A simple existence criterion for normal spanning trees in infinite graphs

TL;DR

This paper simplifies the criteria for the existence of normal spanning trees in infinite graphs by removing the need for the second condition previously thought necessary.

Contribution

It proves that the second condition in Halin's criterion is unnecessary for the existence of a normal spanning tree in connected graphs.

Findings

The second condition in Halin's theorem is redundant.

Normal spanning trees exist under weaker conditions than previously known.

The result broadens the class of graphs known to have normal spanning trees.

Abstract

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the following two conditions: (i) $G$ contains no subdivision of a `fat' $K_{\aleph_0}$, one in which every edge has been replaced by uncountably many parallel edges; and (ii) $G$ has no $K_{\aleph_0}$ subgraph. We show that the second condition is unnecessary.