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Augmentation Lemma for Halin Conjecture

Jerzy Wojciechowski

Abstract

The longstanding conjecture of Halin characterizing the existence of normal spanning trees in infinite graphs has been recently proved by Max Pitz [3]. A critical step in the proof involves the construction of dominated torsos, whose properties are essential to the overall proof. In this note, we provide a correction to the proof of a key property of this construction.

Augmentation Lemma for Halin Conjecture

Abstract

The longstanding conjecture of Halin characterizing the existence of normal spanning trees in infinite graphs has been recently proved by Max Pitz [3]. A critical step in the proof involves the construction of dominated torsos, whose properties are essential to the overall proof. In this note, we provide a correction to the proof of a key property of this construction.

Paper Structure

This paper contains 2 sections, 4 theorems, 10 equations, 2 figures.

Table of Contents

  1. Introduction.
  2. Proof of the Augmentation Lemma.

Key Result

Theorem 1

A connected graph has a normal spanning tree if and only if every minor of it has countable coloring number.

Figures (2)

  • Figure 1: $F$ separates $U$ from $S'$ in $K$
  • Figure 2: $X$ does not separated $U$ from $S$ in $G$

Theorems & Definitions (8)

  • Theorem 1
  • Lemma 2: Decomposition Lemma
  • Claim 3
  • Example 4
  • Lemma 5: Augmentation Lemma
  • Lemma 6
  • proof
  • Remark