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The Erdős-Pósa property for infinite graphs

Thilo Krill

Abstract

We investigate which classes of infinite graphs have the Erdős-Pósa 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 $κ$-EPP, where $κ$ is an infinite cardinal, if for any graph $Γ$ there are either $κ$ disjoint graphs from $\mathcal{G}$ in $Γ$ or there is a set $X$ of vertices of $Γ$ of size less than $κ$ such that $Γ- X$ contains no graph from $\mathcal{G}$. In particular, we study the ($κ$-)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 $κ$-EPP. Furthermore, we show that the class of all subdivisions of any tree $T$ has the $κ$-EPP for every uncountable cardinal $κ$, and if $T$ is rayless, also the $\aleph_0$-EPP and the EPP.

The Erdős-Pósa property for infinite graphs

Abstract

We investigate which classes of infinite graphs have the Erdős-Pósa property (EPP). In addition to the usual EPP, we also consider the following infinite variant of the EPP: a class of graphs has the -EPP, where is an infinite cardinal, if for any graph there are either disjoint graphs from in or there is a set of vertices of of size less than such that contains no graph from . In particular, we study the (-)EPP for classes consisting of a single infinite graph . We obtain positive results when the set of induced subgraphs of is labelled well-quasi-ordered, and negative results when 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 for some has the EPP and the -EPP. Furthermore, we show that the class of all subdivisions of any tree has the -EPP for every uncountable cardinal , and if is rayless, also the -EPP and the EPP.

Paper Structure

This paper contains 22 sections, 48 theorems, 5 figures.

Table of Contents

  1. Introduction
  2. The Erdős-Pósa property
  3. The $\kappa$-Erdős-Pósa property
  4. Graphs without the $\kappa$-EPP for uncountable $\kappa$
  5. Trees without the $\kappa$-EPP for uncountable $\kappa$
  6. The ($\kappa$-)EPP for classes defined by topological minors and minors
  7. Comparing and evaluating the results
  8. Ubiquity
  9. Preliminaries
  10. Minors
  11. Tree order
  12. Tree-decompositions
  13. Compactness
  14. Propositions
  15. Constructing counterexamples to Seymour's Self-Minor Conjecture
  16. ...and 7 more sections

Key Result

Theorem 1.1

There is a function $f: \mathbb N \to \mathbb N$ such that for every graph $\Gamma$ and every $k \in \mathbb N$ one of the following holds:

Figures (5)

  • Figure 1: The graph $\Gamma$ from the proof of Proposition \ref{['prop:finitely_many_rays']}. Adhesion sets in the graphs $G_i$ are depicted by squares and each part $V_{r_i}$ consists of the respective labelled area together with its adjacent squares. Note that every ray in $\Gamma$ meets all but finitely many of the thickly drawn adhesion sets.
  • Figure 2: The infinite comb.
  • Figure 3: A countable subgraph-antichain of finite trees.
  • Figure 4: The tree $T^*_{\alpha, \beta}$ from the proof of Theorem \ref{['thm:antichain_of_trees']}.
  • Figure 5: Consider the tree $S$ from the figure rooted in its unique vertex of infinite degree with tree-decomposition $(S, {\mathcal{W}})$ where $W_s = \{ t \in V(S) : t \leq s \}$ for all $s \in V(S)$. Then the $\aleph_0$-closure of the root of $S$ in $S$ is $S$ itself. However, when we delete all leaves of $S$, then $\aleph_0$-closure of the root of $S$ in $S$ is just the root of $S$.

Theorems & Definitions (83)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.4
  • Corollary 1.4
  • Definition 1.5
  • Theorem 1.7
  • Theorem 1.8
  • Corollary 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 73 more