All graphs have tree-decompositions displaying their topological ends

TL;DR

This paper proves that every connected graph has a spanning tree that captures all its topological ends, resolving longstanding conjectures and problems in graph theory.

Contribution

It establishes that all connected graphs possess a spanning tree displaying all topological ends, confirming a 1964 conjecture and a 1992 problem.

Findings

Proves every connected graph has a spanning tree displaying all topological ends.

Resolves a 1964 conjecture of Halin.

Settles a problem posed by Diestel in 1992.

Abstract

We show that every connected graph has a spanning tree that displays all its topological ends. This proves a 1964 conjecture of Halin in corrected form, and settles a problem of Diestel from 1992.