Embedding spanning subgraphs in uniformly dense and inseparable graphs

Ebsen, Oliver; Maesaka, Giulia S.; Reiher, Christian; Schacht, Mathias; Schülke, Bjarne

doi:10.1002/rsa.20957

Cited by 6 publications

(15 citation statements)

References 13 publications

(36 reference statements)

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“…Additional note. Since this paper was first submitted, Ebsen, Maesaka, Reiher, Schacht, and Schülke [14] have built on our work to generalise Theorem 1.2. Indeed, they replace the minimum degree condition on with an inseparability condition.…”

Section: Introduction and Resultsmentioning

confidence: 99%

The bandwidth theorem for locally dense graphs

Staden

Treglown

2020

Forum of Mathematics, Sigma

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The bandwidth theorem of Böttcher, Schacht, and Taraz [Proof of the bandwidth conjecture of Bollobás andKomlós, Mathematische Annalen, 2009] gives a condition on the minimum degree of an n-vertex graph G that ensures G contains every r-chromatic graph H on n vertices of bounded degree and of bandwidth $o(n)$ , thereby proving a conjecture of Bollobás and Komlós [The Blow-up Lemma, Combinatorics, Probability, and Computing, 1999]. In this paper, we prove a version of the bandwidth theorem for locally dense graphs. Indeed, we prove that every locally dense n-vertex graph G with $\delta (G)> (1/2+o(1))n$ contains as a subgraph any given (spanning) H with bounded maximum degree and sublinear bandwidth.

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Section: Introduction and Resultsmentioning

confidence: 99%

The bandwidth theorem for locally dense graphs

Staden

Treglown

2020

Forum of Mathematics, Sigma

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“…This result was further generalised by Ebsen, Maesaka, Reiher, Schacht and Schülke [22] by replacing the minimum degree with a condition of inseparability. For µ > 0, we say a graph G is µ-inseparable if e(X, Y ) µ|X||Y | for every partition {X, Y } of the vertex set of G. Ebsen et al proved Hamiltonicity for powers of cycles in locally dense and inseparable graphs, and extended this to the bandwidth setting.…”

Section: Locally Dense and Inseparable Graphs Another Type Of Suffici...mentioning

confidence: 74%

“…For the (ordinary) Hamilton cycles, this was done by Kühn, Osthus and Treglown [52] by means of the notion of robust expanders. Ebsen, Maesaka, Reiher, Schacht and Schülke [22] raised the question of generalising the concept of robust expanders to handle powers of cycles and other k-colourable graphs.…”

Section: Introductionmentioning

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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“…For comparison to Theorem 1.2, we note that 1 − 1 χ = k−2 k−1 (1 + α k−2 ). For example, the required minimum degree in the general case for a K 1,2,2 -tiling (or indeed a C 5 -tiling) is 3 5 n, while in the regular case it may be 8 15 n. It is also interesting to note that the new lower bound on δ has derivative tending to 0 as a function of χ when χ approaches k − 1 (equivalently when α approaches 0) unlike the analogous function derived from Theorem 1.2.…”

Section: Discussionmentioning

confidence: 99%

“…Uniformly dense graphs have been investigated in the context of (powers of) Hamilton cycles and graphs of sublinear bandwidth by Staden and Treglown [33] as well as Ebsen, Maesaka, Reiher, Schacht and Schülke [8]. However, in order to guarantee such connected guest graphs one needs to impose further assumptions on the host graphs, which they call 'robust inseparability' and which are stronger than linear minimum degree.…”

Section: Introductionmentioning

confidence: 99%

Sufficient conditions for perfect mixed tilings

Hurley¹,

Joos²,

Lang³

2022

Preprint

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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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Embedding spanning subgraphs in uniformly dense and inseparable graphs

Cited by 6 publications

References 13 publications

The bandwidth theorem for locally dense graphs

The bandwidth theorem for locally dense graphs

On sufficient conditions for Hamiltonicity in dense graphs

Sufficient conditions for perfect mixed tilings

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