Ubiquity in graphs II: Ubiquity of graphs with nowhere-linear end structure

TL;DR

This paper establishes a sufficient structural condition on graphs' ends that guarantees their ubiquity under the minor relation, confirming the ubiquity of the full grid and advancing understanding of graph minors.

Contribution

It provides a new structural criterion for $ ext{ extbackslash preceq}$-ubiquity in graphs, including the full grid, supporting Andreae's conjecture.

Findings

Full grid is $ ext{ extbackslash preceq}$-ubiquitous.

A structural condition on ends implies ubiquity.

Supports Andreae's conjecture for locally finite connected graphs.

Abstract

A graph $G$ is said to be $\preceq$-ubiquitous, where $\preceq$ is the minor relation between graphs, if whenever $\Gamma$ is a graph with $nG \preceq \Gamma$ for all $n \in \mathbb{N}$, then one also has $\aleph_0 G \preceq \Gamma$, where $\alpha G$ is the disjoint union of $\alpha$ many copies of $G$. A well-known conjecture of Andreae is that every locally finite connected graph is $\preceq$-ubiquitous. In this paper we give a sufficient condition on the structure of the ends of a graph~$G$ which implies that $G$ is $\preceq$-ubiquitous. In particular this implies that the full grid is $\preceq$-ubiquitous.