Ubiquity of graphs with nowhere‐linear end structure

Abstract

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.

Ubiquity of oriented rays

“…Andreae conjectured that every locally finite connected graph is minor‐ubiquitous after studying minor‐ubiquity in [1, 2]. Noteworthy progress towards this conjecture was recently achieved by Bowler, Elbracht, Erde, Gollin, Heuer, Pitz and Teegen in a series of papers [5–7], in which they proved, among several other results, that all trees are topological‐minor‐ubiquitous. Throughout the years several results proving and disproving the ubiquity of certain graphs have been published, including results concerning different notions of ubiquity as in [4, 11].…”