Ubiquity of graphs with nowhere‐linear end structure - Bowler, Nathan; Elbracht, Christian; Erde, Joshua; Gollin, J. Pascal; Heuer, Karl; Pitz, Max; Teegen, Maximilian (Journal of graph theory, 2023)
A graph G $G$ is said to be ≼ $\preccurlyeq $‐ubiquitous, where ≼ $\preccurlyeq $ is the minor relation between graphs, if whenever Γ ${\rm{Γ }}$ is a graph with nG≼Γ $nG\preccurlyeq {\rm{Γ }}$ for all n∈N $n\in {\mathbb{N}}$, then one also has ℵ0G≼Γ ${\aleph }_{0}G\preccurlyeq {\rm{Γ }}$, where αG $α G$ is the disjoint union of α $α $ 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.