A Degree Sequence Version of the Kühn–Osthus Tiling Theorem

Hyde, Joseph; Treglown, Andrew

doi:10.37236/8986

Cited by 7 publications

(8 citation statements)

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“…Hyde, Liu and Treglown [14] proved an extension of Komlós' theorem, which states that a graph G on n vertices with a (χ cr (F ), µ)-strong degree sequence contains any F -tiling on n − o(n) vertices for fixed F ∈ F cr . In another paper, Hyde and Treglown [15] subsequently strengthened this to perfect F -tilings, which generalises the results of Kühn and Osthus [21]. We use the first result of Hyde, Liu and Treglown [14] as a black box to give the following extension of the latter result [15] (and Theorem 1.1).…”

Section: Introductionmentioning

confidence: 61%

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Sufficient conditions for perfect mixed tilings

Hurley¹,

Joos²,

Lang³

2022

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We develop a method to study sufficient conditions for perfect mixed tilings. Our framework allows the embedding of bounded degree graphs H with components of sublinear order. As a corollary, we recover and extend the work of Kühn and Osthus regarding sufficient minimum degree conditions for perfect F -tilings (for an arbitrary fixed graph F ) by replacing the F -tiling with the aforementioned graphs H. Moreover, we obtain analogous results for degree sequences and in the setting of uniformly dense graphs. Finally, we asymptotically resolve a conjecture of Komlós in a strong sense. Date: 12th January 2022. The research leading to these results was partially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -428212407 and 450397222. 1 To illustrate this parameter further, consider the complete k-partite graph K a,(k−1) * b whose parts have sizes a, b, . . . , b with a b. (In the literature, K a,(k−1) * b is often called a bottle graph.) Then χcr(K a,(k−1) * b ) = k −1+ a b . So by sliding a b within [0, 1], the critical chromatic number transitions smoothly between k − 1 and k. Conversely, for every graph F with χ(F ) = k, there exist a, b such that χcr(F ) = χcr(K a,(k−1) * b ).

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Section: Introductionmentioning

confidence: 61%

“…In another paper, Hyde and Treglown [15] subsequently strengthened this to perfect F -tilings, which generalises the results of Kühn and Osthus [21]. We use the first result of Hyde, Liu and Treglown [14] as a black box to give the following extension of the latter result [15] (and Theorem 1.1).…”

Section: Introductionmentioning

confidence: 61%

Sufficient conditions for perfect mixed tilings

Hurley¹,

Joos²,

Lang³

2022

Preprint

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“…We remark that each of these applications benefits from already established insights on clique factors and, in case of uniformly dense and inseparable graphs, the connectivity We remark that the degree conditions in Theorem 2.5 are best possible apart from the term µn. This result further was extended by Hyde, Liu and Treglown [34] and Hyde and Treglown [35] to degree conditions which ensure factors and partial factors of arbitrary graphs (not just cliques).…”

Section: Applicationsmentioning

confidence: 78%

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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“…Furthermore, since there are in total between |V 3/9 |+|V 4/9 |−γn and |V 3/9 |+|V 4/9 | small and medium vertices in V (M ), we have that |V 3/9 |+ |V 4/9 |− γn ≤ 3|E bbb |+ 2|E bbB |+ |E bBB | ≤ |V 3/9 | + |V 4/9 |. Thus there exists(7)…”

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confidence: 94%

“…Asymptotically answering a conjecture of Balogh, Kostochka and Treglown [2], Treglown [23] proved a Pósa-type degree sequence version of Hajnal and Szemerédi's [5] perfect K r -tiling result (as well as a degree sequence strengthening of Alon and Yuster's [1] perfect H-tiling result for general graphs H). Using ideas from [23], Hyde, Liu and Treglown [8] proved a Pósa-type degree sequence strengthening of Komlós' [14] almost-perfect tiling theorem which was then utilised by Hyde and Treglown [7] to give a Pósa-type degree sequence version of Kühn and Osthus' [15] perfect tiling theorem. See [13,17,22] for further examples of degree sequence results.…”

Section: Introductionmentioning

confidence: 99%

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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A Degree Sequence Version of the Kühn–Osthus Tiling Theorem

Cited by 7 publications

References 19 publications

Sufficient conditions for perfect mixed tilings

Sufficient conditions for perfect mixed tilings

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

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