The Erd\H{o}s-P\'osa property for infinite graphs

TL;DR

This paper explores the Erdős-Pósa property in infinite graphs, establishing conditions under which classes of such graphs have the property and its variants, with results for graphs excluding long paths and subdivisions of trees.

Contribution

It introduces the infinite variant of the Erdős-Pósa property, providing positive and negative results for classes of infinite graphs based on their structural properties.

Findings

Graphs excluding long paths have the EPP and its variants.

Subdivisions of any tree have the κ-EPP for all uncountable κ.

Rayless trees have the EPP and the countable EPP.

Abstract

We investigate which classes of infinite graphs have the Erd\H{o}s-P\'osa property (EPP). In addition to the usual EPP, we also consider the following infinite variant of the EPP: a class $\mathcal{G}$ of graphs has the $\kappa$-EPP, where $\kappa$ is an infinite cardinal, if for any graph $\Gamma$ there are either $\kappa$ disjoint graphs from $\mathcal{G}$ in $\Gamma$ or there is a set $X$ of vertices of $\Gamma$ of size less than $\kappa$ such that $\Gamma - X$ contains no graph from $\mathcal{G}$. In particular, we study the ($\kappa$-)EPP for classes consisting of a single infinite graph $G$. We obtain positive results when the set of induced subgraphs of $G$ is labelled well-quasi-ordered, and negative results when $G$ is not a proper subgraph of itself (both results require some additional conditions). As a corollary, we obtain that every graph which does not contain a path of length $n$ for some $n \in \mathbb{N}$ has the EPP and the $\kappa$-EPP. Furthermore, we show that the class of all subdivisions of any tree $T$ has the $\kappa$-EPP for every uncountable cardinal $\kappa$, and if $T$ is rayless, also the $\aleph_0$-EPP and the EPP.