On Andreae's Ubiquity Conjecture

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.