On Andreae's Ubiquity Conjecture

TL;DR

This paper challenges Andreae's Ubiquity Conjecture by providing a disconnected counterexample, leaving open the question of whether all connected locally finite graphs are ubiquitous.

Contribution

The paper constructs a disconnected counterexample to Andreae's conjecture, advancing understanding of graph ubiquity and highlighting the distinction between connected and disconnected cases.

Findings

Disproved Andreae's conjecture with a disconnected counterexample

Open problem remains whether all connected locally finite graphs are ubiquitous

Highlights difference between connected and disconnected graph ubiquity

Abstract

A graph $H$ is ubiquitous if for every graph $G$ that for every natural number $n$ contains $n$ vertex-disjoint $H$-minors contains infinitely many vertex-disjoint $H$-minors. Andreae conjectured that every locally finite graph is ubiquitous. We give a disconnected counterexample to this conjecture. It remains open whether every connected locally finite graph is ubiquitous.