Obstructions for normally spanned sets of vertices

Abstract

Halin conjectured that a graph has a normal spanning tree if and only if every minor of it has countable colouring number. This has recently been proven by the second author. In this paper, we strengthen this result by establishing the following local version of it: Given a prescribed set of vertices $U$ in a connected graph $G$, there is a normal tree in $G$ that includes $U$ if and only if every $U$-rooted minor of $G$ (i.e. a minor every branch set of which meets $U$) has countable colouring number. Our proof relies on a novel approach that combines normal partition trees as introduced by Brochet and Diestel with a suitable closure argument developed by Robertson, Seymour and Thomas in their discussion of infinite graphs of finite tree width.