Normal trees of digraphs

TL;DR

This paper explores the concept of normal trees in directed graphs, establishing conditions for their existence and generalizing known results from undirected graphs to directed ones.

Contribution

It introduces the notion of normal trees for directed graphs, proves their existence under certain topological conditions, and compares them with normal arborescences.

Findings

Directed graphs have normal spanning trees iff their topological space is metrizable.

Existence of normal arborescences implies normal trees, but not vice versa.

Generalizes Diestel's result from undirected to directed graphs.

Abstract

In this paper, we investigate normal trees of directed graphs, which extend the fundamental concept of normal trees of undirected graphs. We prove that a directed graph $D$ has a normal spanning tree if and only if the topological space $|D|$ is metrizable, which generalises Diestel's result for undirected graphs. Furthermore, we show that the existence of normal arborescences implies the existence of normal trees in directed graphs, and that the converse is generally not true.