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The immersion-minimal infinitely edge-connected graph

Paul Knappe, Jan Kurkofka

Abstract

We show that there is a unique immersion-minimal infinitely edge-connected graph: every such graph contains the halved Farey graph, which is itself infinitely edge-connected, as an immersion minor. By contrast, any minimal list of infinitely edge-connected graphs represented in all such graphs as topological minors must be uncountable.

The immersion-minimal infinitely edge-connected graph

Abstract

We show that there is a unique immersion-minimal infinitely edge-connected graph: every such graph contains the halved Farey graph, which is itself infinitely edge-connected, as an immersion minor. By contrast, any minimal list of infinitely edge-connected graphs represented in all such graphs as topological minors must be uncountable.
Paper Structure (15 sections, 23 theorems, 3 equations, 4 figures)

This paper contains 15 sections, 23 theorems, 3 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Tools and terminology
  3. Farey graph
  4. Grain lines
  5. Typical sets with regard to the topological minor relation
  6. Proof of Key Lemma 3.4
  7. Proof of Key Lemma 3.5
  8. Excluding k-bounded Farey graph minors
  9. Typical graph with regard to the immersion relation
  10. Overview of the proof of Theorem 1
  11. Compound separations
  12. Faithful nested sets of unitary compound-separation s
  13. Finding the halved Farey graph in wild structures
  14. Decomposing along compound-separation s
  15. The halved Farey graph is everywhere

Key Result

Theorem 1.1

The Farey graph is one of two infinitely edge-connected graphs such that every infinitely edge-connected graph contains at least one of the two as a minor.

Figures (4)

  • Figure 1: The Farey graph
  • Figure 2: The halved Farey graph
  • Figure 3: A generalised halved Farey graph
  • Figure 4: A generalised halved Farey graph in which its vertex $v$ together with the edges (red) in depth at most the depth of $v$ separate $u$ and $w$.

Theorems & Definitions (50)

  • Theorem 1.1
  • Theorem 1
  • Theorem 2
  • Example 2.1
  • Lemma 2.2
  • proof
  • Example 2.3
  • Theorem 3.1
  • Lemma 3.2
  • Example 3.3
  • ...and 40 more