Ubiquity of graphs with nowhere‐linear end structure

• Published in: Journal of graph theory Vol. 103; no. 3; pp. 564 - 598
• Main Authors: Bowler, Nathan, Elbracht, Christian, Erde, Joshua, Gollin, J. Pascal, Heuer, Karl, Pitz, Max, Teegen, Maximilian
• Format: Journal Article
• Language: English
• Published: United States Wiley Subscription Services, Inc 01.07.2023; John Wiley and Sons Inc
• Subjects: graph minors; Graphs; infinite graphs; ubiquity; infinite graphs; graph minors; ubiquity
• ISSN: 0364-9024; 1097-0118
• DOI: 10.1002/jgt.22936

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.