Ubiquity in graphs I: Topological ubiquity of trees

TL;DR

This paper proves that all trees are topologically ubiquitous in infinite graphs, confirming a special case of the Ubiquity Conjecture and advancing understanding of graph minors.

Contribution

It establishes that all trees are topologically ubiquitous regardless of size, addressing a question posed by Andreae in 1979.

Findings

All trees are topologically ubiquitous in infinite graphs.

Supports the Ubiquity Conjecture for a class of graphs.

Advances the theory of graph minors and infinite graph properties.

Abstract

Let $\triangleleft$ be a relation between graphs. We say a graph $G$ is \emph{$\triangleleft$-ubiquitous} if whenever $\Gamma$ is a graph with $nG \triangleleft \Gamma$ for all $n \in \mathbb{N}$, then one also has $\aleph_0 G \triangleleft \Gamma$, where $\alpha G$ is the disjoint union of $\alpha$ many copies of $G$. The \emph{Ubiquity Conjecture} of Andreae, a well-known open problem in the theory of infinite graphs, asserts that every locally finite connected graph is ubiquitous with respect to the minor relation. In this paper, which is the first of a series of papers making progress towards the Ubiquity Conjecture, we show that all trees are ubiquitous with respect to the topological minor relation, irrespective of their cardinality. This answers a question of Andreae from 1979.