Subdivisions of oriented cycles in digraphs with large chromatic number

TL;DR

This paper investigates the relationship between the presence of subdivisions of oriented cycles and the chromatic number in digraphs, revealing conditions under which such subdivisions must exist or can be absent.

Contribution

It establishes that certain oriented cycles are necessarily contained as subdivisions in highly chromatic, strongly connected digraphs, contrasting with the existence of digraphs with large chromatic number lacking such subdivisions.

Findings

Existence of digraphs with large chromatic number lacking subdivisions of certain cycles.

Strongly connected digraphs with large chromatic number contain subdivisions of cycles with two blocks.

Similar results hold for the antidirected cycle on four vertices.

Abstract

An oriented cycle is an orientation of a undirected cycle. We first show that for any oriented cycle $C$, there are digraphs containing no subdivision of $C$ (as a subdigraph) and arbitrarily large chromatic number. In contrast, we show that for any $C$ a cycle with two blocks, every strongly connected digraph with sufficiently large chromatic number contains a subdivision of $C$. We prove a similar result for the antidirected cycle on four vertices (in which two vertices have out-degree $2$ and two vertices have in-degree $2$).