Embedding into Bipartite Graphs

Böttcher, Julia; Heinig, Peter Christian; Taraz, Anusch

doi:10.1137/090765481

Cited by 11 publications

(8 citation statements)

References 38 publications

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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 [9,13,25,31].…”

Section: Introductionmentioning

confidence: 72%

The Bandwidth Theorem in Sparse Graphs

Allen¹,

Böttcher²,

Ehrenmüller³

et al. 2016

Preprint

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The bandwidth theorem [Mathematische Annalen, 343(1):175-205, 2009] states that any n-vertex graph G with minimum degree k−1 k + o(1) n contains all n-vertex k-colourable graphs H with bounded maximum degree and bandwidth o(n). We provide sparse analogues of this statement in random graphs as well as pseudorandom graphs.More precisely, we show that for pwith minimum degree k−1 k +o(1) pn contains all n-vertex k-colourable graphs H with maximum degree ∆, bandwidth o(n), and at least Cp −2 vertices not contained in any triangle. A similar result is shown for sufficiently bijumbled graphs, which, to the best of our knowledge, is the first resilience result in pseudorandom graphs for a rich class of subgraphs. Finally, we provide improved results for H with small degeneracy, which in particular imply a resilience result in G(n, p) with respect to the containment of spanning bounded degree trees for p ≫ log n n 1/3 .

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Section: Introductionmentioning

confidence: 72%

The Bandwidth Theorem in Sparse Graphs

Allen¹,

Böttcher²,

Ehrenmüller³

et al. 2016

Preprint

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Section: Introductionmentioning

confidence: 78%

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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“…The following corollary of the Blow-up Lemma (see Böttcher, Heinig and Taraz [6,7]) asserts that in the setup of Definition 4 graphs H of bounded degree can be embedded into G, if G admits a partition being sufficiently regular on T and super-regular on M . Lemma 9 (Embedding Lemma [6,7]) For all d, ∆ > 0 there is a constant ǫ = ǫ(d, ∆) > 0 such that the following holds. Let G = (V, E) be an N -vertex graph that has a partition ∪ t i=1 V i with (ǫ, d)-reduced graph H on [t] which is (ǫ, d)-super-regular on a graph M ⊂ T .…”

Section: Preliminariesmentioning

confidence: 99%

Three-color Ramsey number of an odd cycle versus bipartite graphs with small bandwidth

You¹,

Lin²

2022

Preprint

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A graph H = (W, E H ) is said to have bandwidth at most b if there exists a labeling of W as w 1 , w 2 , . . . , w n such that |i−j| ≤ b for every edge w i w j ∈ E H . We say H is a balanced (β, ∆)graph if it is a bipartite graph with bandwidth at most β|W | and maximum degree at most ∆, and it also has a proper 2-coloring χIn this paper, we prove that for every γ > 0 and every natural number ∆, there exist a constant β > 0 such that for every balanced (β, ∆)-graph H on n vertices we havefor all sufficiently large odd n. The inequality is asymptotically tight if H is a connected balanced (β, ∆)-graph on n vertices. Let θ n,t be the graph consisting of t internally disjoint paths of length n all sharing the same endpoints. As a corollary, for each fixed t ≥ 1, R(θ n,t , θ n,t , C nt+λ ) = (3t + o(1))n, where λ = 0 if nt is odd and λ = 1 if nt is even. In particular, R(C 2n , C 2n , C 2n+1 ) = (6 + o(1))n.

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Embedding into Bipartite Graphs

Cited by 11 publications

References 38 publications

The Bandwidth Theorem in Sparse Graphs

The Bandwidth Theorem in Sparse Graphs

The Bandwidth Theorem in sparse graphs

Three-color Ramsey number of an odd cycle versus bipartite graphs with small bandwidth

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