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 ${\aleph }_{0}G\preccurlyeq {\rm{\Gamma }}$, 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.