−1 k n − 1 for all xy / ∈ E(G). Then G contains a k-clique factor.Note that the result is tight, as witnessed, for instance, by slightly imbalanced complete kpartite graphs. As an extension of this, Kühn, Osthus and Treglown [51] showed an Ore-type result for general factors (not just cliques). For sufficiently large graphs, Châu [13] proved a generalisation of Ore's theorem for squares of Hamilton cycles, and also conjectured generalisations of this for all k 3. For k = 2, Knox and Treglown [41] were able to strengthen Ore's theorem to (β, ∆, 2)-Hamiltonian graphs. They also conjectured corresponding extensions for all k 3. This was confirmed for k = 3 by Böttcher and Müller [6]. The following result proves these conjectures in a strong sense for all k 2. Theorem 2.4 (Bandwidth theorem under Ore-type conditions). For k, ∆ ∈ N and µ > 0, there are z, β > 0 and n 0 ∈ N with the following property. Let G be a graph on n n 0 vertices with deg(x) + deg(y) 2 k−1 k n + µn for all xy / ∈ E(G). Then G is (z, β, ∆, k)-Hamiltonian. 2.2. Pósa-type conditions. Balogh, Kostochka and Treglown [3, 4] studied degree conditions that guarantee the existence of clique factors and powers of Hamilton cycles. Treglown [71] proved the following Pósa-type result for clique factors. Theorem 2.5 (Clique factors under Pósa-type conditions). For k ∈ N and µ > 0, there is an n 0 ∈ N with the following property. Let G be a graph with degree sequence d 1 , . . . , d n where n n 0 is divisible by k. Suppose that d i k−2 k n + i + µn for every i n/k. Then G contains a k-clique factor.

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Section: Discussionmentioning

confidence: 99%

On sufficient conditions for Hamiltonicity in dense graphs

Lang¹,

Sanhueza‐Matamala²

2021

Preprint

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“…It is natural to consider what degree sequence results may be obtained which improve on known minimum degree thresholds for perfect matchings and other spanning structures, such as Hamiltonian cycles, and tilings of subgraphs other than K k k , in any k-graphs, not just for k = 3. Schulke [21] asks about vertex degree sequences that guarantee the existence of a Hamilton cycle in a 3-graph. Though the existence of a perfect matching does not imply the existence of a Hamilton cycle, our extremal examples containing no perfect matchings do clearly imply degree sequences for which a Hamilton cycle is not guaranteed.…”

Section: Concluding Discussion and Remarksmentioning

confidence: 99%

“…Furthermore, for n ∈ N and a 3-graph H = ([n], E) we define d(i, j) to be the number of edges of H containing both vertex i and vertex j. Generalising a result of Rödl, Ruciński and Szemerédi [20] on the asymptotic minimum co-degree threshold for a tight Hamilton cycle in a 3-graph, Schulke [21] proved the following: Theorem 1.2. For all γ > 0, there exists an n 0 ∈ N such that for n ≥ n 0 the following holds: If…”

Section: Introductionmentioning

confidence: 99%

“…Whilst considerable progress has been made with respect to degree sequence results in graphs, not as much headway has been made for hypergraphs. Very recently, Schulke [21] proved a Pósatype degree sequence result related to finding tight Hamilton cycles in 3-graphs. We say that a 3-graph H on n vertices contains a tight Hamilton cycle if there exists an ordering (v…”

Section: Introductionmentioning

confidence: 99%

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A degree sequence strengthening of the vertex degree threshold for a perfect matching in 3-uniform hypergraphs

Bowtell¹,

Hyde²

2020

Preprint

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The study of asymptotic minimum degree thresholds that force matchings and tilings in hypergraphs is a lively area of research in combinatorics. A key breakthrough in this area was a result of Hàn, Person and Schacht [6] who proved that the asymptotic minimum vertex degree threshold for a perfect matching in an n-vertex 3-graph is 5 9 + o(1) n 2 . In this paper we improve on this result, giving a family of degree sequence results, all of which imply the result of Hàn, Person and Schacht, and additionally allow one third of the vertices to have degree 1 9 n 2 below this threshold. Furthermore, we show that this result is, in some sense, tight.

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“…The oldest variant are Berge cycles [3], whose minimum degree conditions were studied by Bermond, Germa, Heydemann and Sotteau [4]. In the last two decades, however, research has increasingly focused on a stricter notion of Hamilton cycles [1,14,18,30,31,33,[37][38][39]43] introduced by Kierstead and Katona [22]. A tight cycle in 𝑘-graph is a cyclically ordered set of at least 𝑘 + 1 vertices such that every interval of 𝑘 subsequent vertices forms an edge.…”

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confidence: 99%

Minimum degree conditions for tight Hamilton cycles

Lang

Sanhueza‐Matamala

2022

Journal of London Math Soc

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We develop a new framework to study minimum 𝑑degree conditions in 𝑘-uniform hypergraphs, which guarantee the existence of a tight Hamilton cycle. Our main theoretical result deals with the typical absorption, path cover and connecting arguments for all 𝑘 and 𝑑 at once, and thus sheds light on the underlying structural problems. Building on this, we show that one can study minimum 𝑑-degree conditions of 𝑘-uniform tight Hamilton cycles by focusing on the inner structure of the neighbourhoods. This reduces the matter to an Erdös-Gallai-type question for (𝑘 − 𝑑)-uniform hypergraphs, which is of independent interest. Once this framework is established, we can easily derive two new bounds. Firstly, we extend a classic result of Rödl, Ruciński and Szemerédi for 𝑑 = 𝑘 − 1 by determining asymptotically best possible degree conditions for 𝑑 = 𝑘 − 2 and all 𝑘 ⩾ 3. This was proved independently by Polcyn, Reiher, Rödl and Schülke. Secondly, we provide a general upper bound of 1 − 1∕(2(𝑘 − 𝑑)) for the tight Hamilton cycle 𝑑-degree threshold in 𝑘-uniform hypergraphs, thus narrowing the gap to the lower bound of 1 − 1∕ √ 𝑘 − 𝑑 due to Han and Zhao.

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A pair degree condition for Hamiltonian cycles in $3$-uniform hypergraphs

Cited by 5 publications

References 12 publications

On sufficient conditions for Hamiltonicity in dense graphs

On sufficient conditions for Hamiltonicity in dense graphs

A degree sequence strengthening of the vertex degree threshold for a perfect matching in 3-uniform hypergraphs

Minimum degree conditions for tight Hamilton cycles

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