A Sharp Upper Bound for the Complexity of Labeled Oriented Trees

TL;DR

This paper proves a sharp upper bound on the complexity of labeled oriented trees, confirming a conjecture, and shows that these structures are always aspherical, impacting the Whitehead Conjecture.

Contribution

It provides a constructive proof of the conjectured complexity bound for labeled oriented graphs and characterizes the structure of maximal complexity trees.

Findings

Complexity of labeled oriented trees is at most (m+1)/2.

The bound is sharp and achieved by specific structures.

Associated 2-complexes are always aspherical, ruling them out as counterexamples.

Abstract

A labeled oriented graph (LOG) is an oriented graph with a labeling function from the edge set into the vertex set. The complexity of a LOG is the minimal cardinality of an initial set $S$ of vertices such that every vertex can be reached successively from $S$ only using edges with labels in $S$ or already visited vertices. We give a constructive proof of a conjecture by Rosebrock stating that for an interior reduced, connected LOG with $m$ vertices the complexity is at most $(m+1) / 2$ and show that this bound is sharp. Due to results of Howie labeled oriented trees (LOTs) yield crucial candidates for counterexamples of the Whitehead Conjecture stating that every subcomplex of an aspherical 2-complex is aspherical. We explicitly describe the structure of LOTs of maximal complexity $(m+1)/2$. We conclude that the 2-complexes associated to these LOTs are always aspherical excluding them from the list of possible counterexamples.