# Proof of Halin's normal spanning tree conjecture

@article{Pitz2020ProofOH, title={Proof of Halin's normal spanning tree conjecture}, author={Max Pitz}, journal={arXiv: Combinatorics}, year={2020} }

Halin conjectured 20 years ago that a graph has a normal spanning tree if and only if every minor of it has countable colouring number. We prove Halin's conjecture. This implies a forbidden minor characterisation for the property of having a normal spanning tree.

#### 5 Citations

Quickly Proving Diestel's Normal Spanning Tree Criterion

- Mathematics
- The Electronic Journal of Combinatorics
- 2021

We present two short proofs for Diestel's criterion that a connected graph has a normal spanning tree provided it contains no subdivision of a countable clique in which every edge has been replaced… Expand

A new obstruction for normal spanning trees

- Mathematics
- 2020

In a paper from 2001 (Journal of the LMS), Diestel and Leader offered a proof that a connected graph has a normal spanning tree if and only if it does not contain a minor from two specific forbidden… Expand

Halin’s End Degree Conjecture

- Mathematics
- 2020

An end of a graph $G$ is an equivalence class of rays, where two rays are equivalent if there are infinitely many vertex-disjoint paths between them in $G$. The degree of an end is the maximum… Expand

A representation theorem for end spaces of infinite graphs

- Mathematics
- 2021

End spaces of infinite graphs sit at the interface between graph theory, group theory and topology. They arise as the boundary of an infinite graph in a standard sense generalising the theory of the… Expand

Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020)

- Computer Science
- 2020

The first edition of ALGOS 2020 has a particular motivation, namely, an opportunity to honour Maurice Pouzet on his 75th birthday, and has particularly welcomed submissions in areas related to Maurice’s many scientific interests. Expand

#### References

SHOWING 1-10 OF 15 REFERENCES

A Cantor-Bernstein-type theorem for spanning trees in infinite graphs

- Computer Science, Mathematics
- J. Comb. Theory, Ser. B
- 2021

It is shown that if a graph admits a packing and a covering both consisting of $\ lambda$ many spanning trees, then the graph also admits a decomposition into $\lambda$Many spanning trees. Expand

A unified existence theorem for normal spanning trees

- Mathematics, Computer Science
- J. Comb. Theory, Ser. B
- 2020

We show that a graph $G$ has a normal spanning tree if and only if its vertex set is the union of countably many sets each separated from any subdivided infinite clique in $G$ by a finite set of… Expand

A Simple Existence Criterion for Normal Spanning Trees

- Mathematics, Computer Science
- Electron. J. Comb.
- 2016

Halin proved in 1978 that there exists a normal spanning tree in every connected graph $G$ that satisfies the following two conditions: (i) $G$ contains no subdivision of a `fat' $K_{\aleph_0}$, one… Expand

Normal Spanning Trees, Aronszajn Trees and Excluded Minors

- Mathematics
- 2001

It is proved that a connected infinite graph has a normal spanning tree (the infinite analogue of a depth-first search tree) if and only if it has no minor obtained canonically from either an (ℵ 0 ,… Expand

The Colouring Number of Infinite Graphs

- Mathematics, Computer Science
- Comb.
- 2019

We show that, given an infinite cardinal $\mu$, a graph has colouring number at most $\mu$ if and only if it contains neither of two types of subgraph. We also show that every graph with infinite… Expand

Normal tree orders for infinite graphs

- Mathematics
- 1994

A well-founded tree T defined on the vertex set of a graph G is called normal if the endvertices of any edge of G are comparable in T. We study how normal trees can be used to describe the structure… Expand

A new obstruction for normal spanning trees

- Mathematics
- 2020

In a paper from 2001 (Journal of the LMS), Diestel and Leader offered a proof that a connected graph has a normal spanning tree if and only if it does not contain a minor from two specific forbidden… Expand

Miscellaneous problems on infinite graphs

- Computer Science
- J. Graph Theory
- 2000

A collection of open problems on in®nite graphs is presented, divided into twelve sections which are largely independent of each other. ß 2000 John Wiley & Sons, Inc. J Graph Theory 35: 128±151, 2000

SET THEORY

- 2007

The study of sets is important and thus popular in the business and economic world for three major reasons: Basic understanding of concepts in sets and set algebra provides a form of logical language… Expand