Unavoidable butterfly minors in digraphs of large cycle rank

TL;DR

This paper proves that digraphs with sufficiently large cycle rank necessarily contain certain complex butterfly minors, establishing a structural threshold related to cycle rank and exploring its connection to directed weak coloring numbers.

Contribution

It introduces a function linking cycle rank to the guaranteed presence of specific butterfly minors and explores the relationship between cycle rank and directed weak coloring numbers.

Findings

Existence of a function f(k) ensuring large cycle rank digraphs contain specific butterfly minors.

Identification of a connection between cycle rank and directed weak coloring number.

Structural characterization of digraphs with high cycle rank.

Abstract

Cycle rank is one of the depth parameters for digraphs introduced by Eggan in 1963. We show that there exists a function $f:\mathbb{N}\to \mathbb{N}$ such that every digraph of cycle rank at least $f(k)$ contains a directed cycle chain, a directed ladder, or a directed tree chain of order $k$ as a butterfly minor. We also investigate a new connection between cycle rank and a directed analogue of the weak coloring number of graphs.