The bandwidth theorem for locally dense graphs

Staden, Katherine; Treglown, Andrew

doi:10.1017/fms.2020.39

Cited by 7 publications

(11 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Moreover, the n ‐vertex graph G obtained from two cliques of size

false(1 false/ 2 + μ false/ 2 false) n

which intersect in

μ n

vertices is

μ

‐inseparable and

false(ϱ, 1 false/ 2 false)

‐dense (for any fixed

ϱ > 0

), while it fails to satisfy property for arbitrary subsets X and Y . This shows that our result is not covered by the observation of Glock and Joos [20, Concluding Remarks].…”

Section: Introductioncontrasting

confidence: 63%

“…It is easy to see that every graph G = ( V , E ) with minimum degree

δ false(G false) \geq false(1 false/ 2 + μ false) false| V false|

μ

‐inseparable and, consequently, Theorem 1.4 strengthens the result of Staden and Treglown for powers of Hamiltonian cycles 20.…”

Section: Introductionsupporting

confidence: 56%

“…Note that this example is ruled out by the following bipartite version of Definition 1.2, which requires

e false(X, Y false) = true| true{false(x, y false) \in X prefix\times Y : x y \in E false(G false) true} true| \geq d false| X false| false| Y false| prefix- ϱ false| V {false|}^{2}

for all subsets X , Y ⊆ V . It was observed by Glock and Joos (see [20, Concluding Remarks]) that imposing property on G allows a further relaxation of the minimum degree condition for G from to

μ false| V false|

for arbitrary

μ > 0

. We show that requiring property for all subsets X and Y is not needed.…”

Section: Introductionmentioning

confidence: 76%

“…In fact, it was shown by Staden and Treglown 20 that such a hereditary density assumption for any d > 0 and sufficiently small

ϱ > 0

allows us to reduce the minimum degree condition for k th powers of Hamiltonian cycles to

δ false(G false) \geq true(\frac{1}{2} + μ true) false| V false(G false) false|

for any

μ > 0

and sufficiently large vertex sets (see also 17 for K k + 1 ‐factors). In particular, the minimum degree condition becomes independent of k .…”

Section: Introductionmentioning

confidence: 99%

“…Staden and Treglown and also Glock and Joos not only considered the embedding problem for powers of Hamiltonian cycles, but also more generally for bounded degree graphs (with and without small bandwidth, see 20 for details). We also obtain a generalization in that direction and establish the following version of the bandwidth theorem from 2 for inseparable and uniformly dense graphs (see Theorem 1.5).…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Embedding spanning subgraphs in uniformly dense and inseparable graphs

Ebsen

Maesaka

Reiher

et al. 2020

Random Struct Algorithms

View full text Add to dashboard Cite

We consider sufficient conditions for the existence of kth powers of Hamiltonian cycles in n-vertex graphs G with minimum degree n for arbitrarily small > 0. About 20 years ago Komlós, Sarközy, and Szemerédi resolved the conjectures of Pósa and Seymour and obtained optimal minimum degree conditions for this problem by showing that = k k+1 suffices for large n. For smaller values of the given graph G must satisfy additional assumptions. We show that inducing subgraphs of density > 0 on linear subsets of vertices and being inseparable, in the sense that every cut has density at least > 0, are sufficient assumptions for this problem and, in fact, for a variant of the bandwidth theorem. This generalizes recent results of Staden and Treglown. KEYWORDS absorption method, bandwidth theorem, powers of Hamiltonian cycles 1 INTRODUCTION We study sufficient conditions for the existence of spanning subgraphs in large finite graphs and begin the discussion with powers of Hamiltonian cycles. For k ∈ N the kth power of a given graph H is the graph H k on the same vertex set with xy being an edge in H k if x and y are distinct vertices of H that are connected in H by a path of at most k edges. For simplicity, we refer to a kth power of a path with at least k vertices as a k-path. Moreover, we refer to the ordered k-tuples of the first and last k vertices This is an open access article under the terms of the Creative Commons Attribution-NonCommercial License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited and is not used for commercial purposes.

show abstract

“…Moreover, the n ‐vertex graph G obtained from two cliques of size

false(1 false/ 2 + μ false/ 2 false) n

which intersect in

μ n

vertices is

μ

‐inseparable and

false(ϱ, 1 false/ 2 false)

‐dense (for any fixed

ϱ > 0

), while it fails to satisfy property for arbitrary subsets X and Y . This shows that our result is not covered by the observation of Glock and Joos [20, Concluding Remarks].…”

Section: Introductioncontrasting

confidence: 63%

“…It is easy to see that every graph G = ( V , E ) with minimum degree

δ false(G false) \geq false(1 false/ 2 + μ false) false| V false|

μ

‐inseparable and, consequently, Theorem 1.4 strengthens the result of Staden and Treglown for powers of Hamiltonian cycles 20.…”

Section: Introductionsupporting

confidence: 56%

“…Note that this example is ruled out by the following bipartite version of Definition 1.2, which requires

e false(X, Y false) = true| true{false(x, y false) \in X prefix\times Y : x y \in E false(G false) true} true| \geq d false| X false| false| Y false| prefix- ϱ false| V {false|}^{2}

for all subsets X , Y ⊆ V . It was observed by Glock and Joos (see [20, Concluding Remarks]) that imposing property on G allows a further relaxation of the minimum degree condition for G from to

μ false| V false|

for arbitrary

μ > 0

. We show that requiring property for all subsets X and Y is not needed.…”

Section: Introductionmentioning

confidence: 76%

“…In fact, it was shown by Staden and Treglown 20 that such a hereditary density assumption for any d > 0 and sufficiently small

ϱ > 0

allows us to reduce the minimum degree condition for k th powers of Hamiltonian cycles to

δ false(G false) \geq true(\frac{1}{2} + μ true) false| V false(G false) false|

for any

μ > 0

and sufficiently large vertex sets (see also 17 for K k + 1 ‐factors). In particular, the minimum degree condition becomes independent of k .…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Embedding spanning subgraphs in uniformly dense and inseparable graphs

Ebsen

Maesaka

Reiher

et al. 2020

Random Struct Algorithms

View full text Add to dashboard Cite

show abstract

On Sufficient Conditions for Hamiltonicity in Dense Graphs

Lang

Sanhueza‐Matamala

2021

Trends in Mathematics

View full text Add to dashboard Cite

On the Relation of Separability, Bandwidth and Embedding

Csaba

Vásárhelyi

2019

Graphs and Combinatorics

View full text Add to dashboard Cite

In this paper we construct a class of bounded degree bipartite graphs with a small separator and large bandwidth. Furthermore, we also prove that graphs from this class are spanning subgraphs of graphs with minimum degree just slightly larger than n/2. * We remark, that the case of k = 2 was first proved by Abbasi [1], and is also a special case of a result by the author [11]. Some of the ideas of the latter proof are used in the present paper as well.

show abstract

The bandwidth theorem for locally dense graphs

Cited by 7 publications

References 29 publications

Embedding spanning subgraphs in uniformly dense and inseparable graphs

Embedding spanning subgraphs in uniformly dense and inseparable graphs

On Sufficient Conditions for Hamiltonicity in Dense Graphs

On the Relation of Separability, Bandwidth and Embedding

Contact Info

Product

Resources

About