Spanning embeddings of arrangeable graphs with sublinear bandwidth

Böttcher, Julia; Taraz, Anusch; Würfl, Andreas

doi:10.1002/rsa.20593

Cited by 6 publications

(12 citation statements)

References 28 publications

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“…We remark that Theorem 1.1 had been conjectured by Bollobás and Komlós [26]. Since the bandwidth theorem was proven, a number of variants of the result have been obtained, including for arrangeable graphs [10] and degenerate graphs [30] and in the setting of random and pseudorandom graphs [1,5,23], as well as for robustly expanding graphs [24]. Very recently, a bandwidth theorem for approximate decompositions was proven by Condon, Kim, Kühn, and Osthus [12], whilst Glock and Joos [20] proved a -bounded edge colouring extension of Theorem 1.1.…”

Section: Theorem 11 (The Bandwidth Theorem Böttcher Schacht and Tmentioning

confidence: 78%

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: Theorem 11 (The Bandwidth Theorem Böttcher Schacht and Tmentioning

confidence: 78%

The bandwidth theorem for locally dense graphs

Staden

Treglown

2020

Forum of Mathematics, Sigma

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“…Indeed, it was shown in [11] that for bounded degree n-vertex graphs, restricting the bandwidth to o(n) is equivalent to restricting the treewidth to o(n) or forbidding linear sized expanding subgraphs, which implies that bounded degree planar graphs, or more generally classes of bounded degree graphs defined by forbidding some fixed minor have bandwidth o(n). Generalisations of Theorem 1 were obtained in [9,13,27,33].…”

Section: Introductionmentioning

confidence: 79%

“…Lemma 27 is a strengthened version of [13,Lemma 8]. The proof of [13,Lemma 8] is deterministic; here we use a probabilistic argument to show the existence of a function f that also satisfies the additional property (H 6), which is required for Theorem 8. However, we still borrow ideas from the proof of [13,Lemma 8].…”

Section: Main Lemmasmentioning

confidence: 99%

The Bandwidth Theorem in sparse graphs

Allen

Böttcher

Ehrenmüller

et al. 2020

AiC

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One of the first results in graph theory was Dirac's theorem which claims that if the minimum degree in a graph is at least half of the number of vertices, then it contains a Hamiltonian cycle. This result has inspired countless other results all stating that in dense graphs we can find sparse spanning subgraphs. Along these lines, one of the most far-reaching results is the celebrated _Bandwidth Theorem_, proved around 10 years ago by Böttcher, Schacht, and Taraz. It states, rougly speaking, that every $n$-vertex graph with minimum degree at least $\left( \frac{r-1}{r} + o(1)\right) n$ contains a copy of all $n$-vertex graphs $H$ such that $\chi(H) \leq r$, $\Delta (H) = O(1)$, and the bandwidth of $H$ is $o(n)$. This was conjectured earlier by Bollobás and Komlós. The proof is using the Regularity method based on the Regularity Lemma and the Blow-up Lemma. Ever since the Bandwith Theorem came out, it has been open whether one could prove a similar statement for sparse random graphs. In this remarkable, deep paper the authors do just that, they establish sparse random analogues of the Bandwidth Theorem. In particular, the authors show that, for every positive integer $\Delta$, if $p \gg \left(\frac{\log{n}}{n}\right)^{1/\Delta}$, then asymptotically almost surely, every subgraph $G\subseteq G(n, p)$ with $\delta(G) \geq \left( \frac{r-1}{r} + o(1)\right) np$ contains a copy of every $r$-colourable spanning (i.e., $n$-vertex) graph $H$ with maximum degree at most $\Delta$ and bandwidth $o(n)$, provided that $H$ contains at least $C p^{-2}$ vertices that do not lie on a triangle (of $H$). (The requirement about vertices not lying on triangles is necessary, as pointed out by Huang, Lee, and Sudakov.) The main tool used in the proof is the recent monumental sparse Blow-up Lemma due to Allen, Böttcher, Hàn, Kohayakawa, and Person.

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“…Lemma 2.3 without Property (H5) is a special case of Lemma 8 in [7] and Property (H5) can be derived from its proof.…”

Section: Outline Of the Proofmentioning

confidence: 98%

Local resilience of spanning subgraphs in sparse random graphs

Allen

Böttcher

Ehrenmüller

et al. 2015

Electronic Notes in Discrete Mathematics

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For each real γ > 0 and integers ∆ ≥ 2 and k ≥ 1, we prove that there exist constants β > 0 and C > 0 such that for all p ≥ C(log n/n) 1/∆ the random graph G(n, p) asymptotically almost surely contains -even after an adversary deletes an arbitrary (1/k − γ)-fraction of the edges at every vertex -a copy of every nvertex graph with maximum degree at most ∆, bandwidth at most βn and at least C max{p −2 , p −1 log n} vertices not in triangles.

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Spanning embeddings of arrangeable graphs with sublinear bandwidth

Cited by 6 publications

References 28 publications

The bandwidth theorem for locally dense graphs

The bandwidth theorem for locally dense graphs

The Bandwidth Theorem in sparse graphs

Local resilience of spanning subgraphs in sparse random graphs

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