Topological ubiquity of trees

Bowler, Nathan; Elbracht, Christian; Erde, Joshua; Gollin, J. Pascal; Heuer, Karl; Pitz, Max; Teegen, Maximilian

doi:10.1016/j.jctb.2022.05.011

Cited by 5 publications

(11 citation statements)

References 13 publications

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“…To this end we use the following notation for such collections of nG in Γ, most of which we established in [4].…”

Section: Half-grid-like Endsmentioning

confidence: 99%

“…To show that a given graph

G

\preccurlyeq

‐ubiquitous, we shall assume that

nG\preccurlyeq {\rm{\Gamma }}

holds for every

n\in {\mathbb{N}}

an show that this implies

{\aleph }_{0}G\preccurlyeq {\rm{\Gamma }}

. To this end we use the following notation for such collections of

nG

{\rm{\Gamma }}

, most of which we established in [4].…”

Section: G $G$‐tribes and Concentration Of G $G$‐tribes Towards An Endmentioning

confidence: 99%

“…Since every countable tree is a minor of the half‐grid, Theorem 1.5 implies that all countable trees are

\preccurlyeq

‐ubiquitous, see Corollary 9.4. We remark that while it has been shown that all trees are ubiquitous with respect to the topological minor relation [4], the question of whether all uncountable trees are

\preccurlyeq

‐ubiquitous has remained open, and we hope to resolve this in a paper in preparation.…”

Section: Introductionmentioning

confidence: 99%

“…In Sections 6 and 7 we prove Theorem 1.2, classifying the thick ends which are nonpebbly. In Section 8 we reintroduce some concepts from [4] and show that we may assume that the

G

‐minors in

{\rm{\Gamma }}

are concentrated towards some end

\epsilon

{\rm{\Gamma }}

. In Section 9 we use the results of the previous section to prove Theorem 1.5 and finally in Section 10 we prove Theorem 1.3 and its corollaries.…”

Section: Introductionmentioning

confidence: 99%

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Ubiquity of graphs with nowhere‐linear end structure

Bowler

Elbracht

Erde

et al. 2023

Journal of Graph Theory

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A graph G $G$ is said to be ≼ $\preccurlyeq $‐ubiquitous, where ≼ $\preccurlyeq $ is the minor relation between graphs, if whenever normalΓ ${\rm{\Gamma }}$ is a graph with nG≼normalΓ $nG\preccurlyeq {\rm{\Gamma }}$ for all n∈double-struckN $n\in {\mathbb{N}}$, then one also has ℵ0G≼normalΓ ${\aleph }_{0}G\preccurlyeq {\rm{\Gamma }}$, where αG $\alpha G$ is the disjoint union of α $\alpha $ many copies of G $G$. A well‐known conjecture of Andreae is that every locally finite connected graph is ≼ $\preccurlyeq $‐ubiquitous. In this paper we give a sufficient condition on the structure of the ends of a graph G $G$ which implies that G $G$ is ≼ $\preccurlyeq $‐ubiquitous. In particular this implies that the full‐grid is ≼ $\preccurlyeq $‐ubiquitous.

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“…To this end we use the following notation for such collections of nG in Γ, most of which we established in [4].…”

Section: Half-grid-like Endsmentioning

confidence: 99%

“…To show that a given graph

G

\preccurlyeq

‐ubiquitous, we shall assume that

nG\preccurlyeq {\rm{\Gamma }}

holds for every

n\in {\mathbb{N}}

an show that this implies

{\aleph }_{0}G\preccurlyeq {\rm{\Gamma }}

. To this end we use the following notation for such collections of

nG

{\rm{\Gamma }}

, most of which we established in [4].…”

Section: G $G$‐tribes and Concentration Of G $G$‐tribes Towards An Endmentioning

confidence: 99%

“…Since every countable tree is a minor of the half‐grid, Theorem 1.5 implies that all countable trees are

\preccurlyeq

\preccurlyeq

‐ubiquitous has remained open, and we hope to resolve this in a paper in preparation.…”

Section: Introductionmentioning

confidence: 99%

“…In Sections 6 and 7 we prove Theorem 1.2, classifying the thick ends which are nonpebbly. In Section 8 we reintroduce some concepts from [4] and show that we may assume that the

G

‐minors in

{\rm{\Gamma }}

are concentrated towards some end

\epsilon

{\rm{\Gamma }}

. In Section 9 we use the results of the previous section to prove Theorem 1.5 and finally in Section 10 we prove Theorem 1.3 and its corollaries.…”

Section: Introductionmentioning

confidence: 99%

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Ubiquity of graphs with nowhere‐linear end structure

Bowler

Elbracht

Erde

et al. 2023

Journal of Graph Theory

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show abstract

“…Andreae proved quite a few cases of his conjecture, particularly that connected graphs of finite tree-width such that all their maximal 2-connected subgraphs are finite are ubiquitous [2]. Recently Bowler, Elbracht, Erde, Gollin, Heuer, Pitz and Teegen put forward a series of papers [5,4,3] in which they prove that a large class of locally finite graphs are ubiquitous, including the 2-dimensional grid! Here we note that the condition of 'connectedness' needs to be added to Andreae's conjecture.…”

mentioning

confidence: 99%