Ubiquity of graphs with nowhere‐linear end structure
A graph G $G$ is said to be ≼ $\preccurlyeq $‐ubiquitous, where ≼ $\preccurlyeq $ is the minor relation between graphs, if whenever Γ ${\rm{\Gamma }}$ is a graph with nG≼Γ $nG\preccurlyeq {\rm{\Gamma }}$ for all n∈N $n\in {\mathbb{N}}$, then one also has ℵ0G≼Γ ${\aleph }_{0}G\preccurlyeq {\rm{\Gamma...
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| Published in | Journal of graph theory Vol. 103; no. 3; pp. 564 - 598 |
|---|---|
| Main Authors | , , , , , , |
| Format | Journal Article |
| Language | English |
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United States
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01.07.2023
John Wiley and Sons Inc |
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| Online Access | Get full text |
| ISSN | 0364-9024 1097-0118 |
| DOI | 10.1002/jgt.22936 |
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| Abstract | A graph G $G$ is said to be ≼ $\preccurlyeq $‐ubiquitous, where ≼ $\preccurlyeq $ is the minor relation between graphs, if whenever Γ ${\rm{\Gamma }}$ is a graph with nG≼Γ $nG\preccurlyeq {\rm{\Gamma }}$ for all n∈N $n\in {\mathbb{N}}$, then one also has ℵ0G≼Γ ${\aleph }_{0}G\preccurlyeq {\rm{\Gamma }}$, 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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| AbstractList | A graph G is said to be ≼-ubiquitous, where ≼ is the minor relation between graphs, if whenever Γ is a graph with nG≼Γ for all n∈N, then one also has ℵ0G≼Γ, where αG is the disjoint union of α many copies of G. A well-known conjecture of Andreae is that every locally finite connected graph is ≼-ubiquitous. In this paper we give a sufficient condition on the structure of the ends of a graph G which implies that G is ≼-ubiquitous. In particular this implies that the full-grid is ≼-ubiquitous.A graph G is said to be ≼-ubiquitous, where ≼ is the minor relation between graphs, if whenever Γ is a graph with nG≼Γ for all n∈N, then one also has ℵ0G≼Γ, where αG is the disjoint union of α many copies of G. A well-known conjecture of Andreae is that every locally finite connected graph is ≼-ubiquitous. In this paper we give a sufficient condition on the structure of the ends of a graph G which implies that G is ≼-ubiquitous. In particular this implies that the full-grid is ≼-ubiquitous. A graph G $G$ is said to be ≼ $\preccurlyeq $‐ubiquitous, where ≼ $\preccurlyeq $ is the minor relation between graphs, if whenever Γ ${\rm{\Gamma }}$ is a graph with nG≼Γ $nG\preccurlyeq {\rm{\Gamma }}$ for all n∈N $n\in {\mathbb{N}}$, then one also has ℵ0G≼Γ ${\aleph }_{0}G\preccurlyeq {\rm{\Gamma }}$, 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. A graph is said to be ‐ ubiquitous , where is the minor relation between graphs, if whenever is a graph with for all , then one also has , where is the disjoint union of many copies of . A well‐known conjecture of Andreae is that every locally finite connected graph is ‐ubiquitous. In this paper we give a sufficient condition on the structure of the ends of a graph which implies that is ‐ubiquitous. In particular this implies that the full‐grid is ‐ubiquitous. A graph is said to be - , where is the minor relation between graphs, if whenever is a graph with for all , then one also has , where is the disjoint union of many copies of . A well-known conjecture of Andreae is that every locally finite connected graph is -ubiquitous. In this paper we give a sufficient condition on the structure of the ends of a graph which implies that is -ubiquitous. In particular this implies that the full-grid is -ubiquitous. A graph G is said to be ≼ ‐ ubiquitous , where ≼ is the minor relation between graphs, if whenever Γ is a graph with n G ≼ Γ for all n ∈ N , then one also has ℵ 0 G ≼ Γ , where α G is the disjoint union of α many copies of G . A well‐known conjecture of Andreae is that every locally finite connected graph is ≼ ‐ubiquitous. In this paper we give a sufficient condition on the structure of the ends of a graph G which implies that G is ≼ ‐ubiquitous. In particular this implies that the full‐grid is ≼ ‐ubiquitous. |
| Author | Heuer, Karl Elbracht, Christian Erde, Joshua Gollin, J. Pascal Bowler, Nathan Teegen, Maximilian Pitz, Max |
| AuthorAffiliation | 3 Discrete Mathematics Group Institute for Basic Science (IBS) Daejeon Republic of Korea 2 Institute of Discrete Mathematics Graz University of Technology Graz Austria 1 Department of Mathematics Universität Hamburg Hamburg Germany 4 Department of Applied Mathematics and Computer Science Technical University of Denmark Kongens Lyngby Denmark |
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| Author_xml | – sequence: 1 givenname: Nathan surname: Bowler fullname: Bowler, Nathan organization: Universität Hamburg – sequence: 2 givenname: Christian orcidid: 0000-0003-2481-364X surname: Elbracht fullname: Elbracht, Christian organization: Universität Hamburg – sequence: 3 givenname: Joshua orcidid: 0000-0003-1129-4270 surname: Erde fullname: Erde, Joshua organization: Graz University of Technology – sequence: 4 givenname: J. Pascal orcidid: 0000-0003-2095-7101 surname: Gollin fullname: Gollin, J. Pascal email: pascalgollin@ibs.re.kr organization: Institute for Basic Science (IBS) – sequence: 5 givenname: Karl orcidid: 0000-0002-6841-9815 surname: Heuer fullname: Heuer, Karl organization: Technical University of Denmark – sequence: 6 givenname: Max surname: Pitz fullname: Pitz, Max organization: Universität Hamburg – sequence: 7 givenname: Maximilian orcidid: 0000-0001-7384-742X surname: Teegen fullname: Teegen, Maximilian organization: Universität Hamburg |
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| Copyright | 2023 The Authors. published by Wiley Periodicals LLC. 2023 The Authors. Journal of Graph Theory published by Wiley Periodicals LLC. 2023. This article is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License. |
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| Snippet | A graph G $G$ is said to be ≼ $\preccurlyeq $‐ubiquitous, where ≼ $\preccurlyeq $ is the minor relation between graphs, if whenever Γ ${\rm{\Gamma }}$ is a... A graph is said to be ‐ ubiquitous , where is the minor relation between graphs, if whenever is a graph with for all , then one also has , where is the... A graph is said to be - , where is the minor relation between graphs, if whenever is a graph with for all , then one also has , where is the disjoint union of... A graph G is said to be ≼-ubiquitous, where ≼ is the minor relation between graphs, if whenever Γ is a graph with nG≼Γ for all n∈N, then one also has ℵ0G≼Γ,... A graph G is said to be ≼ ‐ ubiquitous , where ≼ is the minor relation between graphs, if whenever Γ is a graph with n G ≼ Γ for all n ∈ N , then one also has... |
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| SubjectTerms | graph minors Graphs infinite graphs ubiquity |
| Title | Ubiquity of graphs with nowhere‐linear end structure |
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